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version update of 12 Feb 2004 [111+ kilobytes]

What numbers can be made with Four Nines? Read on!"The Weekly Dispatch" of 4 February 1900, which ran a puzzle column by Dudeney, introduced a problem which is still provoking interest today. Why? If you live at 9 Oak Avenue, and you can buy numerals and letters made out of aluminum at your local hardware store for ten cents each, then you can nail your address on your door for 70 cents (10 cents for 9, 10 cents for O, ten cents for A, ten cents for K, ten cents for A, ten cents for V, ten cents for E, and the spaces are free). So if you live at 99 Oak Avenue, you can put your address on your door for 80 cents, and if you live at 999 Oak Avenue you can put your address on your door for 90 cents, and if you live at 9999 Oak Avenue you can put your address on your door for a dollar. But if you can also buy plus signs, minus signs, X's for multiplication signs, and letters I turned 45 degrees to be division signs / then what other numbers can you nail on your door for 40 cents worth of numerals 9, plus as much as you want to spend on those other symbols? How can we construct the smaller whole numbers, under 100 for instance, using only all four nines as digits, parentheses, and the arithmetic operators## "

This old puzzle can be updated if we allow the use of exponentiation, radicals (especially the square root "+","-","x","/"?sqrt"), factorial "!", and the floor function "|_N_|" and ceiling function "|-N-|". (note to self: get proper typographic characters!)(it's so hard to see that the ceiling function, as I show it now, differs from the floor function, that I say "ceiling" after any expression that uses it].## Jump to a discussion of some deeper mathematical issues related to the Four Nines Puzzle

Can you improve these below, and/or fill in some blanks? The first hard ones are 22, 38, 40, 41, 47, 49, 50, 52... The first blanks still to be solved (as of 10 Feb 2004) are 286, 287, 311, 313, 314, 316, 322, 323... My son, when he was 9 years old, worked with me 6 years ago to come up with this beginning... with a few more added on his 15th birthday and the next 2 weeks...0 = 99 - 99 = (9/9) - (9/9) 1 = 99/99 = (9/9) x (9/9) = (9+9) / (9+9) = (9x9) / (9x9) 2 = (9/9) + (9/9) = (99/9) - 9 3 = (9 + 9 + 9) / 9 4 = 9 - (sqrt 9)! + 9/9 = (9/sqrt 9) + (9/9) = |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt 63! _| _| _| _| _| _| _| {and, as we'll see later, 63 can be made with a single nine, hence 4 can be made with a single nine} 5 = (9 - sqrt 9) - 9/9 = |_ sqrt |_ sqrt ((sqrt 9)!)! _| _| {that is, (sqrt 9)! = 3! = 6, and 6! = 720, and sqrt 720 = 26.83281573..., and sqrt 26.83281573 = 5.18004012822, which rounds down to 5} 6 = sqrt (9+9+9+9) = (9 - sqrt 9) x 9/9 7 = 9 - ((9+9)/9) = (9 - sqrt 9) + 9/9 = |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt 50! _| _| _| _| _| _| {and, as we'll see later, 50 can be made with a single nine, hence 7 can be made with a single nine} 8 = ((9x9)-9)/9 = (sqrt 9 x sqrt 9) - 9/9 = (99/9) - sqrt 9 9 = (9 + 9 + 9) / sqrt 9 = sqrt 9 x sqrt 9 x (9/9) 10 = (9 + (9x9))/9 = (sqrt 9 x sqrt 9) + 9/9 = (99 - 9) / 9 = |_ sqrt 5! _| {that is, 10 = the rounded-down square root of 5!=120, and we can make a 5 from a single 9; hence we make 10 using only one nine} 11 = 9 + ((9+9)/9) = 9 + sqrt 9 - (9/9) = 99 / (sqrt 9 x sqrt 9) 12 = (9 + 99)/9 = (9 + sqrt 9) x (9/9) 13 = 9 + sqrt 9 + (9/9) 14 = (99 - 9) + sqrt 9 = (9 + (sqrt 9)!) - (9/9) 15 = (9 + (sqrt 9)!) x (9/9) 16 = (9 + (sqrt 9)!) + (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 99! _| {the rounded-down 128th-root of 99!, giving us 16 with only two nines} 17 = 9 + 9 - (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 100! _| {the rounded-down 128th-root of 100!; since we can make 100 with four nines, this gives us another way to make 17 with four nines} 18 = 9 + 9 + 9 - 9 = (9 + 9) x (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 102! _| {the rounded-down 128th-root of 102!; since we can make 102 with four nines, this gives us another way to make 18 with four nines} 19 = 9 + 9 + 9/9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 103! _| {the rounded-down 128th-root of 103!; since we can make 103 with four nines, this gives us another way to make 19 with four nines} 20 = (99/9) + 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 105! _| {the rounded-down 128th-root of 105!; since we can make 103 with four nines, this gives us another way to make 20 with four nines} 21 = ((9 x sqrt 9) - 9) + sqrt 9 = ((sqrt 9)! x (sqrt 9)!) - 9 - (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 106! _| {the rounded-down 128th-root of 106!; since we can make 106 with four nines, this gives us another way to make 21 with four nines} 22 = (9/9) x |_ sqrt (sqrt (9!)))_| - |_ sqrt (sqrt ( sqrt ( sqrt (9!))))_| = |_ 99/(sqrt 9)!) + (sqrt 9)!)_| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 107! _| {the rounded-down 128th-root of 107!; since we can make 107 with four nines, this gives us another way to make 22 with four nines} 23 = (((sqrt 9)!)!)/((sqrt 9)! x (sqrt 9)! + sqrt 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 109! _| {the rounded-down 128th-root of 109!; since we can make 109 with four nines, this gives us another way to make 23 with four nines} 24 = 9 + 9 + sqrt 9 + sqrt 9 = (99/sqrt 9) - 9 = (sqrt 9)! + (sqrt 9)! + (sqrt 9)! + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 103! _| {the rounded-down 128th-root of 110!; since we can make 110 with four nines, this gives us another way to make 24 with four nines} 25 = ((9 x 9) - (sqrt 9)!)/sqrt 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 111! _| {the rounded-down 128th-root of 111!; since we can make 111 with four nines, this gives us another way to make 125 with four nines} 26 = (9 x sqrt 9) - (9/9) = |_ sqrt ((sqrt 9)!)! _| {rounded down square root of (3!)! = 6! =720, to make 26 with a single nine} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 103! _| {the rounded-down 128th-root of 112!; since we can make 112 with four nines, this gives us another way to make 26 with four nines} 27 = (9 x sqrt 9) x (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 113! _| {the rounded-down 128th-root of 113!; since we can make 113 with four nines, this gives us another way to make 27 with four nines} 28 = (9 x sqrt 9) + (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 114! _| {the rounded-down 128th-root of 114!; since we can make 114 with four nines, this gives us another way to make 28 with four nines} 29 = (((sqrt 9)!)! / ((sqrt 9)! x sqrt 9)!) + 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 115! _| {the rounded-down 128th-root of 115!; since we can make 115 with four nines, this gives us another way to make 29 with four nines} 30 = 9 + 9 + 9 + sqrt 9 = (9 + (9/9)) x sqrt 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 116! _| {the rounded-down 128th-root of 116!; since we can make 116 with four nines, this gives us another way to make 30 with four nines} 31 = (99 - (sqrt 9)!) / sqrt 9 = |_ sqrt (999 - 9) _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 117! _| {the rounded-down 128th-root of 117!; since we can make 117 with four nines, this gives us another way to make 31 with four nines} 32 = (99 - sqrt 9) / sqrt 9 = |- sqrt 999 -| 33 = (99/9) x sqrt 9 = 9 + 9 + 9 + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 118! _| {the rounded-down 128th-root of 118!; since we can make 118 with four nines, this gives us another way to make 33 with four nines} 34 = (99 + sqrt 9) / sqrt 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 119! _| {the rounded-down 128th-root of 119!; since we can make 119 with four nines, this gives us another way to make 34 with four nines} 35 = ((sqrt 9)! x (sqrt 9)!) - 9/9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 120! _| {the rounded-down 128th-root of 103!; since we can make 120 with a single nine, ((sqrt 9)!)! this gives us another way to make 35 with one nine} 36 = 9 + 9 + 9 + 9 = ((sqrt 9)! x (sqrt 9)!) x 9/9 37 = ((sqrt 9)! x (sqrt 9)!) + 9/9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 121! _| {the rounded-down 128th-root of 121!; since we can make 121 with four nines, this gives us another way to make 19 with four nines} 38 = |- sqrt 999-| + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 122! _| {the rounded-down 128th-root of 122!; since we can make 122 with four nines, this gives us another way to make 38 with four nines} 39 = (99 / sqrt 9) + (sqrt 9)! 40 = |_ sqrt 999 _| + 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 123! _| {the rounded-down 128th-root of 123!; since we can make 123 with four nines, this gives us another way to make 40 with four nines} 41 = |- sqrt 999 -| + 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 124! _| {the rounded-down 128th-root of 124!; since we can make 124 with four nines, this gives us another way to make 41 with four nines} 42 = (99/sqrt 9) + 9 = ((sqrt 9)! x (sqrt 9)!) + sqrt 9 + sqrt 9 43 = (9 + ((sqrt 9)! x (sqrt 9)!)) / sqrt 9 = |_ sqrt |_ sqrt 10! _| _| {and, as we see above, 10 can be made with a single nine, hence 43 can be made with a single nine} {that is, the rounded-down square root of 10! = sqrt 3628800 = 1904.940943... which rounds down to 1904, and sqrt 1904 = 43.6348484585..., which rounds down to 43} = |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt (((sqrt 9)!)!)! _| _| _| _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 125! _| {the rounded-down 128th-root of 125!; since we can make 125 with four nines, this gives us another way to make 43 with four nines} 44 = (((sqrt 9)!)! / 9) - ((sqrt 9)! x (sqrt 9)!)) = |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt 43! _| _| _| _| _| {and, as we see above, 43 can be made with a single nine, hence 44 can be made with a single nine} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 126! _| {the rounded-down 128th-root of 126!; since we can make 126 with four nines, this gives us another way to make 44 with four nines} 45 = (9 x 9) - ((sqrt 9)! x (sqrt 9)!)) 46 = ((9 + ((sqrt 9)!)!) / sqrt 9) + sqrt 9 = |_ sqrt |_ sqrt |_ sqrt |_ sqrt 26! _| _| _| _| {and, as we see above, 26 can be made with a single nine, hence 46 can be made with a single nine} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 127! _| {the rounded-down 128th-root of 127!; since we can make 127 with four nines, this gives us another way to make 46 with four nines} 47 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| - ((sqrt 9)x(sqrt 9)) 48 = ( 9 - (9/9)) x (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 128! _| {the rounded-down 128th-root of 128!; since we can make 128 with four nines, this gives us another way to make 48 with four nines} 49 = ((9 + ((sqrt 9)!)!) / (sqrt 9)) + (sqrt 9)! 50 = |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| x (9/9) [2 ceilings] = |_ sqrt |_ sqrt |_ sqrt |_ sqrt 44! _| _| _| _| {and, as we see above, 44 can be made with a single nine, hence 50 can be made with a single nine} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 129! _| {the rounded-down 128th-root of 129!; since we can make 129 with four nines, this gives us another way to make 50 with four nines} 51 = ((sqrt 9)! x (sqrt 9)!) + 9 + (sqrt 9)! 52 = |_ sqrt((sqrt 9)!)! _| + |_ sqrt((sqrt 9)!)! _| x (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 130! _| {the rounded-down 128th-root of 130!; since we can make 130 with four nines, this gives us another way to make 52 with four nines} 53 = (9 x (sqrt 9)!) - (9/9) 54 = (9 x (sqrt 9)!) x (9/9) {eliminate the (9/9) and we make it with two nines} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 131! _| {the rounded-down 128th-root of 131!; since we can make 131 with four nines, this gives us another way to make 54 with four nines} 55 = (9 x (sqrt 9)!) + (9/9) 56 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| [uses only two nines], so = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| x (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 132! _| {the rounded-down 128th-root of 132!; since we can make 132 with four nines, this gives us another way to make 56 with four nines} 57 = (9 x (sqrt 9)!) + 9 - (sqrt 9)! 58 = |- sqrt sqrt sqrt sqrt (9 x (sqrt 9))! -| + (9/9) [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 133! _| {the rounded-down 128th-root of 133!; since we can make 133 with four nines, this gives us another way to make 58 with four nines} 59 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| + (9/sqrt 9) 60 = (9 + (9/9)) x (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 134! _| {the rounded-down 128th-root of 134!; since we can make 134 with four nines, this gives us another way to make 60 with four nines} 61 = |- sqrt sqrt sqrt sqrt (9 x (sqrt 9))! -| + (9/sqrt 9) [ceiling] 62 = ((((sqrt 9)!)!/9) - (9 + 9) 63 = (9x9) - (9+9) = |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt 46! _| _| _| _| _| {and, as we see above, 46 can be made with a single nine, hence 63 can be made with a single nine} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 135! _| {the rounded-down 128th-root of 135!; since we can make 135 with four nines, this gives us another way to make 63 with four nines} 64 = |- (sqrt(9!))/9 -| - (9 / sqrt 9) [ceiling] 65 = ((((sqrt 9)!)!/9) - (9 + sqrt 9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 136! _| {the rounded-down 128th-root of 136!; since we can make 136 with four nines, this gives us another way to make 65 with four nines} 66 = ((9x9) - 9) - (sqrt 9)! = |_ (sqrt(9!))/9 _| uses only two nines 67 = |- (sqrt(9!))/9 -| uses only two nines [ceiling] so: = |- (sqrt(9!))/9 -| x (9/9) [ceiling] 68 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| + 9 + (sqrt 9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 137! _| {the rounded-down 128th-root of 137!; since we can make 137 with four nines, this gives us another way to make 68 with four nines} 69 = ((9x9) - 9) - sqrt 9 70 = (9x9) - 9 - |_ sqrt sqrt sqrt sqrt (9!) _| = |_ sqrt 7! _| {and, as we see above, 7 can be made with a single nine, hence 70 can be made with a single nine} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 138! _| {the rounded-down 128th-root of 138!; since we can make 138 with four nines, this gives us another way to make 70 with four nines} 71 = ((((sqrt 9)!)!/9) - ((sqrt 9) x (sqrt 9)) = |- sqrt 7! -| {that is, the ceiling function of sqrt 5040} {and, as we see above, 7 can be made with a single nine, hence 70 can be made with a single nine} 72 = 9 x (9 - (9/9)) = (9x9) - ((sqrt 9) + (sqrt 9)!) 73 = |- (sqrt(9!))/9 -| + 9 - sqrt 9 [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 139! _| {the rounded-down 128th-root of 139!; since we can make 139 with four nines, this gives us another way to make 73 with four nines} 74 = ((((sqrt 9)!)!/9) - ((sqrt 9) + (sqrt 9)) 75 = (9x9) - ((sqrt 9) + (sqrt 9)) 76 = |- (sqrt(9!))/9 -| + ((sqrt 9) x (sqrt 9)) [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 140! _| {the rounded-down 128th-root of 140!; since we can make 140 with four nines, this gives us another way to make 76 with four nines} 77 = |_ Sqrt(999 x (sqrt 9)!) _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt (9x9)! _| {that is, 77 is the 64th root of 81!, rounded down, using just two nines} 78 = |- Sqrt(999 x (sqrt 9)!) -| 79 = |- (sqrt(9!))/9 -| + 9 + sqrt 9 [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 141! _| {the rounded-down 128th-root of 141!; since we can make 141 with four nines, this gives us another way to make 79 with four nines} 80 = (9x9) - (9/9) = ((sqrt 9)!)! / 9 [uses just two nines] 81 = 9 x 9 x (9/9) 82 = (9x9) + (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 142! _| {the rounded-down 128th-root of 142!; since we can make 142 with four nines, this gives us another way to make 82 with four nines} 83 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| + (9 x (sqrt 9)) 84 = (99 - 9) - (sqrt 9)! 85 = (9x9) + |_ sqrt sqrt sqrt sqrt (9!) _| + |_ sqrt sqrt sqrt sqrt (9!) _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 143! _| {the rounded-down 128th-root of 143!; since we can make 143 with four nines, this gives us another way to make 85 with four nines} 86 = (9 x 9) + (sqrt 9)! - |_ sqrt sqrt 9 _| 87 = (((sqrt 9)!)! + ((sqrt 9)!)!) + 9 + (sqrt 9)! 88 = 99 - (9 + |_ sqrt sqrt sqrt sqrt (9!) _| ) 89 = |_ sqrt (9 x 9 x 99) _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 144! _| {the rounded-down 128th-root of 144!; since we can make 144 with four nines, this gives us another way to make 89 with four nines} 90 = 9 x (9 + (9/9)) = 99 - ((sqrt 9) x (sqrt 9)) 91 = (99 - 9) + |_ sqrt sqrt 9 _| 92 = (99 - 9) + |_ sqrt sqrt sqrt sqrt (9!) _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 145! _| {the rounded-down 128th-root of 145!; since we can make 145 with four nines, this gives us another way to make 92 with four nines} 93 = (99 - 9) + sqrt 9 94 = |_ sqrt (9 x 999) _| 95 = |- sqrt (9 x 999) -| 96 = (99 - 9) + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 146! _| {the rounded-down 128th-root of 146!; since we can make 146 with four nines, this gives us another way to make 96 with four nines} 97 = |- (sqrt 9)th-root-of-9!) -| - (9 x sqrt 9) [ceiling] 98 = 99 - (9/9) 99 = 9 + 9 + (9x9) = 99 / (9/9) = 99 + 9 - 9 = (sqrt 99) x (sqrt 99) = |_ sqrt 9999 _| 100 = 99 + (9/9) = |- sqrt 9999 -| [ceiling] = |- sqrt sqrt sqrt sqrt sqrt (9-to-the-99) -| / 9 [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 147! _| {the rounded-down 128th-root of 147!; since we can make 147 with four nines, this gives us another way to make 100 with four nines} 101 = 99 + |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| 102 = |_ (sqrt(9!))/9 _| + ((sqrt 9)! x (sqrt 9)!) 103 = |- (sqrt(9!))/9 -| + ((sqrt 9)! x (sqrt 9)!) [ceiling] = |_ sqrt |_ sqrt |_ sqrt |_ sqrt |_ sqrt 50! _| _| _| _| _| {and, as we see above, 50 can be made with a single nine, hence 103 can be made with a single nine} 104 = 99 + (sqrt 9)! - |_ sqrt sqrt 9 _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 148! _| {the rounded-down 128th-root of 148!; since we can make 148 with four nines, this gives us another way to make 104 with four nines} 105 = 99 + sqrt 9 + sqrt 9 106 = 99 + (sqrt 9)! + |_ sqrt sqrt 9 _| 107 = 99 + 9 - |_ sqrt sqrt 9 _| 108 = 99 + ((sqrt 9) x (sqrt 9)) = 99 + sqrt 9 + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 149! _| {the rounded-down 128th-root of 149!; since we can make 149 with four nines, this gives us another way to make 108 with four nines} 109 = 99 + |- sqrt 99 -| 110 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| + (9 x (sqrt 9)!) 111 = 999/9 = 99 + 9 + sqrt 9 112 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| x ( |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| ) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 103! _| {the rounded-down 128th-root of 150!; since we can make 150 with four nines, this gives us another way to make 112 with four nines} 113 = ( |- sqrt sqrt sqrt sqrt (9 x (sqrt 9))! -| x [ceiling] |_ sqrt sqrt sqrt sqrt (9!) _| ) - |_ sqrt sqrt 9 _| 114 = 99 + 9 + (sqrt 9)! 115 = ( |- sqrt sqrt sqrt sqrt (9 x (sqrt 9))! -| x |_ sqrt sqrt sqrt sqrt (9!) _| ) + |_ sqrt sqrt 9 _| 116 = |- sqrt sqrt sqrt sqrt (9 x (sqrt 9))! -| x [ceiling] ( |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| ) 117 = 99 + 9 + 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 151! _| {the rounded-down 128th-root of 151!; since we can make 151 with four nines, this gives us another way to make 117 with four nines} 118 = |_ (9-to-the-9th / 9!) / 9! _| 119 = ( (sqrt 9)! - |_ sqrt sqrt 9 _| )! - (9/9) 120 = ((9 - sqrt 9) - 9/9)! 121 = ( (sqrt 9)! - |_ sqrt sqrt 9 _| )! + (9/9) |- (sqrt(9!))/9 -| + (9 x (sqrt 9)!) [ceiling] 122 = |_ sqrt(sqrt((9-to-the-9))) _| - (9 + 9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 152! _| {the rounded-down 128th-root of 152!; since we can make 152 with four nines, this gives us another way to make 122 with four nines} 123 = ( (sqrt 9)! - |_ sqrt sqrt 9 _| )! + (9/(Sqrt 9)) 124 = ( (sqrt 9)! - |_ sqrt sqrt 9 _| )! + (sqrt 9) + |_ sqrt sqrt 9 _| 125 = |_ sqrt(sqrt((9-to-the-9))) _| - (9 + (sqrt 9)!) 126 = 99 + (9 x sqrt 9) 127 = ( (sqrt 9)! - |_ sqrt sqrt 9 _| )! + (sqrt 9)! + |_ sqrt sqrt 9 _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 153! _| {the rounded-down 128th-root of 153!; since we can make 153 with four nines, this gives us another way to make 127 with four nines} 128 = |_ sqrt(sqrt((9-to-the-9))) _| - (9 + (sqrt 9)) 129 = ( (sqrt 9)! - |_ sqrt sqrt 9 _| )! + ((sqrt 9)x(sqrt 9)) 130 = ( (sqrt 9)! - |_ sqrt sqrt 9 _| )! + 9 + |_ sqrt sqrt 9 _| 131 = |_ sqrt(sqrt((9-to-the-9))) _| - ((sqrt 9)x(sqrt 9)) 132 = |- sqrt(sqrt((9-to-the-9))) -| - ((sqrt 9)x(sqrt 9)) [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 154! _| {the rounded-down 128th-root of 154!; since we can make 154 with four nines, this gives us another way to make 132 with four nines} 133 = |- (sqrt 9)th-root-of-9!) -| that is, ceiling of cube root of 9! which uses only two nines. Hence: 133 = |- (sqrt 9)th-root-of-9!) -| x (9/9) [ceiling] 134 = |_ sqrt(sqrt((9-to-the-9))) _| - ((sqrt 9) + (sqrt 9)) 135 = |- sqrt(sqrt((9-to-the-9))) -| - ((sqrt 9) + (sqrt 9)) [ceiling] 136 = |_ sqrt(sqrt((9-to-the-9))) _| - (sqrt 9) - |_ sqrt sqrt 9 _| 137 = |_ sqrt(sqrt((9-to-the-9))) _| - sqrt(sqrt(9x9)) = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| + (9x9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 155! _| {the rounded-down 128th-root of 155!; since we can make 155 with four nines, this gives us another way to make 137 with four nines} 138 = |- sqrt(sqrt((9-to-the-9))) -| - sqrt(sqrt(9x9)) [ceiling] 139 = |_ sqrt(sqrt((9-to-the-9))) _| - (9/9) 140 = |_ sqrt(sqrt((9-to-the-9))) _| [uses just two nines, so]: 140 = |_ sqrt(sqrt((9-to-the-9))) _| x (9/9) 141 = |_ sqrt(sqrt((9-to-the-9))) _| + (9/9) 142 = |- sqrt(sqrt((9-to-the-9))) -| + (9/9) [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 156! _| {the rounded-down 128th-root of 156!; since we can make 156 with four nines, this gives us another way to make 142 with four nines} 143 = |_ sqrt(sqrt((9-to-the-9))) _| + (9 - (sqrt 9)!) 144 = |- sqrt(sqrt((9-to-the-9))) -| + (9 - (sqrt 9)!) [ceiling] 145 = |_ sqrt(sqrt((9-to-the-9))) _| + (sqrt 9)! - |_ sqrt sqrt 9 _| 146 = |_ sqrt(sqrt((9-to-the-9))) _| + 9 - (sqrt 9) 147 = |- sqrt(sqrt((9-to-the-9))) -| + 9 - (sqrt 9) [ceiling] 148 = |- (sqrt(9!))/9 -| + (9x9) [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 157! _| {the rounded-down 128th-root of 157!; since we can make 157 with four nines, this gives us another way to make 148 with four nines} 149 = |_ sqrt(sqrt((9-to-the-9))) _| + ((sqrt 9)x(sqrt 9)) = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| / (sqrt 9)! 150 = |_ sqrt(sqrt((9-to-the-9))) _| + 9 + |_ sqrt sqrt 9 _| 150 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt 9999! _| {that is, 150 is the 16384th root of 9999!, rounded down} 151 =|_ sqrt(sqrt((9-to-the-9))) _| + 9 + |- sqrt sqrt 9 -| [ceiling] 152 = |_ sqrt(sqrt((9-to-the-9))) _| + 9 + (sqrt 9) 153 = (9 x (9+9)) - 9 = 99 + (9 x (sqrt 9)!) 154 = ((((sqrt 9)!)! / 9) x |- sqrt sqrt 9 -| ) - (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 158! _| {the rounded-down 128th-root of 158!; since we can make 158 with four nines, this gives us another way to make 154 with four nines} 154 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {91!} _| {that is, 154 = the 64th root of 91!, so insert an existing four nines solution for 91 here} 155 = |_ sqrt sqrt sqrt sqrt (9 x (sqrt 9))! _| + 99 = |_ sqrt(sqrt((9-to-the-9))) _| + 9 + (sqrt 9)! 156 = ( 9 x 9 x |- sqrt sqrt 9 -| ) - (sqrt 9)! [ceiling] 157 = ((((sqrt 9)!)! / 9) x |- sqrt sqrt 9 -| ) - (sqrt 9) 158 = |_ sqrt(sqrt((9-to-the-9))) _| + 9 + 9 159 = ( 9 x 9 x |- sqrt sqrt 9 -| ) - (sqrt 9) [ceiling] 160 = (((sqrt 9)!)! / 9) x |- sqrt sqrt 9 -| [ceiling] = (((sqrt 9)!)! / 9) x ( |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| ) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 159! _| {the rounded-down 128th-root of 159!; since we can make 159 with four nines, this gives us another way to make 160 with four nines} 161 = ( 9 x 9 x |- sqrt sqrt 9 -| ) - |_ sqrt sqrt 9 _| [ceiling then floor] 162 = (9x9) + (9x9) 163 = ( 9 x 9 x |- sqrt sqrt 9 -| ) + |_ sqrt sqrt 9 _| [ceiling then floor] 164 = ( 9 x 9 x |- sqrt sqrt 9 -| ) + |- sqrt sqrt 9 -| [ceiling then ceiling] 165 = |_ (sqrt(9!))/9 _| + 99 166 = |- (sqrt(9!))/9 -| + 99 [ceiling] = |_ 999/(sqrt 9)! _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt {92!} _| {that is, 166 = the 64th root of 92!, so insert an existing four nines solution for 92 here} 167 = |_ sqrt(sqrt((9-to-the-9))) _| + (9 x sqrt 9) = |- 999/(sqrt 9)! -| [ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 160! _| {the rounded-down 128th-root of 160!; since we can make 160 with four nines, this gives us another way to make 167 with four nines} 168 = |- sqrt(sqrt((9-to-the-9))) -| + (9 x sqrt 9) [ceiling] 169 = ((((sqrt 9)!)! / 9) x |- sqrt sqrt 9 -| ) + 9 170 = |- sqrt sqrt sqrt (99-to-the-9th) -| - (sqrt 9)! [ceiling] 171 = 9 + (9 x (9+9)) = ((9 x (sqrt 9)!) + sqrt 9) x sqrt 9 172 = |_ sqrt sqrt sqrt sqrt sqrt sqrt (99 - (sqrt 9)!)!) _| - (sqrt 9)! 173 = |_ sqrt sqrt sqrt (99-to-the-9th) _| - |- sqrt sqrt 9-| [floor then ceiling] = |_ ( |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| ) / sqrt 9 _| 174 = |_ sqrt sqrt sqrt (99-to-the-9th) _| - |_ sqrt sqrt 9_| = |- |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| / sqrt 9 -| [floor in ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 161! _| {the rounded-down 128th-root of 161!; since we can make 161 with four nines, this gives us another way to make 174 with four nines} 175 = |_ sqrt sqrt sqrt sqrt sqrt sqrt (99 - (sqrt 9)!)!) _| - sqrt 9 = |_ sqrt sqrt sqrt (99-to-the-9th) _| [with just three nines] 176 = |_ (((sqrt 9)!)-to-the-9) / (9 + 9) _| = |- sqrt sqrt sqrt (99-to-the-9th) -| [ceiling] [with just three nines] 177 = |_ sqrt sqrt sqrt sqrt sqrt sqrt (99 - (sqrt 9)!)!) _| - |_ sqrt sqrt 9 _| 178 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {93!} _| {that is, 178 = the 64th root of 93!, so insert an existing four nines solution for 93 here} {note that 93 = 99 - (sqrt 9)!} 179 = (((sqrt 9)!)! / 9) + 99 180 = 99 + (9x9) 181 = |_ sqrt sqrt sqrt sqrt sqrt sqrt (99 - (sqrt 9)!)!) _| + sqrt 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 162! _| {the rounded-down 128th-root of 162!; since we can make 162 with four nines, this gives us another way to make 181 with four nines} 182 = |- sqrt sqrt sqrt (99-to-the-9th) -| + (sqrt 9)! [ceiling] 183 = ( |_ sqrt |_ sqrt |_ sqrt ((sqrt 9)!)! _| _| _| x (9+9) ) + sqrt 9 {the above is a fancy way of saying 183 = (10 x 18) + 3 } 184 = |_ sqrt sqrt sqrt sqrt sqrt sqrt (99 - (sqrt 9)!)!) _| + (sqrt 9)! 185 = |- sqrt sqrt sqrt (99-to-the-9th) -| + 9 [ceiling] 186 = ( |_ sqrt |_ sqrt |_ sqrt ((sqrt 9)!)! _| _| _| x (9+9) ) + (sqrt 9)! {the above is a fancy way of saying 183 = (10 x 18) + 6 } 187 = |_ sqrt sqrt sqrt sqrt sqrt sqrt (99 - (sqrt 9)!)!) _| + 9 188 = |_ sqrt (9 - |_ sqrt sqrt 9 _| _|)! _| - 9 - sqrt 9 {see 200 for an explanation, as this is 200 - 9 - 3} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 163! _| {the rounded-down 128th-root of 163!; since we can make 163 with four nines, this gives us another way to make 188 with four nines} 189 = ( |_ (sqrt(9!))/9 _| x sqrt 9 ) - 9 190 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-((sqrt 9)!-to-the-((sqrt 9)))) _| - (sqrt 9)! 191 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {94!} _| {that is, 191 = the 64th root of 94!, so insert an existing four nines solution for 94 here} 192 = ( |_ (sqrt(9!))/9 _| x sqrt 9 ) - (sqrt 9)! 193 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-((sqrt 9)!-to-the-((sqrt 9)))) _| - sqrt 9 194 = |_ sqrt(sqrt((9-to-the-9))) _| + (9 x (sqrt 9)!) 195 = ( |_ (sqrt(9!))/9 _| x sqrt 9 ) - (sqrt 9) 196 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-((sqrt 9)!-to-the-((sqrt 9)))) _| [with just three nines] { that is, 196 = 2-to-the-22nd root of 9^(6^3)) rounded down } = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 164! _| {the rounded-down 128th-root of 164!; since we can make 164 with four nines, this gives us another way to make 196 with four nines} 197 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-((sqrt 9)!-to-the-((sqrt 9)))) -| [with just three nines] { that is, 197 = 2-to-the-22nd root of 9^(6^3)) rounded up } 198 = 99 + 99 199 = |_ (sqrt(9!))/9 _| x (9/(sqrt 9)) 200 = ( |_ (sqrt(9!))/9 _| x sqrt 9 ) + |_ sqrt sqrt 9 _| = |_ sqrt 8! _| {and, as we can near the top, there are several ways to make 8 with four nines, hence we can make 200 with four nines} {be we can make 8 with two nines, so we can make 200 with two nines}: = |_ sqrt (9 - |_ sqrt sqrt 9 _| _|)! _| 201 = |- (sqrt(9!))/9 -| x (9/(sqrt 9)) [ceiling] = ( |_ (sqrt(9!))/9 _| x sqrt 9 ) + (sqrt 9) 202 = |- (sqrt(9!))/9 -| x (sqrt 9) + |_ sqrt sqrt 9 _| [ceiling then floor] 203 = ( |- (sqrt(9!))/9 -| x (sqrt 9) ) + |- sqrt sqrt 9 -| [ceiling then ceiling] 204 = ( |_ (sqrt(9!))/9 _| x sqrt 9 ) + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 165! _| {the rounded-down 128th-root of 165!; since we can make 165 with four nines, this gives us another way to make 204 with four nines} 205 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {95!} _| {that is, 205 = the 64th root of 95, so insert an existing four nines solution for 95 here, rounded down} 206 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-((sqrt 9)!-to-the-((sqrt 9)))) -| + 9 [ceiling] { that is, 206 = 2-to-the-22nd root of 9^(6^3)) rounded up, + 9 } 207 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) - 9 208 = |_ sqrt (9 - |_ sqrt sqrt 9 _| _|)! _| + 9 - |_ sqrt sqrt 9 _| {fancy way of saying 200 + 9 - 1 ... see 200 for details} 209 = |_ sqrt (9 - |_ sqrt sqrt 9 _| _|)! _| + (9/sqrt 9) = |_ sqrt (9 - |_ sqrt sqrt 9 _| _|)! _| + 9 {that makes 209 with 3 nines, see 200 for details} 210 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) - (sqrt 9)! 211 = |_ sqrt (9 - |_ sqrt sqrt 9 _| _|)! _| + 9 + |- sqrt sqrt 9 -| {fancy way of saying 200 + 9 + 2 ... see 200 for details} 212 = |_ sqrt (9 - |_ sqrt sqrt 9 _| _|)! _| + 9 + 3 {fancy way of saying 200 + 9 + 3 ... see 200 for details} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 166! _| {the rounded-down 128th-root of 166!; since we can make 166 with four nines, this gives us another way to make 212 with four nines} 213 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) - (sqrt 9) 214 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) - |- sqrt sqrt 9 -| [ceiling] 215 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) - |_ sqrt sqrt 9 _| 216 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) x |_ sqrt sqrt 9 _| = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - (9+9) 217 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) + |_ sqrt sqrt 9 _| 218 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) + |- sqrt sqrt 9 -| [ceiling] 219 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) + (sqrt 9) 220 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {96!} _| {that is, 220 = the 64th root of 96!, so insert an existing four nines solution for 96 here, rounded down} 221 = |_ sqrt(sqrt((9-to-the-9))) _| + (9x9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 167! _| {the rounded-down 128th-root of 167!; since we can make 167 with four nines, this gives us another way to make 221 with four nines} 222 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) + (sqrt 9)! 223 = = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - 9 - |- sqrt sqrt 9 -| [floor then ceiling] 224 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - 9 - |_ sqrt sqrt 9 _| 225 = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) + 9 = (((sqrt 9)!)! / (sqrt 9)) - 9 - (sqrt 9)! 226 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - (sqrt 9) - |- sqrt sqrt 9 -| [floor then ceiling] 227 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - (sqrt 9) - |_ sqrt sqrt 9 _| 228 = (((sqrt 9)!)! / (sqrt 9)) - 9 - sqrt 9 229 = (((sqrt 9)!)! / (sqrt 9)) - 9 - |- sqrt sqrt 9 -| [ceiling] 230 = (((sqrt 9)!)! / (sqrt 9)) - 9 - |_ sqrt sqrt 9 _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 168! _| {the rounded-down 128th-root of 168!; since we can make 168 with four nines, this gives us another way to make 230 with four nines} 231 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - (9/sqrt 9) 232 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - |_ sqrt sqrt 9 _| - |_ sqrt sqrt 9 _| 233 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) - (9/9) 234 = |_ sqrt ((sqrt 9)!)! _| x 9 {26 x 9 using just two nines) 235 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) + (9/9) 236 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {97!} _| {that is, 236 = the 64th root of 97!, so insert an existing four nines solution for 97 here, rounded down} 237 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) + (9/sqrt 9) 238 = ( |_ sqrt ((sqrt 9)!)! _| x 9 ) + (sqrt 9) + |_ sqrt sqrt 9 _| 239 = |_ sqrt(sqrt((9-to-the-9))) _| + 99 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 169! _| {the rounded-down 128th-root of 169!; since we can make 169 with four nines, this gives us another way to make 239 with four nines} 240 = |- sqrt(sqrt((9-to-the-9))) -| + 99 [ceiling] = ((sqrt 9)!)! / (sqrt 9) uses only two nines, so 240 = (((sqrt 9)!)! / (sqrt 9)) + 9/9 241 = (((sqrt 9)!)! / (sqrt 9)) + (9/9) 242 = (((sqrt 9)!)! / (sqrt 9)) + |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| 243 = 9 x (9 + 9 + 9) = (((sqrt 9)!)! / (sqrt 9)) + 9/(sqrt 9) 244 = (((sqrt 9)!)! / (sqrt 9)) + |- sqrt sqrt 9 -| + |- sqrt sqrt 9 -| [two ceilings] 245 = (((sqrt 9)!)! / (sqrt 9)) + (sqrt 9)! - |_ sqrt sqrt 9 _| 246 = (((sqrt 9)!)! / (sqrt 9)) + 9 - sqrt 9 247 = (((sqrt 9)!)! / (sqrt 9)) + 9 - |- sqrt sqrt 9 -| [ceiling] 248 = (((sqrt 9)!)! / (sqrt 9)) + 9 - |_ sqrt sqrt 9 _| 249 = (((sqrt 9)!)! / (sqrt 9)) + ((sqrt 9) x (sqrt 9)) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 170! _| {the rounded-down 128th-root of 170!; since we can make 170 with four nines, this gives us another way to make 249 with four nines} 250 = (((sqrt 9)!)! / (sqrt 9)) + 9 + |_ sqrt sqrt 9 _| 251 = (((sqrt 9)!)! / (sqrt 9)) + 9 + |- sqrt sqrt 9 -| [ceiling] 252 = (((sqrt 9)!)! / (sqrt 9)) + 9 + sqrt 9 253 = |_ sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99_| - |-sqrt sqrt 9-| [floor then ceiling] 254 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {98!} _| {that is, 254 = the 64th root of 98!, rounded down, so insert an existing four nines solution for 98 here} 255 = (((sqrt 9)!)! / (sqrt 9)) + 9 + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99 _| [with just 3 nines] {that is, the 32nd root of 6-to-the-99 is 255.50854...} 256 = |- sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99 -| [ceiling] [with just 3 nines] 257 = |_ sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99 _| + |-sqrt sqrt 9-| [floor then ceiling] 258 = (((sqrt 9)!)! / (sqrt 9)) + 9 + 9 259 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 171! _| {the rounded-down 128th-root of 171!; since we can make 171 with four nines, this gives us another way to make 259 with four nines} 260 = |_ ( |_ ( |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| ) / (|_ sqrt sqrt 9 _| ) _| 261 = |_ sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99 _| + (sqrt 9)! 262 = |- sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99 -| + (sqrt 9)! [ceiling] 263 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| - 9 - |_ sqrt sqrt 9 _| 264 = |_ sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99 _| + 9 265 = |- sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-99 -| + 9 [ceiling] 267 = (((sqrt 9)!)! / (sqrt 9)) + (9 x sqrt 9) 268 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| + |_ sqrt sqrt 9 _| - (sqrt 9)! 269 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| - |- sqrt sqrt 9 -| - |- sqrt sqrt 9 -| {floor followed by two ceilings} 270 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| - (9/sqrt 9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 172! _| {the rounded-down 128th-root of 172!; since we can make 172 with four nines, this gives us another way to make 270 with four nines} 271 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| - |_ sqrt sqrt 9 _| - |_ sqrt sqrt 9 _| 272 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| - (9/9) 273 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| {that is, 273 is the 64th root of 99!, rounded down, using just two nines} 274 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| + (9/9) 275 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| + |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| 276 = (((sqrt 9)!)! / (sqrt 9)) + ((sqrt 9)! x (sqrt 9)!) 277 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| + |- sqrt sqrt 9 -| + |- sqrt sqrt 9 -| {floor followed by two ceilings} 278 = 277 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| + (sqrt 9) + |- sqrt sqrt 9 -| {floor followed by ceiling} 279 = 9 x |_ sqrt 999 _| 280 = |_ sqrt(sqrt((9-to-the-9))) _| + |_ sqrt(sqrt((9-to-the-9))) _| 281 = |_ sqrt(sqrt((9-to-the-9))) _| + |- sqrt(sqrt((9-to-the-9))) -| [floor then ceiling] = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 173! _| {the rounded-down 128th-root of 173!; since we can make 173 with four nines, this gives us another way to make 281 with four nines} 282 = |- sqrt(sqrt((9-to-the-9))) -| + |- sqrt(sqrt((9-to-the-9))) -| [ceiling then ceiling] 283 = |_ sqrt sqrt sqrt sqrt sqrt sqrt 99! _| + 9 + |_ sqrt sqrt 9 _| 284 = |_ 9 x sqrt 999 _| 285 = |- 9 x sqrt 999 -| [ceiling] 286 = 287 = 288 = 9 x |- sqrt 999 -| [ceiling] 289 = {300 made with two nines} - 9 - |- sqrt sqrt 9 -| [ceiling] 290 = {300 made with two nines} - 9 - |_ sqrt sqrt 9 _| 291 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) - 9 292 = {300 made with two nines} - 9 + |_ sqrt sqrt 9 _| 293 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {100!} _| {that is, 293 = the 64th root of 100!, rounded down, so insert an existing four nines solution for 100 here} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 174! _| {the rounded-down 128th-root of 174!; since we can make 174 with four nines, this gives us another way to make 293 with four nines} 294 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) - (sqrt 9)! 295 = {300 made with two nines} - (sqrt 9) - |- sqrt sqrt 9 -| [ceiling] 296 = {300 made with two nines} - (sqrt 9) - |_ sqrt sqrt 9 _| 297 = 99 x (9 - (sqrt 9)!) 298 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| / sqrt 9 299 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) - |_ sqrt sqrt 9 _| 300 = 50 x (sqrt 9)! {note that 50 can be made with just one nine, and a whole lot of square roots, factorials, and floors so we can make 300 with just two nines} = ( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)! {300 made with three nines} [2 ceilings] 301 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) + |_ sqrt sqrt 9 _| [2 ceilings then a floor] 302 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) + |- sqrt sqrt 9 -| [3 ceilings] 303 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) + sqrt 9 [2 ceilings] 304 = {300 made with two nines} + ( sqrt 9) + |_ sqrt sqrt 9 _| 305 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 175! _| {the rounded-down 128th-root of 175!; since we can make 175 with four nines, this gives us another way to make 305 with four nines} 306 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) + (sqrt 9)! [2 ceilings] 307 = {300 made with two nines} + (sqrt 9)! + |_ sqrt sqrt 9 _| 308 = {300 made with two nines} + 9 - |_ sqrt sqrt 9 _| 309 = (( |- sqrt(sqrt(9!)) -| + |- sqrt(sqrt(9!)) -| ) x (sqrt 9)!) + 9 [2 ceilings] = 103 x (sqrt 9), and we see above that we can make 103 with a single nine, hence we can make 309 with just two nines. 310 = {300 made with two nines} + 9 + |_ sqrt sqrt 9 _| 311 = {300 made with two nines} + 9 + |- sqrt sqrt 9 -| [ceiling] 312 = {300 made with two nines} + 9 + sqrt 9 314 = 315 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {101!} _| {that is, 315 = the 64th root of 101!, rounded down, so insert an existing four nines solution for 101 here} 316 = 317 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 176! _| {the rounded-down 128th-root of 176!; since we can make 176 with four nines, this gives us another way to make 317 with four nines} 318 = {300 made with two nines} + 9 + 9 319 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt 999! _| {that is, 319 is the 1024th root of 999!, rounded down, using just three nines} 320 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt 999! -| {that is, 319 is the 1024th root of 999!, rounded up, using just three nines} 321 = {309 made with two nines} + 9 + sqrt 9 322 = 323 = 324 = (9+9) x (9+9) 325 = 326 = 327 = {309 made with two nines} + 9 + 9 328 = 329 = 330 = 331 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 177! _| {the rounded-down 128th-root of 177!; since we can make 177 with four nines, this gives us another way to make 331 with four nines} 332 = 333 = 999/sqrt 9 334 = 335 = 336 = {309 made with two nines} + (9 x sqrt 9) 339 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {102!} _| {that is, 339 = the 64th root of 102!, rounded down, so insert an existing four nines solution for 102 here} 340 = 341 = 342 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| - 9 - 9 [ceiling] 343 = |_ (((sqrt 9)!)-to-the-9) / 9 _| - 9 344 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 178! _| {the rounded-down 128th-root of 178!; since we can make 178 with four nines, this gives us another way to make 344 with four nines} 345 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| - 9 - (sqrt 9)! [ceiling] = {309 made with two nines} + ((sqrt 9)! x (sqrt 9)!) 346 = |_ (((sqrt 9)!)-to-the-9) / 9 _| - (sqrt 9)! 347 = 348 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| - 9 - sqrt 9 [ceiling] 349 = |_ (((sqrt 9)!)-to-the-9) / 9 _| - (sqrt 9) 350 = |- (((sqrt 9)!)-to-the-9) / 9 -| - (sqrt 9) [ceiling] 351 = 352 = |_ (((sqrt 9)!)-to-the-9) / 9 _| [uses just three nines] 353 = |- (((sqrt 9)!)-to-the-9) / 9 -| [ceiling] [uses just three nines] 354 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| - (sqrt 9) - (sqrt 9) [ceiling] 355 = |_ (((sqrt 9)!)-to-the-9) / 9 _| + (sqrt 9) 356 = |- (((sqrt 9)!)-to-the-9) / 9 -| + (sqrt 9) [ceiling] 357 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| - (9 / sqrt 9) [ceiling] 358 = |_ (((sqrt 9)!)-to-the-9) / 9 _| + (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 179! _| {the rounded-down 128th-root of 179!; since we can make 179 with four nines, this gives us another way to make 358 with four nines} 359 = |- (((sqrt 9)!)-to-the-9) / 9 -| + (sqrt 9)! [ceiling] 360 = ((sqrt 9)!)! / ((sqrt 9) - (9/9)) 361 = |_ (((sqrt 9)!)-to-the-9) / 9 _| + 9 362 = |- (((sqrt 9)!)-to-the-9) / 9 -| + 9 [ceiling] 363 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + (9 / sqrt 9) [ceiling] 364 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + |- sqrt sqrt 9 -| + |- sqrt sqrt 9 -| [three ceilings] 365 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + (sqrt 9)! - |_ sqrt sqrt 9 _| = |_ sqrt sqrt sqrt sqrt sqrt sqrt {103!} _| {that is, 365 = the 64th root of 103!, rounded down, so insert an existing four nines solution for 103 here} 366 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + (sqrt 9) + (sqrt 9) 367 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 9! - |- sqrt sqrt 9 -| [two ceilings] 368 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 9! - |_ sqrt sqrt 9 _| 369 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + ((sqrt 9) x (sqrt 9)) 370 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 9 + |_ sqrt sqrt 9 _| 371 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 9 + |- sqrt sqrt 9 -| [two ceilings] 372 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 9 + sqrt 9 373 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 180! _| {the rounded-down 128th-root of 180!; since we can make 180 with four nines, this gives us another way to make 373 with four nines} 374 = 375 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 9 + (sqrt 9)! 376 = 377 = 378 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 9 + 9 379 = 380 = 381 = {300 made with two nines} + (9x9) ... 387 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + (9 x (sqrt 9)!) 388 = 389 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 181! _| {the rounded-down 128th-root of 181!; since we can make 181 with four nines, this gives us another way to make 389 with four nines} 390 = {309 made with two nines} + (9x9) 391 = 392 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {104!} _| {that is, 392 = the 64th root of 104!, rounded down, so insert an existing four nines solution for 104 here} 393 = 394 = 395 = |_ sqrt sqrt sqrt sqrt ((sqrt 9)! x (sqrt 9)!) _| - (9/9) 396 = |_ (sqrt(9!))/9 _| x (sqrt 9)! [66x6 with just 3 nines], so = |_ (sqrt(9!))/9 _| x (sqrt 9)! x |_ sqrt sqrt 9 _| = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + ((sqrt 9)! x (sqrt 9)!) = |_ sqrt sqrt sqrt sqrt ((sqrt 9)! x (sqrt 9)!) _| {that is to say. 396 = the 16th root of 36! and that uses just two nines} so: 396 = |_ sqrt sqrt sqrt sqrt ((sqrt 9)! x (sqrt 9)!) _| x (9/9) 397 = ( |_ (sqrt(9!))/9 _| x (sqrt 9)!) + |_ sqrt sqrt 9 _| = |_ sqrt sqrt sqrt sqrt ((sqrt 9)! x (sqrt 9)!) _| + (9/9) 398 = 399 = ( |_ (sqrt(9!))/9 _| x (sqrt 9)!) + (sqrt 9) = {300 made with two nines} + 99 400 = 401 = 402 = ( |_ (sqrt(9!))/9 _| x (sqrt 9)!) + (sqrt 9)! 403 = 404 = 405 = ( |_ (sqrt(9!))/9 _| x (sqrt 9)!) + 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 182! _| {the rounded-down 128th-root of 182!; since we can make 182 with four nines, this gives us another way to make 405 with four nines} 406 = 407 = |_ 999 / sqrt((sqrt 9)!) _| 408 = |- 999 / sqrt((sqrt 9)!) -| [ceiling] = {309 made with two nines} + 99 ... 422 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {105!} _| {that is, 422 = the 64th root of 105!, rounded down, so insert an existing four nines solution for 105 here} = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 183! _| {the rounded-down 128th-root of 183!; since we can make 183 with four nines, this gives us another way to make 422 with four nines} ... 438 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9!)) _| {that is, with 17 sqare roots nested, 9 to the power of 9! = 438.402046.... with just two nines} 439 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 184! _| {the rounded-down 128th-root of 184!; since we can make 184 with four nines, this gives us another way to make 439 with four nines} 440 = 441 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + (9x9) ... 454 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {106!} _| {that is, 454 = the 64th root of 106!, rounded down, so insert an existing four nines solution for 106 here} 455 = 456 = 457 = 458 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 185! _| {the rounded-down 128th-root of 185!; since we can make 185 with four nines, this gives us another way to make 458 with four nines} 459 = ((sqrt 9)!)! / |- sqrt sqrt 9 -| + 99 ... 477 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 186! _| {the rounded-down 128th-root of 186!; since we can make 186 with four nines, this gives us another way to make 477 with four nines} ... 482 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-((sqrt 9)!) _| {that is, the 256th root of 9-to-the-6! = 256th root of 9-to-the-720 = 482.8442... with just two nines} 488 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {107!} _| {that is, 488 = the 64th root of 107!, rounded down, so insert an existing four nines solution for 107 here} ... 494 = |_ sqrt 9! _| - 99 - 9 495 = 496 = 497 = |_ sqrt 9! _| - 99 - (sqrt 9)! = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 187! _| {the rounded-down 128th-root of 187!; since we can make 187 with four nines, this gives us another way to make 497 with four nines} 498 = 499 = |_ (999/|-sqrt sqrt 9 -|) _| [ceiling inside floor] 500 = |- (999/|-sqrt sqrt 9 -|) -| [ceiling inside ceiling] 501 = |_ sqrt 9! _| - 99 - |- sqrt sqrt 9 -| [floor then ceiling] 502 = |_ sqrt 9! _| - 99 - |_ sqrt sqrt 9 _| 503 = |_ sqrt 9! _| - 99 {makes 503 with just three nines} 504 = |_ sqrt 9! _| - 99 + |_ sqrt sqrt 9 _| 505 = |_ sqrt 9! _| - 99 + |- sqrt sqrt 9 -| [floor then ceiling] 506 = |_ sqrt 9! _| - 99 + sqrt 9 507 = 508 = 506 = |_ sqrt 9! _| - 99 + (sqrt 9)! 510 = 511 = 512 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| - 9 = |_ sqrt 9! _| - 99 + 9 513 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) -| - 9 [ceiling] 514 = 515 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| - (sqrt 9)! 516 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) -| - (sqrt 9)! [ceiling] 517 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 188! _| {the rounded-down 128th-root of 188!; since we can make 188 with four nines, this gives us another way to make 517 with four nines} 518 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| - (sqrt 9) 519 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) -| - (sqrt 9) [ceiling] 520 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9)! ) - 9 521 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| [with just three nines] {that is, 521 = rounded down 256th root of 9^(9^3))} = |_ sqrt 9! _| - (9x9) {makes 521 with just three nines} 522 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) -| [with just three nines] {that is, 522 = rounded up 256th root of 9^(9^3))} 523 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9)! ) - (sqrt 9)! 524 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| + (sqrt 9) 525 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {108!} _| {that is, 525 = the 64th root of 108!, rounded down, so insert an existing four nines solution for 108 here} 526 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9)! ) - (sqrt 9) 527 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| + (sqrt 9)! 528 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) -| + (sqrt 9)! [ceiling] 529 = |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9)! [uses just three nines] 530 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) _| + 9 531 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-(sqrt 9))) -| + 9 [ceiling] 532 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9)! ) + (sqrt 9) 533 = 534 = 535 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9)! ) + (sqrt 9)! 536 = 537 = 538 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9)! ) + 9 539 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 189! _| {the rounded-down 128th-root of 189!; since we can make 189 with four nines, this gives us another way to make 539 with four nines} ... 562 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 190! _| {the rounded-down 128th-root of 190!; since we can make 190 with four nines, this gives us another way to make 562 with four nines} 563 = 564 = 565 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {109!} _| {that is, 565 = the 64th root of 109!, rounded down, so insert an existing four nines solution for 109 here} ... 575 = |_ sqrt 9! _| - ((sqrt 9) x (sqrt 9) x (sqrt 9)) 576 = |_ 999/sqrt sqrt 9 _| 577 = |- 999/sqrt sqrt 9 -| [ceiling] 578 = |_ sqrt 9! _| - 9 - 9 - (sqrt 9)! 579 = 580 = 581 = |_ sqrt 9! _| - 9 - 9 - sqrt 9 582 = 583 = 584 = |_ sqrt 9! _| - 9 - sqrt 9 - sqrt 9 585 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 191! _| {the rounded-down 128th-root of 191!; since we can make 191 with four nines, this gives us another way to make 585 with four nines} ... 593 = |_ sqrt 9! _| - 9 {makes 593 with just two nines} 594 = 99 x (9 - sqrt 9) = ( |_ (sqrt(9!))/9 _| x 9 ) [66x9 with just 3 nines], so, next: 595 = ( |_ (sqrt(9!))/9 _| x 9 ) + |_ sqrt sqrt 9 _| 596 = |_ sqrt 9! _| - 9 + (9/sqrt 9) = |_ sqrt 9! _| - (sqrt 9)! {uses just two nines} 597 = ( |_ (sqrt(9!))/9 _| x 9 ) + (sqrt 9) 598 = |_ sqrt 9! _| - (sqrt 9) - (9/9) 599 = |_ sqrt 9! _| - 9 + (sqrt 9) + (sqrt 9) = |_ sqrt 9! _| - sqrt 9 {uses just two nines} 600 = ( |_ (sqrt(9!))/9 _| x 9 ) + (sqrt 9)! |_ sqrt 9! _| - |_ sqrt sqrt sqrt sqrt (9!) _| 601 = |_ sqrt 9! _| + |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| - sqrt 9 602 = |_ sqrt 9! _| {uses just one nine} 603 = ( |_ (sqrt(9!))/9 _| x 9 ) + 9 604 = |_ sqrt 9! _| + (sqrt 9) - |_ sqrt sqrt 9 _| - |_ sqrt sqrt 9 _| 605 = |_ sqrt 9! _| + (sqrt 9) x (9/9) 606 = |_ sqrt 9! _| + (sqrt 9) + (9/9) 607 = |_ sqrt 9! _| + (sqrt 9)! - (9/9) 608 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {110!} _| {that is, 608 = the 64th root of 110!, rounded down, so insert an existing four nines solution for 110 here} = |_ sqrt 9! _| + (sqrt 9)! x (9/9) 609 = |_ sqrt 9! _| + 9 - |_ sqrt sqrt 9 _| - |_ sqrt sqrt 9 _| 610 = |_ sqrt 9! _| + 9 - (9/9) = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 192! _| {the rounded-down 128th-root of 192!; since we can make 192 with four nines, this gives us another way to make 610 with four nines} 611 = |_ sqrt 9! _| + 9 x (9/9) 612 = ((sqrt 9)!)! - 99 - 9 = |_ sqrt 9! _| + 9 + (9/9) 613 = |_ sqrt 9! _| + 9 + |_ sqrt sqrt 9 _| + |_ sqrt sqrt 9 _| 614 = |_ sqrt 9! _| + 9 + (9/sqrt 9) 615 = ((sqrt 9)!)! - 99 - (sqrt 9)! 616 = |_ sqrt 9! _| + 9 + (sqrt 9) + |- sqrt sqrt 9 -| [floor then ceiling] 617 = |_ sqrt 9! _| + 9 + (sqrt 9) + (sqrt 9) 618 = ((sqrt 9)!)! - 99 - sqrt 9 = 103 x (sqrt 9)!, and we see above that we can make 103 with a single nine, hence we can make 618 with just two nines. 619 = ((sqrt 9)!)! - 99 - |- sqrt sqrt 9 -| [ceiling] 620 = ((sqrt 9)!)! - 99 - |_ sqrt sqrt 9 _| 621 = 622 = ((sqrt 9)!)! - 99 + |_ sqrt sqrt 9 _| 623 = ((sqrt 9)!)! - 99 + |- sqrt sqrt 9 -| [ceiling] 624 = ((sqrt 9)!)! - 99 + sqrt 9 625 = 626 = 627 = ((sqrt 9)!)! - 99 + (sqrt 9)! 628 = 629 = 630 = 9-to-the-(sqrt 9)th - 99 = ((sqrt 9)!)! - 99 + 9 631 = 632 = 633 = ((sqrt 9)!)! - (9x9) - 9 634 = 635 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 193! _| {the rounded-down 128th-root of 193!; since we can make 193 with four nines, this gives us another way to make 635 with four nines} 636 = ((sqrt 9)!)! - (9x9) - sqrt 9 637 = ((sqrt 9)!)! - (9x9) - |- sqrt sqrt 9 -| [ceiling] 638 = ((sqrt 9)!)! - (9x9) - |_ sqrt sqrt 9 _| 639 = 640 = ((sqrt 9)!)! - (9x9) + |_ sqrt sqrt 9 _| 641 = ((sqrt 9)!)! - (9x9) + |- sqrt sqrt 9 -| [ceiling] 642 = ((sqrt 9)!)! - (9x9) + sqrt 9 643 = 644 = 645 = ((sqrt 9)!)! - (9x9) + (sqrt 9)! 646 = 647 = 648 = 9 x ((9x9) - 9) = ((sqrt 9)! x (sqrt 9)! x (sqrt 9)!) x (sqrt 9) = ((sqrt 9)!)! - (9x9) + 9 ... 655 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {111!} _| {that is, 655 = the 64th root of 111!, rounded down, so insert an existing four nines solution for 111 here} ... 662 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 194! _| {the rounded-down 128th-root of 194!; since we can make 194 with four nines, this gives us another way to make 662 with four nines} ... 690 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 195! _| {the rounded-down 128th-root of 195!; since we can make 195 with four nines, this gives us another way to make 690 with four nines} ... 705 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {112!} _| {that is, 705 = the 64th root of 112!, rounded down, so insert an existing four nines solution for 112 here} 706 = |_ 999/(sqrt(|-sqrt sqrt 9-|) _| [ceiling inside floor] 706 = |- 999/(sqrt(|-sqrt sqrt 9-|) -| [ceiling inside ceiling] ... 720 = ((sqrt 9)!)! {with just one nine , because (3!)! = 6! = 720} = (9 x (9x9)) - 9 ... 728 = 9-to-the-(sqrt 9)th - (9/9) 729 = 9-to-the-(sqrt 9)th x (9/9) 730 = 9-to-the-(sqrt 9)th + (9/9) = |_ (9-to-the-9)/((sqrt 9)!)! _| - (sqrt 9) 731 = 732 = 733 = |_ (9-to-the-9)/((sqrt 9)!)! _| [uses just 3 nines] = |_ (9-to-the-9)/((sqrt 9)! + (sqrt 9))! _| [uses all 4] 734 = |- (9-to-the-9)/((sqrt 9)! + (sqrt 9))! -| [ceiling function] 735 = 736 = |_ (9-to-the-9)/((sqrt 9)!)! _| + (sqrt 9) 737 = |- (9-to-the-9)/((sqrt 9)!)! -| + (sqrt 9) 738 = 9 + (9 x 9 x 9) 739 = |_ (9-to-the-9)/((sqrt 9)!)! _| + (sqrt 9)! 740 = |- (9-to-the-9)/((sqrt 9)!)! -| + (sqrt 9)! 741 = 9-to-the-(sqrt 9)th + (sqrt 9)! + (sqrt 9)! 742 = (9-to-the-9)/((sqrt 9)!)! + 9 ... 744 = 9-to-the-(sqrt 9)th + 9 + (sqrt 9)! ... 747 = 9-to-the-(sqrt 9)th + 9 + 9 748 = 749 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 197! _| {the rounded-down 128th-root of 197!; since we can make 197 with four nines, this gives us another way to make 749 with four nines} ... 759 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {113!} _| {that is, 759 = the 64th root of 113!, rounded down, so insert an existing four nines solution for 113 here} ... 781 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 198! _| {the rounded-down 128th-root of 198!; since we can make 198 with four nines, this gives us another way to make 781 with four nines} ... 810 = 9 x (9 + (9x9)) = 9-to-the-(sqrt 9)th + (9x9) 811 = 812 = 813 = 814 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 199! _| {the rounded-down 128th-root of 199!; since we can make 199 with four nines, this gives us another way to make 814 with four nines} 815 = 816 = 817 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {114!} _| {that is, 817 = the 64th root of 114!, rounded down, so insert an existing four nines solution for 114 here} ... 821 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-((sqrt 9)!))) _| / 9 { that is, 821 = 2-to-the-17th root of 9^(9^6)) rounded down, / 0 } ... 828 = 9-to-the-(sqrt 9)th + 99 ... 848 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 200! _| {the rounded-down 128th-root of 200!; since we can make 200 with four nines, this gives us another way to make 848 with four nines} ... 880 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {115!} _| {that is, 880 = the 64th root of 115!, rounded down, so insert an existing four nines solution for 115 here} 881 = 882 = (9 x 99) - 9 883 = 884 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 201! _| {the rounded-down 128th-root of 201!; since we can make 201 with four nines, this gives us another way to make 884 with four nines} 885 = 886 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| - 9 887 = 888 = 889 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| - (sqrt 9)! 890 = 891 = 892 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| - sqrt 9 893 = 894 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| - |_sqrt sqrt 9_| 895 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| [with just three nines], so: = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| x |_sqrt sqrt 9_| 896 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| + |_sqrt sqrt 9_| 897 = 898 = 899 = 900 = 9 + (9 x 99) 901 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| + (sqrt 9)! 902 = 903 = 904 = |_ sqrt sqrt sqrt sqrt sqrt (9-to-the-99) _| + 9 ... 922 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 202! _| {the rounded-down 128th-root of 2!; since we can make 202 with four nines, this gives us another way to make 922 with four nines} 923 = 924 = 925 = 926 = 927 = 103 x 9, and we see above that we can make 103 with a single nine, hence we can make 927 with just two nines. ... 948 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {116!} _| {that is, 948 = the 64th root of 116!, rounded down, so insert an existing four nines solution for 116 here} ... 961 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 203! _| {the rounded-down 128th-root of 203!; since we can make 203 with four nines, this gives us another way to make 961 with four nines} ... 972 = 9 x (9 + 99) ... 990 = 999 - 9 991 = 992 = 993 = 999 - (sqrt 9)! 994 = 995 = 996 = 999 - sqrt 9 997 = 998 = 999 = 1000 = 1001 = 1002 = 999 + sqrt 9 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 204! _| {the rounded-down 128th-root of 204!; since we can make 204 with four nines, this gives us another way to make 1002 with four nines} 1003 = 1004 = 1005 = 999 + (sqrt 9)! 1006 = 1007 = 1008 = 999 + 9 ... 1021 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {117!} _| {that is, 1021 = the 64th root of 117!, rounded down, so insert an existing four nines solution for 117 here} ... 1044 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 205! _| {the rounded-down 128th-root of 205!; since we can make 205 with four nines, this gives us another way to make 1044 with four nines} ... 1058 = |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9) [uses just three nines] 1059 = 1060 = 1061 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9) ) + (sqrt 9) 1062 = ( |- ((sqrt 9)!)-to-the-9 -| / (sqrt 9) ) + (sqrt 9) [ceiling] 1063 = 1064 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9) ) + (sqrt 9)! 1065 = ( |- ((sqrt 9)!)-to-the-9 -| / (sqrt 9) ) + (sqrt 9)! [ceiling] 1066 = 1067 = ( |_ ((sqrt 9)!)-to-the-9 _| / (sqrt 9) ) + 9 1067 = |_ 9-to-the-9 / (9!) _| [uses just three nines] 1068 = 1069 = 1070 = |_ 9-to-the-9 / (9!) _| + (sqrt 9) 1071 = |- 9-to-the-9 / (9!) -| + (sqrt 9) [ceiling] 1072 = 1073 = |_ 9-to-the-9 / (9!) _| + (sqrt 9)! 1074 = |- 9-to-the-9 / (9!) -| + (sqrt 9)! [ceiling] 1075 = 1076 = |_ 9-to-the-9 / (9!) _| + 9 1077 = |_ 9-to-the-9 / (9!) _| + 9 [ceiling] ... 1087 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-999 _| {that is, the 256the root of 6-to-the-999 = 1087.96185... } 1088 = |- sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt ((sqrt 9)!)-to-the-999 -| [ceiling] 1089 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 206! _| {the rounded-down 128th-root of 206!; since we can make 206 with four nines, this gives us another way to make 1089 with four nines} ... 1100 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {118!} _| {that is, 1100 = the 64th root of 118!, rounded down, so insert an existing four nines solution for 118 here} ... 1135 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 207! _| {the rounded-down 128th-root of 207!; since we can make 207 with four nines, this gives us another way to make 1135 with four nines} ... 1140 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (sqrt 9)-to-the-power-((sqrt 9)-to-the-power-(|_sqrt sqrt 9_|- to-the-power-(sqrt 9)))) _| {that is 1140 = rounded down the 2^10th root of 3^(3^(2^3))} ... 1183 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 208! _| {the rounded-down 128th-root of 208!; since we can make 208 with four nines, this gives us another way to make 1183 with four nines} 1184 = 1185 = 1186 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {119!} _| {that is, 1186 = the 64th root of 119!, rounded down, so insert an existing four nines solution for 119 here} ... 1232 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-((sqrt 9)!))) _| / ((sqrt 9)!) { that is, 1232 = 2-to-the-17th root of 9^(9^6)) rounded down, / 6 } 1233 = 1234 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 209! _| {the rounded-down 128th-root of 209!; since we can make 209 with four nines, this gives us another way to make 1234 with four nines} ... 1278 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {120!} _| {that is, 1278 = the 64th root of 120!, rounded down, so insert an existing four nines solution for 120 here} { note that 120 = 5! } ... 1286 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 210! _| {the rounded-down 128th-root of 210!; since we can make 210 with four nines, this gives us another way to make 1286 with four nines} ... 1296 = (sqrt 9)! x (sqrt 9)! x (sqrt 9)! x (sqrt 9)! ... 1341 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 211! _| {the rounded-down 128th-root of 211!; since we can make 211 with four nines, this gives us another way to make 1341 with four nines} ... 1377 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {121!} _| {that is, 1377 = the 64th root of 121!, rounded down, so insert an existing four nines solution for 121 here} ... 1399 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 212! _| {the rounded-down 128th-root of 212!; since we can make 212 with four nines, this gives us another way to make 1399 with four nines} ... 1458 = 9 x (9 x (9+9)) 1459 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 213! _| {the rounded-down 128th-root of 213!; since we can make 213 with four nines, this gives us another way to make 1459 with four nines} ... 1485 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {122!} _| {that is, 1485 = the 64th root of 122!, rounded down, so insert an existing four nines solution for 122 here} ... 1521 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 214! _| {the rounded-down 128th-root of 214!; since we can make 214 with four nines, this gives us another way to make 1521 with four nines} ... 1586 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 215! _| {the rounded-down 128th-root of 215!; since we can make 215 with four nines, this gives us another way to make 1586 with four nines} ... 1601 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {123!} _| {that is, 1601 = the 64th root of 123!, rounded down, so insert an existing four nines solution for 123 here} ... 1654 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 216! _| {the rounded-down 128th-root of 216!; since we can make 216 with four nines, this gives us another way to make 1654 with four nines} ... 1725 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 217! _| {the rounded-down 128th-root of 217!; since we can make 217 with four nines, this gives us another way to make 1725 with four nines} 1726 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {124!} _| {that is, 1726 = the 64th root of 124!, rounded down, so insert an existing four nines solution for 124 here} ... 1782 = 99 x (9+9) ... 1800 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 218! _| {the rounded-down 128th-root of 218!; since we can make 218 with four nines, this gives us another way to make 1800 with four nines} ... 1861 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {125!} _| {that is, 1861 = the 64th root of 125!, rounded down, so insert an existing four nines solution for 125 here} ... 1877 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 219! _| {the rounded-down 128th-root of 219!; since we can make 219 with four nines, this gives us another way to make 1877 with four nines} ... 1904 = |_ sqrt 10! _| {but, as we see near the top, we can make 10 with just a single 9, so we can make 1904 with a single 9} {that is, the rounded-down square root of 10! = sqrt 3628800 = 1904.940943... which rounds down to 1904} ... 1944 = 9 x (sqrt 9)! x (sqrt 9)! x (sqrt 9)! ... 1958 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 220! _| {the rounded-down 128th-root of 220!; since we can make 220 with four nines, this gives us another way to make 1958 with four nines} ... 2007 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {126!} _| {that is, 2007 = the 64th root of 126!, rounded down, so insert an existing four nines solution for 126 here} ... 2042 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 221! _| {the rounded-down 128th-root of 221!; since we can make 221 with four nines, this gives us another way to make 2042 with four nines} ... 2130 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 222! _| {the rounded-down 128th-root of 222!; since we can make 222 with four nines, this gives us another way to make 2130 with four nines} ... 2165 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {127!} _| {that is, 2165 = the 64th root of 127!, rounded down, so insert an existing four nines solution for 127 here} ... 2187 = 9 x 9 x 9 x (sqrt 9) ... 2222 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 223! _| {the rounded-down 128th-root of 223!; since we can make 223 with four nines, this gives us another way to make 2222 with four nines} ... 2318 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 224! _| {the rounded-down 128th-root of 224!; since we can make 224 with four nines, this gives us another way to make 2318 with four nines} ... 2336 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {128!} _| {that is, 2336 = the 64th root of 128!, rounded down, so insert an existing four nines solution for 128 here} ... 2368 = |_ 999-to-the-9 _| 2368 = |- 999-to-the-9 -| [ceiling] ... 2418 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 225! _| {the rounded-down 128th-root of 225!; since we can make 225 with four nines, this gives us another way to make 2418 with four nines} ... 2465 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-((sqrt 9)!))) _| / (sqrt 9) { that is, 3698 = 2-to-the-17th root of 9^(9^6)) rounded down, / 3 } ... 2520 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {129!} _| {that is, 2520 = the 64th root of 129!, rounded down, so insert an existing four nines solution for 129 here} 2521 = 2522 = 2523 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 226! _| {the rounded-down 128th-root of 226!; since we can make 226 with four nines, this gives us another way to make 2523 with four nines} ... 2632 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 227! _| {the rounded-down 128th-root of 227!; since we can make 227 with four nines, this gives us another way to make 2632 with four nines} ... 2719 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {130!} _| {that is, 2719 = the 64th root of 130!, rounded down, so insert an existing four nines solution for 130 here} ... 2746 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 228! _| {the rounded-down 128th-root of 228!; since we can make 228 with four nines, this gives us another way to make 2746 with four nines} ... 2865 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 229! _| {the rounded-down 128th-root of 229!; since we can make 229 with four nines, this gives us another way to make 2865 with four nines} ... 2935 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {131!} _| {that is, 2935 = the 64th root of 131!, rounded down, so insert an existing four nines solution for 131 here} ... 2990 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 230! _| {the rounded-down 128th-root of 230!; since we can make 230 with four nines, this gives us another way to make 2990 with four nines} ... 2997 = 999 x sqrt 9 ... 3120 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 231! _| {the rounded-down 128th-root of 231!; since we can make 230 with four nines, this gives us another way to make 3120 with four nines} ... 3167 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {132!} _| {that is, 3167 = the 64th root of 132!, rounded down, so insert an existing four nines solution for 132 here} ... 3173 = |_ ((sqrt 9)!)-to-the-9 _| - (9/9) 3174 = |_ ((sqrt 9)!)-to-the-9 _| = [1296 sqrt 6] [uses just two nines] = |_ ((sqrt 9)!)-to-the-9 _| x (9/9) 3175 = |_ ((sqrt 9)!)-to-the-9 _| + (9/9) ... 3255 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 232! _| {the rounded-down 128th-root of 232!; since we can make 232 with four nines, this gives us another way to make 3255 with four nines} ... 3397 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 233! _| {the rounded-down 128th-root of 233!; since we can make 233 with four nines, this gives us another way to make 3397 with four nines} ... 3419 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {133!} _| {that is, 3419 = the 64th root of 133!, rounded down, so insert an existing four nines solution for 133 here} ... 3545 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 234! _| {the rounded-down 128th-root of 234!; since we can make 234 with four nines, this gives us another way to make 2990 with four nines} ... 3691 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {134!} _| {that is, 3691 = the 64th root of 134!, rounded down, so insert an existing four nines solution for 134 here} ... 3698 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-((sqrt 9)!))) _| / |- sqrt sqrt 9 -| [floor divided by ceiling] { that is, 3698 = 2-to-the-17th root of 9^(9^6)) rounded down, / 2 } 3699 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 235! _| {the rounded-down 128th-root of 235!; since we can make 235 with four nines, this gives us another way to make 3699 with four nines} ... 3861 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 236! _| {the rounded-down 128th-root of 236!; since we can make 236 with four nines, this gives us another way to make 3861 with four nines} ... 3985 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {135!} _| {that is, 3985 = the 64th root of 135!, rounded down, so insert an existing four nines solution for 135 here} ... 4029 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 237! _| {the rounded-down 128th-root of 237!; since we can make 237 with four nines, this gives us another way to make 4029 with four nines} ... 4205 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 238! _| {the rounded-down 128th-root of 238!; since we can make 238 with four nines, this gives us another way to make 4205 with four nines} ... 4303 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {136!} _| {that is, 4303 = the 64th root of 136!, rounded down, so insert an existing four nines solution for 136 here} ... 4374 = 9 x 9 x 9 x (sqrt 9)! ... 4389 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 239! _| {the rounded-down 128th-root of 239!; since we can make 239 with four nines, this gives us another way to make 4389 with four nines} ... 4581 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 240! _| {the rounded-down 128th-root of 240!; since we can make 240 with four nines, this gives us another way to make 4581 with four nines} ... 4647 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {137!} _| {that is, 4647 = the 64th root of 137!, rounded down, so insert an existing four nines solution for 137 here} ... 4782 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 241! _| {the rounded-down 128th-root of 241!; since we can make 241 with four nines, this gives us another way to make 4782 with four nines} ... 4991 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 242! _| {the rounded-down 128th-root of 242!; since we can make 242 with four nines, this gives us another way to make 4991 with four nines} ... 5019 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {138!} _| {that is, 5019 = the 64th root of 138!, rounded down, so insert an existing four nines solution for 138 here} ... 5210 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt 243! _| {the rounded-down 128th-root of 243!; since we can make 243 with four nines, this gives us another way to make 5210 with four nines} ... 5421 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {139!} _| {that is, 5421 = the 64th root of 139!, rounded down, so insert an existing four nines solution for 139 here} ... 5856 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {140!} _| {that is, 5856 = the 64th root of 140!, rounded down, so insert an existing four nines solution for 140 here} ... 5994 = 999 x (sqrt 9)! ... 6317 = |_ sqrt 11! _| {but, as we see near the top, we can make 11 with just three nines, so we can make 6317 with three nines}: = |_ sqrt (9 + (9/9))! _| {with three nines} ... 6327 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {141!} _| {that is, 6327 = the 64th root of 141!, rounded down, so insert an existing four nines solution for 141 here} ... 6561 = 9 x 9 x 9 x 9 ... 6836 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {142!} _| {that is, 6836 = the 64th root of 142!, rounded down, so insert an existing four nines solution for 142 here} ... 7388 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {143!} _| {that is, 7388 = the 64th root of 143!, rounded down, so insert an existing four nines solution for 143 here} ... 7396 = |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt (9-to-the-(9-to-the-((sqrt 9)!))) _| [with just three nines] { that is, 7396 = 2-to-the-17th root of 9^(9^6)) rounded down } ... 7984 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {144!} _| {that is, 7984 = the 64th root of 144!, rounded down, so insert an existing four nines solution for 144 here} ... 8019 = 9 x 9 x 99 ... 8630 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {145!} _| {that is, 8630 = the 64th root of 145!, rounded down, so insert an existing four nines solution for 145 here} ... 8991 = 999 x 9 ... 9329 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {146!} _| {that is, 9329 = the 64th root of 146!, rounded down, so insert an existing four nines solution for 146 here} ... 9801 = 99 x 99 ... 10,085 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {147!} _| {that is, 10,085 = the 64th root of 147!, rounded down, so insert an existing four nines solution for 147 here} ... 10,904 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {148!} _| {that is, 10,904 = the 64th root of 148!, rounded down, so insert an existing four nines solution for 148 here} ... 11,791 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {149!} _| {that is, 11,791 = the 64th root of 149!, rounded down, so insert an existing four nines solution for 149 here} ... 12,751 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {150!} _| {that is, 12,751 = the 64th root of 150!, rounded down, so insert an existing four nines solution for 150 here} ... 13,791 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {151!} _| {that is, 13,791 = the 64th root of 151!, rounded down, so insert an existing four nines solution for 151 here} ... 14,918 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {152!} _| {that is, 14,918 = the 64th root of 152!, rounded down, so insert an existing four nines solution for 152 here} ... 16,138 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {153!} _| {that is, 16,138 = the 64th root of 153!, rounded down, so insert an existing four nines solution for 153 here} ... 17,459 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {154!} _| {that is, 17,459 = the 64th root of 154!, rounded down, so insert an existing four nines solution for 154 here} ... 18,890 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {155!} _| {that is, 18,890 = the 64th root of 155!, rounded down, so insert an existing four nines solution for 155 here}... ... 19,682 = |_ sqrt(9-to-the-9) _| - (9/9) 19,683 = |_ sqrt(9-to-the-9) _| = 3-to-the-9 = |_ sqrt((sqrt 9)-to-the-9)) _| which uses just two nines, so: 19,683 = |_ sqrt(9-to-the-9) _| x (9/9) 19,684 = |_ sqrt(9-to-the-9) _| + (9/9) ... 20,441 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {156!} _| {that is, 20,441 = the 64th root of 156!, rounded down, so insert an existing four nines solution for 156 here} ... 21,886 = |_ sqrt 12! _| {but, as we see near the top, we can make 12 with just two nines, so we can make 21,886 with two nines}: = |_ sqrt (9 + sqrt 9)! _| ... 22,122 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {157!} _| {that is, 22,122 = the 64th root of 157!, rounded down, so insert an existing four nines solution for 157 here} ... 23,943 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {158!} _| {that is, 23,943 = the 64th root of 158!, rounded down, so insert an existing four nines solution for 158 here} ... 25,916 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {159!} _| {that is, 25,916 = the 64th root of 159!, rounded down, so insert an existing four nines solution for 159 here} ... 28,055 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {160!} _| {that is, 28,055 = the 64th root of 160!, rounded down, so insert an existing four nines solution for 160 here} ... 30,374 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {161!} _| {that is, 30,374 = the 64th root of 161!, rounded down, so insert an existing four nines solution for 161 here} ... 30,915 = |_ sqrt sqrt 99-to-the-9 _| [uses just 3 nines] ... 32,887 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {162!} _| {that is, 32,887 = the 64th root of 162!, rounded down, so insert an existing four nines solution for 162 here} ... 35,611 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {163!} _| {that is, 35,611 = the 64th root of 163!, rounded down, so insert an existing four nines solution for 163 here} ... 38,565 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {164!} _| {that is, 38,565 = the 64th root of 164!, rounded down, so insert an existing four nines solution for 164 here} ... 41,768 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {165!} _| {that is, 41,768 = the 64th root of 165!, rounded down, so insert an existing four nines solution for 165 here} ... 45,241 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {166!} _| {that is, 45,241 = the 64th root of 166!, rounded down, so insert an existing four nines solution for 166 here} ... 49,008 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {167!} _| {that is, 49,008 = the 64th root of 167!, rounded down, so insert an existing four nines solution for 167 here} ... 53,093 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {168!} _| {that is, 53,093 = the 64th root of 168!, rounded down, so insert an existing four nines solution for 168 here} ... 57,523 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {169!} _| {that is, 57,523 = the 64th root of 169!, rounded down, so insert an existing four nines solution for 169 here} ... 62,330 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {170!} _| {that is, 62,330 = the 64th root of 170!, rounded down, so insert an existing four nines solution for 170 here} ... 67,544 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {171!} _| {that is, 62,330 = the 64th root of 171!, rounded down, so insert an existing four nines solution for 171 here} ... 73,201 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {172!} _| {that is, 73,201 = the 64th root of 172!, rounded down, so insert an existing four nines solution for 172 here} ... 78,911 = |_ sqrt 13! _| {but, as we see near the top, we can make 13 with four nines, so we can make 78,911 with four nines} ... 79,339 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {173!} _| {that is, 79,339 = the 64th root of 173!, rounded down, so insert an existing four nines solution for 173 here} ... 86,000 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {174!} _| {that is, 86,000 = the 64th root of 174!, rounded down, so insert an existing four nines solution for 174 here} {exact multiple of 1000} ... 93,227 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {175!} _| {that is, 93,227 = the 64th root of 175!, rounded down, so insert an existing four nines solution for 175 here} ... 101,072 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {176!} _| {that is, 101,072 = the 64th root of 176!, rounded down, so insert an existing four nines solution for 176 here} ... 109,586 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {177!} _| {that is, 109,586 = the 64th root of 177!, rounded down, so insert an existing four nines solution for 177 here} ... 118,828 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {178!} _| {that is, 118,828 = the 64th root of 178!, rounded down, so insert an existing four nines solution for 178 here} ... 128,860 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {179!} _| {that is, 128,860 = the 64th root of 179!, rounded down, so insert an existing four nines solution for 179 here} ... 139,752 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {180!} _| {that is, 139,752 = the 64th root of 180!, rounded down, so insert an existing four nines solution for 180 here} ... 151,577 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {181!} _| {that is, 151,577 = the 64th root of 181!, rounded down, so insert an existing four nines solution for 181 here} ... 164,417 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {182!} _| {that is, 164,417 = the 64th root of 182!, rounded down, so insert an existing four nines solution for 182 here} ... 178,360 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {183!} _| {that is, 178,360 = the 64th root of 183!, rounded down, so insert an existing four nines solution for 183 here} ... 193,502 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {184!} _| {that is, 193,502 = the 64th root of 184!, rounded down, so insert an existing four nines solution for 184 here} ... 209,948 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {185!} _| {that is, 209,948 = the 64th root of 185!, rounded down, so insert an existing four nines solution for 185 here} ... 227,810 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {186!} _| {that is, 227,810 = the 64th root of 186!, rounded down, so insert an existing four nines solution for 186 here} ... 247,212 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {187!} _| {that is, 247,212 = the 64th root of 187!, rounded down, so insert an existing four nines solution for 187 here} ... 268,289 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {188!} _| {that is, 268,289 = the 64th root of 188!, rounded down, so insert an existing four nines solution for 188 here} ... 291,188 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {189!} _| {that is, 291,188 = the 64th root of 189!, rounded down, so insert an existing four nines solution for 189 here} ... 295,259 = |_ sqrt 14! _| {but, as we see near the top, we can make 14 with four nines, so we can make 295,259 with four nines} ... 316,067 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {190!} _| {that is, 316,067 = the 64th root of 190!, rounded down, so insert an existing four nines solution for 190 here} ... 343,099 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {191!} _| {that is, 343,099 = the 64th root of 191!, rounded down, so insert an existing four nines solution for 191 here} ... 372,475 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {192!} _| {that is, 372,475 = the 64th root of 192!, rounded down, so insert an existing four nines solution for 192 here} ... 404,398 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {193!} _| {that is, 404,398 = the 64th root of 193!, rounded down, so insert an existing four nines solution for 193 here} ... 439,092 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {194!} _| {that is, 439,092 = the 64th root of 194!, rounded down, so insert an existing four nines solution for 194 here} ... 476,801 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {195!} _| {that is, 476,801 = the 64th root of 195!, rounded down, so insert an existing four nines solution for 195 here} ... 517,790 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {196!} _| {that is, 517,790 = the 64th root of 196!, rounded down, so insert an existing four nines solution for 196 here} ... 531,342 = (9-to-the-((sqrt 9)!)) - 99 ... 531,360 = (9-to-the-((sqrt 9)!)) - (9x9) ... 531,440 = (9-to-the-((sqrt 9)!)) - (9/9) 531,441 = (9-to-the-((sqrt 9)!)) x (9/9) 531,441 = (9-to-the-((sqrt 9)!)) + (9/9) ... 531,441 = (9-to-the-((sqrt 9)!)) + (9x9) ... 531,540 = (9-to-the-((sqrt 9)!)) + 99 ... 562,348 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {197!} _| {that is, 562,348 = the 64th root of 197!, rounded down, so insert an existing four nines solution for 197 here} ... 610,788 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {198!} _| {that is, 610,788 = the 64th root of 198!, rounded down, so insert an existing four nines solution for 198 here} ... 663,453 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {199!} _| {that is, 663,453 = the 64th root of 199!, rounded down, so insert an existing four nines solution for 199 here} ... 720,715 = |_ sqrt sqrt sqrt sqrt sqrt sqrt {200!} _| {that is, 720,715 = the 64th root of 200!, rounded down, so insert an existing four nines solution for 200 here} ... 970,299 = 99-to-the-(sqrt 9) - sqrt 9 970,297 = 99-to-the-(sqrt 9) - |- sqrt sqrt 9 -| [ceiling] 970,298 = 99-to-the-(sqrt 9) - |_ sqrt sqrt 9 _| 970,299 = 99-to-the-(sqrt 9) [with just 3 nines] {99 cubed} 970,300 = 99-to-the-(sqrt 9) + |_ sqrt sqrt 9 _| 970,301 = 99-to-the-(sqrt 9) + |- sqrt sqrt 9 -| [ceiling] 970,302 = 99-to-the-(sqrt 9) + sqrt 9 ... 43,046,712 = ((9-to-the-9)/9) - 9 ... 43,046,715 = ((9-to-the-9)/9) - (sqrt 9)! ... 43,046,718 = ((9-to-the-9)/9) - (sqrt 9) ... 43,046,721 = (9-to-the-9)/9 (9-to-the-8th only uses 3 9s so far) ... 43,046,724 = ((9-to-the-9)/9) + (sqrt 9) ... 43,046,727 = ((9-to-the-9)/9) + (sqrt 9)! ... 43,046,730 = ((9-to-the-9)/9) + 9 ... 387,420,488 = (9-to-the-9) - (9/9) 387,420,489 = (9-to-the-9) x (9/9) 387,420,490 = (9-to-the-9) + (9/9) 387,420,491 = 387,420,492 = (9-to-the-9) + (9/(sqrt 9)) 387,420,493 = 387,420,494 = 387,420,495 = (9-to-the-9) + 9 - (sqrt 9) 387,420,496 = 387,420,497 = 387,420,498 = (9-to-the-9) + ((sqrt 9)x(sqrt 9)) 387,420,499 = 387,420,500 = (9-to-the-9) + 9 + (sqrt 9) 387,420,501 = 387,420,502 = 387,420,503 = (9-to-the-9) + 9 + (sqrt 9)! 387,420,504 = 387,420,505 = 387,420,506 = (9-to-the-9) + 9 + 9 ... 387,420,570 = (9-to-the-9) + (9 x 9) ... 387,420,588 = (9-to-the-9) + 99 ... 774,840,978 = (9-to-the-9) + (9-to-the-9) ...## A Discussion of Some Deeper Mathematical Issues Related to the Four Nines Puzzle

There is, of course, no upper limit to the numbers which we can build with nines and the operators mentioned at the top of this web page. Consider the infinite series: 9 9! (9!)! ((9!)!)! (((9!)!)!)! ((((9!)!)!)!)! ... and that just uses one 9. Playing around with the puzzle, it soon becomes obvious that there are integers that can be represented by four nines in an infinite number of different ways. It is NOT obvious whether ALL numbers can be represented in at least one way. We return to this later.Note: the floor function, symbolized as floor(x) = |_ x _| and ceiling function , symbolized as ceiling(x) = |- x -| work on numbers which are not integers, by rounding up or rounding down. That is: |_ 0.5 _| = 0 |- 0.5 -| = 1 |_ 1.2345 _| = 1 |- 1.2345 -| = 2 |_ pi _| = |_ 3.141592653689793... _| = 3 |- pi- | = |_ 3.141592653689793... _| = 4 These last examples suggest the "four pi's" problem. 0 = (pi x pi) - (pi x pi) 1 = (pi x pi) / (pi x pi) 2 = (pi / pi) + (pi / pi) 3 = (pi + pi + pi)/pi 4 = |- pi -| {using just one pi} 5 = (|_ pi _| + |_ pi _|) - (pi/pi) 6 = (|_ pi _| + |_ pi _|) x (pi/pi) 7 = (|_ pi _| + |_ pi _|) + (pi/pi) 8 = (|_ pi _| x |_ pi _|) - (pi/pi) 9 = |_ pi _| x |_ pi _| x (pi/pi) 10 = (|_ pi _| x |_ pi _|) + (pi/pi) 11 = (|_ pi _| x |- pi -|) - (pi/pi) 12 = (|_ pi _| x |- pi -|) x (pi/pi) 13 = (|_ pi _| x |- pi -|) + (pi/pi) 14 = |_ pi _| + |- pi -| + |_ pi _| + |- pi -| 15 = |_ pi _| + |- pi -| + |- pi -| + |- pi -| 16 = |- pi -| + |- pi -| + |- pi -| + |- pi -| etc.... Note that we don't actually need the ceiling(x) = |- x -| because the same thing can be done with the floor function as follows: ceiling(x) = |- x -| = the least integer greater than or equal to x = - floor(-x) = - |_ -x _| By more advanced mathematics, it might be shown that every integer can be represented by a sufficiently long sequence of the operators we use here.## Definition of Floor Function and Ceiling Function

In "The Weekly Dispatch" of 4 February 1900, the puzzle column by Dudeney introduced this problem. But Professor Donald Knuth comments on Dudeney's Solution Number 310, which gives a table. Knuth criticizes: "he disallows (sqrt 9)! for completely illogical reasons; also, he fails to express 38, 41, 43, ... with fewer than five 9s." Knuth on Dudeney## Dudeney Invented the Four Nines; Knuth Criticizes Dudeney

## Let us note that there are some unsolved mathematical questions about the Factorial Function N! The well-known definition is: 1! = 1 2! = 1 x 2 = 2 3! = 1 x 2 x 3 = 6 4! = 1 x 2 x 3 x 4 = 24 5! = 1 x 2 x 3 x 4 x 5 = 120 6! = 1 x 2 x 3 x 4 x 5 x 6 = 720 7! = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40,320 9! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362,880 10! =1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3,628,800 ... and so on

Definition of Factorial## Believe it or not, your telephone number, your Social Security Number, or any other whole number important to you can be found as the beginning digits of some enormous factorial. John E. Maxfield proved this theorem: "If A is any positive integer having M digits, there exists a positive integer N such that the first M digits of N! constitute the integer A." ["A Note on N!", John E. Maxfield. Kansas State University, Mathematics Magazine, March-April 1970, pp.64-67; supported in part by a National Science Foundation grant].

Any Integer is the First Digits of Some Factorial## For non-integers, and for complex numbers, the universally accepted generalization of the Factorial Function is the Gamma Function. Discussion of that shall be reserved until I provide my analysis of the Four Nines Puzzle as applied to Complex Numbers, such as first appear with sqrt (-9) = 3

Factorials of Negative, Fractional, and Complex Numbersi.## In our solutions to the Four Nines Puzzle, we often add to or subtract something from a factorial, and then take a square root. Typically, this is not a whole number, so we round down, or round up (floor or ceiling). This leads to the question: When is N! + 1 a perfect square? Or, equivalently, When is sqrt (N! + 1) a whole number? The only solutions known are: 25 = 4! + 1 = 5 x 5 121 = 5! + 1 = 11 x 11 5041 = 7! + 1 = 71 x 71 Nobody knows if there are any other perfect squares one more than factorials. Clearly related is a paper by Berend and Osgood: [Journal of Number Theory, vol. 42, 1992] which proves that for any polynomial P of degree > 1 the set of positive integers N for which P(X) = N! has an integral solution X, is of zero density. This paper explicitly says that it is not known if the equation X^2 - 1 = N! has only finitely many solutions. We also don't know if there are an infinite number of primes of the form n! + 1. We also don't know if there are an infinite number of primes of the form n! - 1. The largest such prime that we know is 3610! - 1. It has 11,277 digits [Caldwell, title to be added here, 1993] By the way, there is a fairly elementary proof that, except for 0!=1 and 1!=1, NO factorials are perfect squares. That fairly elementary proof, though, uses a heavy mathematical result known as Bertrand's Postulate, also known as Chebychev's Theorem (after the man who first proved it). This Theorem says that there always exists at least one prime between N and 2N, if N>2. Erdos gave a genuinely elementary (although neither short nor obvious) proof. This will all be inserted here or linked to in a later version of this web page.

Square-less-one Factorials## The Four Nines Puzzle itself is, to be sure, very elementary stuff. So I stand meekly in the shadow of the great mathematician's who were my Teachers' teachers' teachers... Gottfried Leibnitz, Jacob Bernoulli, Johann Bernoulli, Leonhard Euler, Joseph Louis Lagrange, Simeon Poisson, Pafnuty Lvovich Chebyshev, Andrei A. Markov, G. H. Hardy, Alonzo Church, David Hilbert, Norbert Wiener, Alan Turing... With them in mind, I remark that Donald Knuth conjectures that

Ergodic HypothesisALLintegers can be made with a sufficiently lengthy combination of square roots and factorials and floors and ceilings ... built around a single 4. I make the related conjecture, based on the number 9. As we see near the top of this web page, we can make a 4 from a single 9, with a lot of square roots, factorials, and floor functions. Hence Knuth's conjecture for 4 immediately applies to 9. In summary of a subtle proof of Knuth's Conjecture, still in progress, factorials make a number bigger, and square roots make it smaller. Iterating sufficiently, we are "folding" the algebraic number line back onto itself recursively, and this is an ergodic property, which carries a number arbitrarily close to any given integer, at which point a final floor or ceiling gets us exactly to that given integer.## The problem becomes (as I shall show in a forthcoming paper co-authored by Andrew Carmichael Post and Dr. George Hockney) how to achieve as many integers as possible with 4 nines -- for a given degree of "complexity" as defined by the number of symbols of a standardized way of expressing the combination of 9, 99, 999, 9999, +, -, x, /, sqrt, factorial, floor, and ceiling (say in Backus-Naur Form). For example: 9999 = 9999 (complexity = 4) 1 = 99/99 (complexity = 5) 1008 = 999 + 9 (complexity = 5) 9801 = 99 x 99 (complexity = 5) 2 = 99/9 - 9 (complexity = 6) 20 = 99/9 + 9 (complexity = 6) 19 = 9 + 9 + 9/9 (complexity 7) 36 = 9 + 9 + 9 + 9 (complexity 7) 13 = 9 + sqrt 9 + 9/9 (complexity 8) 40 = |_ sqrt 999 _| + 9 (complexity 8) and so on. The complexity function creates an order on the solutions to the four nines problem. Of interest are such functions as the smallest number whose complexity exceeds a given value, and upper and lower bounds on the ratio of a number to its complexity. Almost all numbers have very high complexity. But details will be revealed in that forthcoming paper. The problem of whether two strings of characters evaluate to the same integer is a very hard problem, in terms of the amount of computation necessary to determine it in general, called the "word problem" in complexity theory.

Complexity Ordering of SolutionsThe "Four Fours Problem" first appeared in: "Mathematical Recreations and Essays", by W. W. Rouse Ball [1892]. In this book the "Four Fours Problem" is called a "traditional recreation." There are several fine sites on the World Wide Web for "Four Fours Problem." But I recommend to the reader: "Mathematical Games", by Martin Gardner, [Scientific America, Jan 1964]. The [Feb 1964] issue has answers to the puzzles posed in January. Martin Gardner was extending the "two fours" problem as first posed by J. A. Tierney in 1944, and extended by others in 1964. ["E631", J. A. Tierney, Amer. Math Monthly, 52(1945)219]. ["64 Ways to Write 64 Using four 4's", M. Bicknell and V. E. Hoggatt, Recreational Mathematics Magazine, 14(1964)13-25]. More recently, we have Knuth's Conjecture: "Representing Numbers Using Only One 4", Donald Knuth, [Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp.308-310]. Knuth shows how (using a computer program he wrote) all integers from 1 through 207 may be represented with only one 4, varying numbers of square roots, varying numbers of factorials, and the floor function. For example: Knuth shows how to make the number 64 using only one 4: |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt |_ sqrt |_ sqrt sqrt sqrt sqrt sqrt (4!)! _| ! _| ! _| ! _| ! _| ! _| ! _| ! _| As to notation in the above example, he means sqrt n! stands for sqrt (n!), not (sqrt n)! Knuth further points out that |_ sqrt |_ X _| _| = |_ sqrt X _| so that the floor function's brackets are only needed around the entire result and before factorials are taken. He CONJECTURES that all integers may be represented that way: "It seems plausible that all positive integers possess such a representation, but this fact (if true) seems to be tied up with very deep propertis of the integers." Your Humble Webmaster believes that Knuth is right, for 9 as well as 4, and will prove that in a forthcoming paper. Knuth comments: "The referee has suggested a stronger conjecture, that a representation may be found in which all factorial operations precede all square root operations; and, moreover, if the greatest integer function [our floor function] is not used, an arbitrary positive real number can probably be approximated as closely as desired in this manner." If we abbreviate the long strings of symbols, we can express this more elegantly. Let NK be the number (...((4!)!)!...)! with K factorial operations. Then the anonymous referee's stronger conjecdture is equivalent to: log NK/(2^|_log(log NK)_|, for K=1,2,3..., are dense in the interval (1,2). Notationally here log means log to the base 2, and ^ means exponentiation. One key to the proof I shall publish is that, for any logarithm base: "The fractional part of log N is dense on the unit interval." From that, it has been proved that: "The fractional part of log N! is dense on [0,1]." ["A Note on N!", John E. Maxfield. Kansas State University, Mathematics Magazine, March-April 1970, pp.64-67; supported in part by a National Science Foundation grant]. Maxfield also proves, in the notation above, if we also define logK is the Kth iterant of Log N, i.e. log2(N) = log log N, that: The fractional part of logK (NK) is dense on the unit interval.## The "Four Nines Problem" is closely related to the "Four Fours Problem"

## In our solutions of the Four Nines Puzzle, we sometimes take a square root of the sum of two slightly different square roots. As noted in Mathpages Mathpages #305: How can we find integer solutions of M = sqrt ( sqrt (N) + sqrt ((KxN)+1)) "This can be viewed as a Pell Equation with an extra solution on the solution." We have: M^2 = sqrt (N) + sqrt ((KxN)+1) in integers, so we know that there exist integers X and Y such that: N = Y^2 KN + 1 = X^2 X + Y = M^2 Eliminating N from the first two equations above gives the Pell equation: X^2 + KY^2 = 1 For any given K we are looking for solutions X,Y such that X+Y is a square. "Of course, for any positive integer K there are infinitely many solutions to the Pell Equation, but solutions with X+Y=square are rare. For example, with K=8, the values of X+Y satisfying the Pell Equation" are: ((16 + 11 sqrt 2)/8) (3 + sqrt 8)^Q + ((16 - 11 sqrt 2)/8) (3 - sqrt 8)^Q This gives the sequence 4, 23, 134, 781, 4552 ..., which satisfies the second-order recurrence: S(j) = 6S(j-1) - S(j-2) "So the question is whether this sequence contains any squares after the initial vale 4. Recall the proof that the only square Fibonacci numbers are 0, 1, and 144." [J. Cohn, "On the Square Fibonacci Numbers", J. London Math Soc., 39 (1964) 537-540] "In general the problem reduces to finding square terms of a general second-order recurring sequence, like the Fibonacci sequence. The best approach might be to apply Cohn's method of proof to the general second-order recurrence."

Square Root of a Sum of Square Roots## In our solutions to the "Four Nines Problem" we often take square roots of factorials, and sometimes square roots of square roots of factorial of factorials, and so forth. As it turns out, there are some solved and some unsolved problems in the Physics of Quantum Wave Functions which involve square roots of factorials. As B. Nagel has written, "Mathematical problems in Quantum Optics", {ref to be done}: "As a continuation of earlier studies of squeezed states and other special harmonic oscillator states of interest in quantum optics I have studied the phase representations of these states. Although the phase observable, which is roughly speaking conjugate to the number operator, does not exist as a hermitian operator -- this is a longstanding and still popular problem, initiated by Dirac in 1927 -- it exists as a so-called general observable and permits a probability interpretation via a phase distribution on the unit circle. The corresponding wave function is obtained simply by substituting the harmonic wave exp(in[[phi]]) for the number state |n>.

Square Roots of Factorials in Quantum MechanicsThe series expansions thus obtained for the coherent and squeezed states contain a square root of a factorial n!, which makes it impossible to get closed analytical expressions. Approximate analytical expressions have been derived, valid for large values of the mean value of the number operator...." For another example, see the computer program given in: GrozinHydrogen Wave Functions and E1 TransitionsProcedure R(n,1); % radial wave function -2/n^2*sqrt(factorial(n+1)/factorial(n-1-1))*exp(-r/n) ... etc. Wouldn't it be interesting if an in-depth analysis of the century-old recreation math puzzle about Four Nines turned out to be useful in solving a problem in 21st Century Quantum Optics with Squeezed States? {more discussion to be added in February 2004} Special Thanks to Dr. George Hockney, NASA/JPL, for informal discussion and review in January-February 2004. Thanks to Forrest Bishop for informal discussion and review in January-February 2004.

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