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Transformations Tables

These grids are filled according to some rules - can you complete them?

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We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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The Great Tiling Count

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.


Age 7 to 11 Challenge Level:

This solution comes from Henry. Well done!

To make a square, we need three more matchsticks.

To make a second square, we need three more matchsticks.

I noticed that for each square we make we need three matchsticks, in addition to the one we had at the beginning. Therefore to make 10 squares we need a total of 31 matchsticks. For 20 squares we need 61 matchsticks and for 50 squares we need 151.

The general pattern isif $n$ is the number of squares, we use $3n + 1$ matchsticks in total.