We had used this idea before on a cushion with 36 squares. It shows where triangular numbers and square numbers coincide.

A ‘triangular number’ is formed by adding the counting numbers.

Examples:

The 4th triangular number = 1 + 2 + 3 + 4 = 10
The 9th triangular number = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

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A ‘square number’ is what you get when you multiply a number by itself.

Examples:

The 4th square number = 4 x 4 = 16
The 9th square number = 9 x 9 = 81

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Occasionally, a triangular number coincides with a square number:

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The 8th triangular number is 36; the 6th square number is also 36.

The next time this happens is the 49th triangular number and the 35th square number. Both give a total of 1225. This is the number of  squares shown on Data Log.

The afghan is 35 squares  x 35 squares. Each coloured section is one square longer than the section before.

The next time it occurs is when the total is 41616.

With the addition of the square grid, the afghan can be used as a counting aid as each block of colour has a different number of squares.

The afghan was constructed using the ‘log cabin’ method, where narrow strips are added to each side in turn. It is not immediately obvious that the strips get longer, by one square, each time so a grid was added to make it possible to easily count the squares.