Puzzling Asked by Joseph Sible-Reinstate Monica on May 5, 2021
How many ways are there to split the numbers 1, 2, 3, 4, 5, and 6 into two groups, such that the product of the numbers in one group is 24 and the sum of the numbers in the other group is 12? An answer should enumerate all of the possibilities (if any) and prove that there are no others. (It’s possible to figure this out by brute force, since there are only 64 possible partitionings, but there’s a more elegant approach that doesn’t require this.)
I came up with this puzzle myself.
The only possibilities are:
Proof:
Correct answer by Jaap Scherphuis on May 5, 2021
Another way:
Answered by trolley813 on May 5, 2021
Yet another way:
Answered by loopy walt on May 5, 2021
There is still another way:
Answered by Toby Mak on May 5, 2021
Here's how I reasoned through this before posting it. Others' methods turned out to be much simpler, though:
Looking back, I think I know why my solution is longer now: I was originally going to specify that there must be three numbers in each group in the description, but I realized this would make the puzzle too easy. I then grafted steps to derive that fact onto my original solution instead of trying to re-solve it from scratch without needing that intermediate step.
As I think about this some more, here's another way to do it, based on loopy walt's answer and its key insight, but where (IMO) each step has a simpler justification:
Answered by Joseph Sible-Reinstate Monica on May 5, 2021
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