[Daily puzzle] If m! x n! = 10! then m x n = ? \ stacker news ~charts_and

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[Daily puzzle] If m! x n! = 10! then m x n = ?

905 sats \ 22 comments \ @south_korea_ln 24 Sep charts_and_numbers

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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542 sats \ 6 replies \ @random_ 24 Sep

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?

21 sats \ 0 replies \ @Undisciplined 24 Sep

OMG! You beat me by less than a minute.

100 sats \ 0 replies \ @OneOneSeven 24 Sep

Nice! It’s always 42!

0 sats \ 2 replies \ @south_korea_ln OP 24 Sep

And yes, 6 and 7 are one of the 3 possible answer sets. Good job!

221 sats \ 1 reply \ @random_ 24 Sep

10, 1 10, 0

0 sats \ 0 replies \ @south_korea_ln OP 24 Sep

Yes, this completes the answer sheet. 0! = 1, hence the third solution.

0 sats \ 0 replies \ @south_korea_ln OP 24 Sep

edit: how to not spoil the puzzle for others?

I suggested a feature for this yesterday: https://github.com/stackernews/stacker.news/issues/1422

200 sats \ 4 replies \ @clarity 24 Sep

10 !

0 sats \ 3 replies \ @south_korea_ln OP 24 Sep

10 is an answer, 10! isn't.

0 sats \ 2 replies \ @clarity 24 Sep

No no, there'e a space. I was indicating genuine excitement that I got the answer. Is it correct?

21 sats \ 1 reply \ @south_korea_ln OP 24 Sep

Yes. 1! x 10! = 10!, so 1x10 = 10. Good job.

The last answer requires some additional knowledge of factorials...

0 sats \ 0 replies \ @clarity 24 Sep

I dont' know how I did it, but I did, and I win, right ?!

1 sat \ 0 replies \ @0xIlmari 24 Sep

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.

0 sats \ 1 reply \ @WeAreAllSatoshi 24 Sep

@remindme in 12 hours

0 sats \ 0 replies \ @WeAreAllSatoshi 25 Sep

This was a lot of fun. Basically I expanded each factorial into their prime factorization, and determined that either m or n would need to be larger than 6 so as to include 7 in the factorization. Then just tried to find the other correct matching pair.

10,1 was an obvious solution.

10,0 was not obvious at first but now is :)

0 sats \ 3 replies \ @Satosora 24 Sep

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.

0 sats \ 2 replies \ @south_korea_ln OP 24 Sep

Can you elaborate on how you get to 18?

0 sats \ 0 replies \ @Satosora 24 Sep

No way, I wasnt able to type because the power went down at my plant!

0 sats \ 0 replies \ @Satosora 24 Sep

I am going off of memory here for 10 and 18. Let me solve write them down.

0 sats \ 2 replies \ @south_korea_ln OP 24 Sep

Disclaimer, these are not my puzzles. But linking to my sources would take the fun out of them.

0 sats \ 1 reply \ @clarity 24 Sep freebie

You got them from a 5th grade algebra exam I bet