A red cube divided int osmaller cubes

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!

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Brainteaser hint

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

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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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4 responses

  1. Anthony Holbourn Avatar
    Anthony Holbourn

    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

  2. Anthony Avatar

    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.

  3. Anthony Avatar

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

  4. Quamera Avatar

    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

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