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Square Numbers

If we have a square which has sides of 3 cms, then its area is 3x3=9 square centimetres. Any square with sides of length x has an area of x2. If x is a whole number, then so is x2. So any whole number which we can write as a number times itself is called a square number. The first few square numbers are:

1 1=1, 2 2 = 4, 3 3 = 9, 4 4 = 16, 5 5 = 25, 6 6 = 36,...

We can picture these numbers using squares, with the same number of squares across as there are down:

2 2 = 22 = 4 3 3 = 32 = 9 4 4 = 42 = 16

The difference between two neighbouring square numbers is always an odd number.

As with triangular numbers, we play the Addition game: Someone challenges you by choosing a number. You then have to write it as an addition sum totalling to the chosen number but you can only use square numbers. Is it always possible (it was with triangular numbers)? If so, what is the most number of squares that you will ever need?

  • 1 is a square number.
  • 2 is 1+1, a sum of two square numbers.
  • 3 is 1+1+1, a sum of three square numbers.
  • 4 is a square number.
  • 5 is 1+4, a sum of two square numbers.
  • 6 is 1+1+4, a sum of three square numbers.
  • 7 is 1+1+1+4, a sum of four square numbers.
  • 8 is 4+4, a sum of two square numbers.
  • 9 is a square number.

From the list above, no more than four squares were needed to express 7. In fact, it has been proved that:

Every number can be written as a sum of at most 4 square numbers.

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