Smallest Number with Five 9's

Logic Level 5

Find the smallest positive integer which requires at least five 9's to be written in an expression which is equal to said number.

Allowed operations:

  • + , − , × , ÷ , ⋅ +, -, \times, \div, \sqrt{ \cdot }
  • Brackets
  • Factorial
  • Power / Exponentiation
  • Concatenation of 9's in the expression

Disallowed operators. Everything else, including but not limited to:

  • Decimal points (since you need a concatenate a 0)
  • Other mathematical operators like cube roots

An example of how to use these operators is using 4's to create 28 would be 28 = 4 × ( 4 + 4 + ( 4 ÷ 4 ) ) 28=4 \times (4+\sqrt{4}+(4 \div 4)) , which uses five 4's, but you can also do this with two 4's as follows: 28 = 4 ! + 4 28=4!+4 , and this is the shortest method. Therefore, 28 can be written using at least two 4's.


The answer is 94.

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Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Sharky Kesa
Apr 6, 2017

Once again, I have no better method of solving this apart from going through all the numbers:

1 = 9 9 2 = ( 9 ) ! 9 3 = 9 4 = 9 + 9 9 5 = 9 + ( 9 ) ! 9 6 = ( 9 ) ! 7 = 9 − ( 9 ) ! 9 8 = 9 − 9 9 9 = 9 10 = 9 + 9 9 11 = 99 9 12 = 9 + 9 13 = 9 + 9 + 9 9 14 = 9 + 9 + ( 9 ) ! 9 15 = 9 + ( 9 ) ! 16 = 9 + 9 − ( 9 ) ! 9 17 = 9 + 9 − 9 9 18 = 9 + 9 19 = 9 + 9 + 9 9 20 = ( ( 9 ) ! ) ! ( 9 ) ! ( 9 ) ! 21 = 9 + 9 + 9 22 = ( ( ( 9 ) ! ) ! + ( 9 ) ! ) × ( 9 ) ! 9 23 = ( ( 9 ) ! ) ! ( 9 ) ! ( 9 ) ! + 9 24 = 9 × 9 − 9 25 = 9 × 9 − ( 9 ) ! 9 26 = 9 × 9 − 9 9 27 = 9 × 9 28 = 9 × 9 + 9 9 29 = 9 × 9 + ( 9 ) ! 9 30 = 9 × 9 + 9 31 = 99 − ( 9 ) ! 9 32 = ( 9 ) ! 9 ( ( 9 ) ! ) ! 9 33 = 99 9 34 = ( 9 ) ! × ( 9 ) ! − ( 9 ) ! 9 35 = ( 9 ) ! × ( 9 ) ! − 9 9 36 = ( 9 ) ! × ( 9 ) ! 37 = ( 9 ) ! × ( 9 ) ! + 9 9 38 = ( 9 ) ! × 9 ) ! + ( 9 ) ! 9 39 = ( 9 ) ! × ( 9 ) ! + 9 40 = ( ( 9 ) ! ) ! 9 ( 9 ) ! 41 = ( ( 9 ) ! ) ! ( 9 ) ! + 9 9 42 = ( 9 ) ! × ( 9 ) ! + ( 9 ) ! 43 = ( ( 9 ) ! ) ! 9 ( 9 ) ! + 9 44 = ( ( 9 ) ! ) ! 9 − ( 9 ) ! × ( 9 ) ! 45 = ( 9 ) ! × 9 − 9 46 = ( ( 9 ) ! ) ! 9 ( 9 ) ! + ( 9 ) ! 47 = 9 ! ( ( 9 ) ! ) ! × 9 − 9 48 = 9 × ( 9 ) ! − ( 9 ) ! 49 = ( ( 9 ) ! ) ! 9 ( 9 ) ! + 9 50 = 9 ! ( ( 9 ) ! ) ! × 9 ) − ( 9 ) ! 51 = 9 × ( 9 ) ! − 9 52 = 9 × ( 9 ) ! − ( 9 ) ! 9 53 = 9 × ( 9 ) ! − 9 9 54 = 9 × ( 9 ) ! 55 = 9 × ( 9 ) ! + 9 9 56 = 9 ! ( ( 9 ) ! ) ! × 9 57 = 9 × ( 9 ) ! + 9 58 = ( ( 9 ) ! 9 ) ( 9 ) ! − ( 9 ) ! 59 = 9 ! ( ( 9 ) ! ) ! × 9 + 9 60 = 9 × ( 9 ) ! + ( 9 ) ! 61 = ( ( 9 ) ! 9 ) ( 9 ) ! − 9 62 = ( ( 9 ) ! ) ! 9 − 9 − 9 63 = 9 × ( 9 ) ! + 9 64 = ( ( 9 ) ! 9 ) ( 9 ) ! 65 = ( ( 9 ) ! ) ! 9 − 9 − ( 9 ) ! 66 = ( ( ( 9 ) ! ) ! + ( 9 ) ! ) × ( 9 ) ! 67 = ( ( 9 ) ! 9 ) ( 9 ) ! + 9 68 = ( ( 9 ) ! ) ! 9 − 9 − 9 69 = 9 × 9 − 9 − 9 70 = ( ( 9 ) ! 9 ) ( 9 ) ! + ( 9 ) ! 71 = ( ( 9 ) ! ) ! 9 − 9 72 = 9 × 9 − 9 73 = ( ( 9 ) ! ) ! − 9 9 − ( 9 ) ! 74 = ( ( 9 ) ! ) ! 9 − ( 9 ) ! 75 = 9 × 9 − ( 9 ) ! 76 = ( ( 9 ) ! ) ! − 9 9 − 9 77 = ( ( 9 ) ! ) ! 9 − 9 78 = 9 × 9 − 9 79 = ( ( 9 ) ! ) ! − 9 9 80 = ( ( 9 ) ! ) ! 9 81 = 9 × 9 82 = 9 × 9 + 9 9 83 = ( ( 9 ) ! ) ! 9 + 9 84 = 9 × 9 + 9 85 = ( ( 9 ) ! ) ! − 9 9 + ( 9 ) ! 86 = ( ( 9 ) ! ) ! 9 + ( 9 ) ! 87 = 9 × 9 + ( 9 ) ! 88 = ( ( 9 ) ! ) ! − 9 9 + 9 89 = ( ( 9 ) ! ) ! 9 + 9 90 = 99 − 9 91 = ( ( 9 ) ! ) ! + 99 9 92 = ( ( 9 ) ! ) ! 9 + 9 + 9 93 = 99 − ( 9 ) ! 94 = 99 − 9 − ( 9 ) ! 9 \begin{aligned} 1&=\dfrac{9}{9}\\ 2&=\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 3&=\sqrt{9}\\ 4&=\sqrt{9}+\dfrac{9}{9}\\ 5&=\sqrt{9}+\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 6&=(\sqrt{9})!\\ 7&=9-\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 8&=9-\dfrac{9}{9}\\ 9&=9\\ 10&=9+\dfrac{9}{9}\\ 11&=\dfrac{99}{9}\\ 12&=9+\sqrt{9}\\ 13&=9+\sqrt{9}+\dfrac{9}{9}\\ 14&=9+\sqrt{9}+\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 15&=9+(\sqrt{9})!\\ 16&=9+9-\frac{(\sqrt{9})!}{\sqrt{9}}\\ 17&=9+9-\dfrac{9}{9}\\ 18&=9+9\\ 19&=9+9+\dfrac{9}{9}\\ 20&=\dfrac{\frac{((\sqrt{9})!)!}{(\sqrt{9})!}}{(\sqrt{9})!}\\ 21&=9+9+\sqrt{9}\\ 22&=\sqrt{\dfrac{(((\sqrt{9})!)!+(\sqrt{9})!)\times (\sqrt{9})!}{9}}\\ 23&=\dfrac{\frac{((\sqrt{9})!)!}{(\sqrt{9})!}}{(\sqrt{9})!}+\sqrt{9}\\ 24&=9\times \sqrt{9}-\sqrt{9}\\ 25&=9\times \sqrt{9}-\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 26&=9\times \sqrt{9}-\dfrac{9}{9}\\ 27&=9\times \sqrt{9}\\ 28&=9\times \sqrt{9}+\dfrac{9}{9}\\ 29&=9\times \sqrt{9}+\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 30&=9\times \sqrt{9}+\sqrt{9}\\ 31&=\dfrac{99-(\sqrt{9})!}{\sqrt{9}}\\ 32&=\sqrt{\sqrt{\sqrt{\sqrt{\dfrac{(\sqrt{9})!}{\sqrt{9}}}}}}^{\dfrac{((\sqrt{9})!)!}{9}}\\ 33&=\dfrac{99}{\sqrt{9}}\\ 34&=(\sqrt{9})!\times (\sqrt{9})!-\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 35&=(\sqrt{9})!\times (\sqrt{9})!-\dfrac{9}{9}\\ 36&=(\sqrt{9})!\times (\sqrt{9})!\\ 37&=(\sqrt{9})!\times(\sqrt{9})!+\dfrac{9}{9}\\ 38&=(\sqrt{9})!\times \sqrt{9})!+\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 39&=(\sqrt{9})!\times(\sqrt{9})!+\sqrt{9}\\ 40&=\dfrac{\frac{((\sqrt{9})!)!}{\sqrt{9}}}{(\sqrt{9})!}\\ 41&=\dfrac{\frac{((\sqrt{9})!)!}{(\sqrt{9})!}+\sqrt{9}}{\sqrt{9}}\\ 42&=(\sqrt{9})!\times (\sqrt{9})!+(\sqrt{9})!\\ 43&=\dfrac{\frac{((\sqrt{9})!)!}{\sqrt{9}}}{(\sqrt{9})!}+\sqrt{9}\\ 44&=\dfrac{((\sqrt{9})!)!}{9}-(\sqrt{9})!\times (\sqrt{9})!\\ 45&=(\sqrt{9})!\times 9-9\\ 46&=\dfrac{\frac{((\sqrt{9})!)!}{\sqrt{9}}}{(\sqrt{9})!}+(\sqrt{9})!\\ 47&=\dfrac{9!}{((\sqrt{9})!)!\times 9}-9\\ 48&=9 \times (\sqrt{9})! - (\sqrt{9})!\\ 49&=\dfrac{\frac{((\sqrt{9})!)!}{\sqrt{9}}}{(\sqrt{9})!}+9\\ 50&=\dfrac{9!}{((\sqrt{9})!)!\times 9)}-(\sqrt{9})!\\ 51&=9\times (\sqrt{9})!-\sqrt{9}\\ 52&=9\times (\sqrt{9})!-\dfrac{(\sqrt{9})!}{\sqrt{9}}\\ 53&=9\times (\sqrt{9})!-\dfrac{9}{9}\\ 54&=9\times (\sqrt{9})!\\ 55&=9\times (\sqrt{9})!+\dfrac{9}{9}\\ 56&=\dfrac{9!}{((\sqrt{9})!)!\times 9}\\ 57&=9\times (\sqrt{9})!+\sqrt{9}\\ 58&=(\dfrac{(\sqrt{9})!}{\sqrt{9}})^{(\sqrt{9})!} - (\sqrt{9})!\\ 59&=\dfrac{9!}{((\sqrt{9})!)!\times 9}+\sqrt{9}\\ 60&=9\times (\sqrt{9})!+(\sqrt{9})!\\ 61&=(\dfrac{(\sqrt{9})!}{\sqrt{9}})^{(\sqrt{9})!} - \sqrt{9}\\ 62&=\dfrac{((\sqrt{9})!)!}{9}-9-9\\ 63&=9\times (\sqrt{9})!+9\\ 64&=(\dfrac{(\sqrt{9})!}{\sqrt{9}})^{(\sqrt{9})!}\\ 65&=\dfrac{((\sqrt{9})!)!}{9} - 9 - (\sqrt{9})!\\ 66&=\sqrt{(((\sqrt{9})!)!+(\sqrt{9})!)\times (\sqrt{9})!}\\ 67&=(\dfrac{(\sqrt{9})!}{\sqrt{9}})^{(\sqrt{9})!}+\sqrt{9}\\ 68&=\dfrac{((\sqrt{9})!)!}{9}-\sqrt{9}-9\\ 69&=9\times 9 - 9 - \sqrt{9}\\ 70&=(\dfrac{(\sqrt{9})!}{\sqrt{9}})^{(\sqrt{9})!}+(\sqrt{9})!\\ 71&=\dfrac{((\sqrt{9})!)!}{9}-9\\ 72&=9\times 9 - 9\\ 73&=\dfrac{((\sqrt{9})!)!-9}{9} - (\sqrt{9})!\\ 74&=\dfrac{((\sqrt{9})!)!}{9}-(\sqrt{9})!\\ 75&=9\times 9 - (\sqrt{9})!\\ 76&=\dfrac{((\sqrt{9})!)!-9}{9} -\sqrt{9}\\ 77&=\dfrac{((\sqrt{9})!)!}{9}-\sqrt{9}\\ 78&=9\times 9-\sqrt{9}\\ 79&=\dfrac{((\sqrt{9})!)!-9}{9}\\ 80&=\dfrac{((\sqrt{9})!)!}{9}\\ 81&=9\times 9\\ 82&=9\times 9+\dfrac{9}{9}\\ 83&=\dfrac{((\sqrt{9})!)!}{9}+\sqrt{9}\\ 84&=9\times 9+\sqrt{9}\\ 85&=\dfrac{((\sqrt{9})!)!-9}{9}+(\sqrt{9})!\\ 86&=\dfrac{((\sqrt{9})!)!}{9}+(\sqrt{9})!\\ 87&=9\times 9+(\sqrt{9})!\\ 88&=\dfrac{((\sqrt{9})!)!-9}{9}+9\\ 89&=\dfrac{((\sqrt{9})!)!}{9}+9\\ 90&=99-9\\ 91&=\dfrac{((\sqrt{9})!)!+99}{9}\\ 92&=\dfrac{((\sqrt{9})!)!}{9}+9+\sqrt{9}\\ 93&=99-(\sqrt{9})!\\ 94&=99-\sqrt{9}-\dfrac{(\sqrt{9})!}{\sqrt{9}} \end{aligned}

For 94, I have not found a better solution (and I doubt there is), and I found that for all numbers under 100, all numbers but 94 require at most four 9's.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

That's an impressive list!

Calvin Lin Staff - 4 years, 2 months ago

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Yeah, it is quite large (and took me two sessions to do in latex), but I had the solutions written up. I have an iOS app called Tchisla which has similar exercises, and so far, no one has come up with a better solution for 94.

Sharky Kesa - 4 years, 2 months ago

@Sharky Kesa By the way, you have a typo in your equation for 39. It uses a 3.

Geoff Pilling - 4 years, 2 months ago

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Thanks. Thought I had already fixed it.

Sharky Kesa - 4 years, 2 months ago

@Sharky Kesa - Man that was quite the challenge!! ,Here are some that i did differently,

22 = ( 9 ) ! ( 9 + ( 9 ) ! 9 ) 23 = ( ( 9 ) ! ) 9 − 9 9 32 = 99 − 9 9 38 = 1 9 ( ( ( 9 ) ! ) ! ( 9 ) ! − ( 9 ) ! ) 57 = 1 9 ( 9 ! ( ( 9 ) ! ) ! + 9 ) 63 = 99 − ( ( 9 ) ! ( 9 ) ! ) 66 = 99 9 ( 9 ) ! 76 = ( ( 9 ) ! ) ! − ( 9 ) ! ( 9 ) ! 9 22=(\sqrt{9})!\left(\sqrt{9}+\dfrac{(\sqrt{9})!}{9}\right)\\ 23=\dfrac{((\sqrt{9})!)^{\sqrt{9}}-9}{9}\\ 32=\dfrac{99-\sqrt{9}}{\sqrt{9}}\\ 38=\dfrac{1}{\sqrt{9}}\left(\dfrac{((\sqrt{9})!)!}{(\sqrt{9})!}-(\sqrt{9})!\right)\\ 57=\dfrac{1}{9}\left(\dfrac{9!}{((\sqrt{9})!)!}+9\right)\\ 63=99-((\sqrt{9})!(\sqrt{9})!)\\ 66=\dfrac{99}{9}(\sqrt{9})!\\ 76=\dfrac{((\sqrt{9})!)!-(\sqrt{9})!(\sqrt{9})!}{9}

Also i think you have to correct 89 89 :)

Anirudh Sreekumar - 4 years, 1 month ago

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I assume in 38 and 57, you don't actually have the 1 there, but should be the expression in the following brackets? Also, great job! You'll find that my solutions give the lowest number of 9's required to create the number.

Sharky Kesa - 4 years, 1 month ago

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