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Penta Place

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

Tessellating Triangles

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

Two by One

An activity making various patterns with 2 x 1 rectangular tiles.


Age 7 to 11 Challenge Level:

We received a solution to the first part of the problem from Andrei of School NO. 205 in Bucharest and from Chong Ching, Chen Wei and Teo Seow from Secondary 1B, River Valley High School in Singapore. Well done to you all.

The diagram below shows how the students from River Valley High School combined the original tetromino, together with 15 copies of itself, to cover the eight by eight chessboard.

River Valley High's solution.

Andrei followed this up with: It is clear that the 4 in a line tetrominoes can be fitted into an 8 by 8 square, because they can be fitted into a 4 by 4 square:

4-in-a-line tertronimoes.

Here there are a lot of possible arrangements, all leading to the same result.

The L-shaped tetromino can also be fitted into an 8 by 8 square for the same reason:

L-shaped tetronimoes.

Here there are also a lot of solutions.

The square tetromino will obviously fit into the 8 by 8 square, but there is only one possible arrangement.

Square tetrominoes.

Finally, I checked the possible combinations for the Z-shaped tetromino, but, as I analysed the possible combinations, I observed that there weren't any.
Now, I must prove it.
I tried several combinations, then I observed that there couldn't have been any solutions because there always remains a place where it isn't possible to put a tetromino. For example:

Z-shaped tetronimoes.


More Z-shaped tetronimoes.

This is the only tetromino that will not cover the 8 by 8 chessboard.