Penta people, the Pentominoes, always build their houses from five
square rooms. I wonder how many different Penta homes you can
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
An interactive activity for one to experiment with a tricky tessellation
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
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:
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:
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.
Finally, I checked the possible
combinations for the Z-shaped tetromino, but, as I analysed the
possible combinations, I observed that there weren't
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.
This is the only tetromino that will not cover
the 8 by 8 chessboard.