 | THE FOUR NINES PUZZLE |
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"+","-","x","/"?
This old puzzle can be updated if we allow the use of exponentiation, radicals (especially the square root "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