# The minimum number that cannot be summed by $11$ or fewer factorials.

What is the smallest positive integer one can find impossible to create by $11$ or less factorials?

I only know how to limit the possibilities, but not how to actually solve this. I'm assuming that this is a simple trick in a logic question, but I can't seem to see how to start, nor figure out what type of question this is. Any ideas?

Thanks!

• $1$? ${}{}{}{}{}{}$ – user384138 Jan 15 '17 at 0:51
• I'm sorry, I edited the question – user406996 Jan 15 '17 at 0:56

## 2 Answers

$<2!$ you need one factorial, $1!=1$

$<3!$ you might need another $2 \times 2!$

$<4!$ you might need another $3 \times 3!$

$<5!$ you might need another $4 \times 4!$

$<2\times 5!$ you might need another $1 \times 5!$ - we could need $11$ factorials at this point

At $3\times 5!-1=359$, then, you should need $12$ factorial to sum to this number.

(see also factorial number system)

• So the strategy is to try numbers 1 less than a multiple of n factorial? – user406996 Jan 15 '17 at 1:06
• @user406996 Such numbers require a lot of factorials to sum to, for sure. – Joffan Jan 15 '17 at 1:07
• Ok, thanks! I understand how you did it now. – user406996 Jan 15 '17 at 1:07

1 cannot be summed as 11 factorials.

Does your question mean "no more than 11 factorials"?

(after your edit)

it looks like task of dynamical programming - calculate for an n the minimal number of the factorials in the sum.

• Yes, thanks for catching that! :) – user406996 Jan 15 '17 at 0:55
• Yes, but is there a way to this with just pencil and paper? – user406996 Jan 15 '17 at 1:03
• Try to precalculate answer(it is 359 ^_^) and create a proof after – kotomord Jan 15 '17 at 1:12
• You can build up to the number knowing that 1 less than factorials is hard to create as a sum of factorials :) Thanks for pointing out the Dynamical Programming approach! – user406996 Jan 15 '17 at 1:14