Happy Binning!
You have 24 balls labelled 1 through 24.
How many different ways can you separate them into four bins such that no two numbers in a bin multiply together to form a square number?
Assumption : The bins are indistinguishable and in no definite order, so, for example, if you put all the balls in one bin, and then took them all out and put them in a different bin, it wouldn't constitute a different arrangement.
Inspired by Brian Charlesworth
The answer is 173946175488.
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1 solution
WLOG you can label the boxes 1, 4, 9 and 16 and put those respective numbers inside.
Now there are 11 numbers which appear to be able to go into any of the 4 bins, namely 7, 10, 11, 13, 14,15, 17, 19, 21, 22, and 23.
So the distribution of these 11 numbers yields 4 1 1 = 4194304 possibilities.
And there are a few pairs of non squares which couldn't be put in the same bin, namely:
The 2-8-18 triad can be put in 4! ways.
Each of the other three pairs can be put in 4*3 ways.
So the total would be 4 1 1 ∗ 4 ! ∗ 1 2 ∗ 1 2 ∗ 1 2 = 1 7 3 9 4 6 1 7 5 4 8 8
Phew! Glad I didn't enumerate them all... :0)