When you multiply a whole number by itself, you get a square number. When you take three copies of a whole number and multiply them all together, you get a cube number.

There’s only one two digit number that is both a square and a cube. Can you work out what number it is?

For those looking for a challenge, can you find the only three digit number that is both a square and a cube?

Scroll down for the answer!


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Brainteaser answer

There are plenty of ways to do this puzzle. One way is to list out all the square numbers with two digits: 16, 25, 36, 49, 64, 81.

Then list out all the cube numbers with two digits: 27, 64.

The answer is the only number in both lists: 64.

There’s a trickier way to solve this puzzle too, which makes it easier to solve the challenge question. It relies on a maths fact: if you multiply two squares together, you get a square! If you’re not sure about this fact, try a few examples to see if you can work out what’s going on.

We’re going to start with a square, and use it to make a square that is also a cube. Let’s start with a small square: 4.

We know 4 x 4 = 16 is a square from the fact above, and also because of what a square means. But we can go one step further. 4 x 16 = 64 is the product of two squares, so it is also a square. But you can also write it as 64 = 4 x 4 x 4, making it a cube.

Using this method we can work out the next square that is a cube by repeating this process. This time, start with the next smallest square: 9. First, 9 x 9 = 81.

9 x 81 = 729, is the product of two squares, so it is also a square. You can also write it as 729 = 9 x 9 x 9, making it a cube.

Therefore the answer to the challenge is 729.

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9 responses

  1. Sreenu Avatar

    The first square and cube number when doubled is

    Please give me reply Urget

  2. What you do from my name Avatar
    What you do from my name

    It is useful for me but I can not understand the second one (729)

    1. David Avatar

      hi there!
      It’s a bit tricky, but the idea is:
      3 x 3 x 3 x 3 x 3 x 3 = 729
      and we can group those to make three terms:
      3×3   x   3×3   x   3×3 = 729
      so it’s a cube

      We can also group them to make two terms:
      3x3x3   x   3x3x3 = 729
      so it’s a square too!

      1. Aarinola Idowu-Titus Avatar
        Aarinola Idowu-Titus


  3. Krishna Manohar Avatar
    Krishna Manohar

    So you don’t understand
    Let me explain in short
    See 4³=64, 8²=64
    Now:- 9³=729, 27²=729
    This also works with all the squares like 16 cube and 81 cube

  4. Big brain Avatar
    Big brain

    I found a little formula for it
    So if x³=y then do
    ((√x)³)² =y and here it is the square of it

  5. bigger brain Avatar
    bigger brain

    But sqrt(x) cubed squared = x cubed because sqrt and squared cancel each other so it’s the same this as (x to the power of 3) = y

  6. toufik Avatar

    is it right 16^3=64^2

    1. David Shaw Avatar
      David Shaw

      That’s definitely a square and cube number – and it’s the only one with 4 digits!

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