Little math problems September 2018

[pier4r]

A little one, that nonetheless is nice (and maybe someone can extend it to make it more challenging).
There are 15 white balls and 13 black balls in a box. Plus spare white and black balls.
One picks 2 balls at random.
If those are both white or black, a white ball is inserted in the box, those picked are removed.
If the colors are different, a black ball is inserted in the box. Those picked are removed.
You do the process until you end up with only one ball in the box. What color is it?

Bonus: do a calculator program that goes through the possible states of the bag and picks.

A second little one:
Find two positive integers such that: the sum of whose squares is a cube and the sum of whose cubes is a square.
I know only one family of them, if you find more, great.

Bonus: a calculator program that checks all those under, say, 1 million (each integer can vary from 1 to 1million). The faster the better (each device used will made its own category).

Credit: mathjam

answers with spoilers like this

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