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

Question

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!

Answer 1

$<2!$ 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)

Answer 2

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.