Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Impossibilities

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.

*With thanks to Don Steward, whose ideas formed the basis of this problem.*
## You may also like

### Marbles

### More Marbles

### Differences

Or search by topic

Age 11 to 14

Challenge Level

- Problem
- Student Solutions

*Impossibilities printable sheet*

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.

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?