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.*

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.