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This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?


Stage: 2 and 3 Short Challenge Level: Challenge Level:1

a solution
The diagram shows that it is possible to fit five T shapes in the square. In order to fit six T shapes into the square, exactly one of the $25$ squares would be left uncovered; hence at least $3$ corner squares must be covered.

We now label a corner square $H$ or $V$ if it is covered by a T shape which has the top part of the T horizontal or vertical respectively. If all four corner squares are covered then there must be at least two cases of an $H$ corner with an adjacent $V$ corner.

Each such combination produces a non-corner square which cannot be covered e.g. the second square from the right on the top row of the diagram. If only $3$ corner squares are covered, there must again be at least one $H$ corner with an adjacent $V$ corner and therefore a non-corner square uncovered, as well as the uncovered fourth corner. In both cases, at least $2$ squares are uncovered, which means that it is impossible to fit six T shapes into the square.

This problem is taken from the UKMT Mathematical Challenges.