## Dolly’s cheese brainteaser

By David Shaw, 10 June 2020

Dolly’s cheese is covered in wax

**Difficulty: Tricky**

Dolly bought a special cheese. It comes in a square block that’s 5 centimetres wide, 5 centimetres deep and 5 centimetres tall. It’s also coated in wax on all sides. Without removing the wax, Dolly cuts the cheese into 1 centimetre cubes.

Can you work out how many cubes have wax on them?

**Scroll down or click for a hint, or the answer!**

## Brainteaser hint

How many cubes does Dolly cut? Can you work out how many cubes don’t have wax on them?

## Brainteaser answer

Dolly’s cheese block is 5 cubes wide, 5 cubes deep and 5 cubes high. So it will contain 5 x 5 x 5 = 125 cubes.

All the cubes on the outside of the cheese have wax on them. If we take them away, we’re left with a 3 x 3 x 3 shape, with 27 unwaxed cubes of cheese.

To work out how many have wax, we can use subtraction. 125 cubes – 27 unwaxed cubes = 98 waxed cubes!

For a different approach, we can break the cheese into different pieces.

• There are 8 cubes in the corners

• There are 12 edges, each with 3 cubes that aren’t corners, and 12 x 3 = 36

• There are 6 faces, and each has 9 cubes that aren’t corners or edges. 6 x 9 = 54

Adding it all together, we get 8 + 36 + 54 = 98 waxed cubes.

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10 June, 2020 at 6:50 pm

At first sight it seems to be 86:

8 corners, each with three sides

24 corners, each with two sides

54 flat surfaces, outward facing

10 June, 2020 at 8:29 pm

We each approached it in our own way and we came to the correct conclusion either way, replicating your suggestions. It was interesting just rationalising it as well as the arithmetic.

A gentle way to spend half an hour or so, dwelling on the discussion.

Gwen and Tony.

10 June, 2020 at 8:31 pm

… and I had corrected my initial omission by the time we got together.

13 June, 2020 at 10:25 pm

A different way is take away 2 faces of 25 waxed cubes then 2 faces of 15 (5×3) waxed cubes and lastly 2 faces of 9 (3×3) waxed cubes. 50 + 30 + 18 = 98