### Ice cream quiz

#### by David Radcliffe

This problem was posted in an ice cream shop in Oxford, England. I found it surprisingly difficult.

Here is my solution. Suppose that and . After substituting,

which implies that . Therefore

But the left side is positive for , so there are no whole number solutions.

*Image credit: Janet McKnight*

*Thanks to @samuelprime and @nodrogadog for pointing out mistakes in earlier versions of this post.*

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After spending some time on unsuccessful brute-force attempts, I stumbled upon a surprisingly simple solution. If $$x+3=a^3$$ and $$x^2+3=b^3$$ then

$$! (ab)^3 = (x+3)(x^2+3) = x^3+3x^2+3x+9 = (x+1)^3+8. $$

Now, it is easily seen that there are only two pairs of integer cubes which have a difference of 8: $$((-2)^3, 0^3)$$ and $$(0^3, 2^3)$$. These imply $$x=-3$$ or $$x=-1$$ (resp.), but then $$x^2+3$$ must be either 12 or 4, none of which is an integer cube.

It’s strange, I have this sudden craving for ice cream.

[resubmitting post with (hopefully) correct latex triggers, please delete my previous post]

After spending some time on unsuccessful brute-force attempts, I stumbled upon a surprisingly simple solution. If and then

Now, it is easily seen that there are only two pairs of integer cubes which have a difference of 8: and . These imply or (resp.), but then must be either 12 or 4, none of which is an integer cube.

It’s strange, I have this sudden craving for ice cream.