The integers 1 through $n$ (inclusive) are separated into four bins.

For each bin, no two different numbers in it can multiply together to form a square number.

What is the largest $n$ for which this is possible?

If you think the answer is infinite, please put 99999 as your answer.

The answer is 24.

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Potential follow-up question: Find the number of ways the numbers 1 - 24 can be divided into the 4 boxes such that the "no square product" condition is satisfied.

Brian Charlesworth
- 4 years, 6 months ago

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Sounds good! Not sure I know the answer yet tho... :-/

Geoff Pilling
- 4 years, 6 months ago

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Yeah, I'm not sure how to tackle it yet either. I think we would first have to identify all the pairs of non-squares whose products are squares and make sure that their components are put in separate boxes.

Brian Charlesworth
- 4 years, 6 months ago

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@Brian Charlesworth – I find only a few combinations which would need to be avoided:

- 2,8
- 2,18
- 3,12
- 5,20
- 6,24
- 8,18

Any I missed?

The 2,8,18 trio is interesting. It means that even if you exclude the square numbers themselves, with 2 bins you could only go up to 17.

Geoff Pilling
- 4 years, 6 months ago

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@Geoff Pilling – Yes, I think you've found all the pairs. So without worrying about how the boxes are labeled, this would yield $24 * 12^{3} = 41472$ possible arrangements.

Brian Charlesworth
- 4 years, 6 months ago

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@Brian Charlesworth – Hmmm... Lemme think about it for a bit... Does that take into account that 1,4,9, and 16 need to be in different bins? And the 2-8-18 triad?

Geoff Pilling
- 4 years, 6 months ago

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@Geoff Pilling – Hi Geoff. I think the answer should be your answer ×4! Because 1,4,9,16 can also be arranged in 4! different ways. What do you think?

Kushagra Sahni
- 4 years, 6 months ago

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@Kushagra Sahni – Well, if it asked how many different ways they could be arranged, then, yes, I believe you would be correct, however it is only asking what is the largest number for which this is possible. And 24 is the largest number. Beyond that you have 5 square numbers which will be impossible to put into 5 bins without two multiplying together in one of the bins to make a square.

Geoff Pilling
- 4 years, 6 months ago

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@Geoff Pilling – Oh yeah, that is perfectly fine. I did solve the question that way only but I was talking about the question Brian Sir gave to you.

Kushagra Sahni
- 4 years, 6 months ago

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@Kushagra Sahni – Ah, ok, got it... Yeah definitely 4! needs to be included in the count... Thanks for your input!

Geoff Pilling
- 4 years, 6 months ago

@Brian Charlesworth – I get over a million... But I may have an issue with my logic. It seems to me that 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.

It seems like the distribution of these numbers alone yields $4^{11}$ = 4194304 possibilities.

And we haven't yet distributed the numbers in the six combinations of non squared numbers above.

Geoff Pilling
- 4 years, 6 months ago

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@Geoff Pilling – Now to finish off, 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 $4^{11}*4!*12*12*12 = 173946175488$

Phew! Glad I didn't enumerate them all...

But did I make a mistake somewhere? (I'm pretty good at doing that! ;) )

Geoff Pilling
- 4 years, 6 months ago

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@Geoff Pilling – Yes, that's correct! I was so preoccupied with the noted pairs and triad that I totally forgot about the other 11 loners that could be placed in any box.

Brian Charlesworth
- 4 years, 6 months ago

@Geoff Pilling very nice question.Expecting more questions like this

Kushal Bose
- 4 years, 6 months ago

nice question. I am so happy I got it correct :)

Hana Wehbi
- 4 years, 6 months ago

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The numbers 1-24 can be divided into 4 bins as follows:

Bin 1: 1,2,3,5,7,11,13,17,19,23,24Bin 2: 4,6,10,15,22Bin 3: 8,9,20,21,12Bin 4: 12,16,18However, none of the numbers 1,4,9,16, and 25 can be in the same bin, since they are square numbers, and the product of square numbers is another square number. Therefore we would need five bins to accommodate them.

Therefore, $\boxed{24}$ is the largest $n$ for which the above is possible.