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Square and cube number brainteaser

By David, 2 May 2018 Brainteaser

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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