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# Impossibilities

##### Age 11 to 14 Challenge Level: Sinead has $10$ pockets and $44$ one pound coins.

She wants to put all these pounds into her pockets so that each pocket contains a different number of coins.

Prove that this is impossible.
What is the minimum number of coins Sinead would need in order to be able to do this? Abbie has a set of $10$ plastic cubes, with edges of lengths $1, 2, 3, 4, 5, 6, 7, 8, 9$ and $10$ cm. She tries to build two towers of the same height using all of the cubes.
Prove that this is impossible.

If Abbie has a set of $n$ plastic cubes, with edges of lengths $1$ to $n$, for which values of $n$ can Abbie build two towers of the same height using all of the cubes?

Eustace is adding sets of four consecutive numbers. He wants to find a set where the total is a multiple of $4$.

Prove that this is impossible.

If you enjoyed this challenge, you may wish to try Summing Consecutive Numbers.

With thanks to Don Steward, whose ideas formed the basis of this problem.