Here today's iteration of my attempt at a daily puzzle.
FYI, n! (n factorial) is calculated as follows: n! = (1)x(2)x...x(n-1)x(n)
Hint, you don't need to calculate 10! And there are 3 valid answers as far as I know.
Previous iteration: #696880
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10! = 10×9×8×7×6×5×4×3×2×1
= 5×2×3×3×2×2×2×7×3×2×5×2×2×3×2×1
= 7×5×5×3×3×3×3×2×2×2×2×2×2×2×2×1
So either m or n must be >= 7
= 7!×5×3×3×2×2×2×2×1
= 7! ×6×5×4×3×2×1
= 7!×6!
m=7, n=6
m×n=42
edit: how to not spoil the puzzle for others?
Nice! It’s always 42!
And yes, 6 and 7 are one of the 3 possible answer sets. Good job!
10, 1
10, 0
Yes, this completes the answer sheet. 0! = 1, hence the third solution.
OMG! You beat me by less than a minute.
I suggested a feature for this yesterday: https://github.com/stackernews/stacker.news/issues/1422
10 !
10 is an answer, 10! isn't.
No no, there'e a space. I was indicating genuine excitement that I got the answer. Is it correct?
Yes. 1! x 10! = 10!, so 1x10 = 10. Good job.
The last answer requires some additional knowledge of factorials...
I dont' know how I did it, but I did, and I win, right ?!
Disclaimer, these are not my puzzles. But linking to my sources would take the fun out of them.
You got them from a 5th grade algebra exam I bet
Considering a generalized version of the puzzle, "If m! x n! = X! then m x n = ?"
(X, 1) and (X, 0) will always be valid "degenerate" solutions (producing answers "X" and "0", respectively). Let's ignore them onward.
Interestingly, for X=10 we have the pair (7, 6) which works only due to the fact that 6! = 8 x 9 x 10. In other words, 6! can be restructured to form the "tail" of a different factorial.
My intuition is that it happens because of 6! containing relatively many prime factors, allowing them to be rearranged to form the "tail".
This made me wonder how often that happens in general (for different X), with the intuition being, it's extremely rare.
So I had ChatGPT write me a program to check this as far as possible (read: until I think I understood the pattern and got fed up waiting for 8! to pop up):
6! = 5! x 3! 10! = 7! x 6! 24! = 23! x 4! 120! = 119! x 5! 720! = 719! x 6! 5040! = 5039! x 7!It seems that "n" always has to be small, which makes sense because we need to tightly control the prime factors involved in the completion of the tail.
But that also means that, from a certain point, the only solutions seem to be of form m = X - 1, n! = X. (In fact, X=10 is the only example that breaks this rule, making it, in my eyes, the only "interesting" solution.)
It's not a rigorous proof, of course, that this is the only asymptotic solution form, but at least a strong heuristic. Also it helped build a wandwavy proof that there are infintely many X for which a non-degenerate solution exist.
@remindme in 12 hours
This was a lot of fun. Basically I expanded each factorial into their prime factorization, and determined that either
mornwould need to be larger than 6 so as to include 7 in the factorization. Then just tried to find the other correct matching pair.10,1was an obvious solution.10,0was not obvious at first but now is :)I actually encountered this before.
there are three answers.
10, 18, and twenty something.
I have to do some number punching to figure the last one out.
Can you elaborate on how you get to 18?
No way, I wasnt able to type because the power went down at my plant!
I am going off of memory here for 10 and 18.
Let me solve write them down.