I am thinking of a number between 1 and 1300. It meets 3 criteria:

- It is a perfect cube.
- It is less than 500.
- It is a perfect square.

Just kidding, it only meets two of those. To make up for lying though, I’ll tell you that it starts with 5, 7, or 9.

Include in your answer how much googling it took you.

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729 = 9^3 = 27^2

I got this by looking at cubes of numbers between 1 and 10. This was the second one that could be re-written as a square of some other number.

The first is 64 = 4^3 = 8^2, which would fit all 3 constraints. But doesn't fit the problem restatement…

No Googling.

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3^2=9<500, satisfying conditions 2 and 3.

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I didn't google. I wrote a quick pari+bash script to look for all the possibilities.

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No googling when it's squares and cubes. Those and primes are the center of my numbers OCD. I used to factor things like the odometer for fun.

I had SJ's answer, but James's is much more elegant. I hazarded 512 as a guess because it was barely over 500, and I thought that might be the trick. But it didn't work, of course, because it's 2^9, and thus not a square. But figuring that cued me immediately that 9^3 was going to switch that and had to work.

But of course, 9 actually does begin with 9.

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SJ definitely got the answer I was looking for, but James answer was quite lovely. Not sure how I missed that one, but I'm glad to be corrected!

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