Did you solve it? That’s numberwang!

Earlier today I set three number puzzles. Here they are again with solutions.

1. Well balanced

What number equals the average of its digits?

Solution 4.5

If you allow trailing zeros, there are three solutions with four digits and one with five.

1.500
2.250
3.750
1.8000

2. The painted cube

A large wooden cube is painted on all of its faces. I divide the cube into a thousand mini cubes by slicing it nine times in each dimension.

How many cubes have no paint on them?

The bad way to calculate the answer is to start with 1000 cubes and subtract the ones with paint: 100 x 2 from the first set of opposing faces, 90 x 2 from what’s left of the next two opposing faces and 64 x 2 from what’s left of the remaining two faces.

The puzzle is to find a much faster way to get a solution. It involves a simpler calculation that you are more likely to be able to do in your head.

Solution 8³ = (2³)³ = 2⁹ = 512

All the outside mini cubes have paint. If you remove them all you will be left with an 8 x 8 x 8 cube. You could try multiplying 8 x 64, but easier would be to double nine times to get the answer

3. Clever countdown

You are playing a game with a friend, in which you start at the number 100 and take turns to deduct a number between 1 and 7. The winner is the person that lands on zero.

Which number do you choose first to be assured of winning?

Solution: 96

The winner is the person who lands on 8. (If you are on 8, your opponent cannot land on 0. But they will be forced to land on a number, from which you can land on zero.)

Thus the winner is the person who lands on 16, since this means they will land on 8, and therefore land on 0. Carry on up the eight times table and the closest number to the start is 96.

I hope you enjoyed today’s puzzles. Have a great Easter and see you in two weeks.

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.