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Trisected Triangle

Age 14 to 16 Short Challenge Level:

If an equilateral triangle is split into a number of smaller identical equilateral triangles then there must be one small triangle on the top row, three small triangles on the row below, five small triangles in the row below that and so on.

So total number of small triangles is $1$ or $4$ or $9$ or $16$ etc. These are exactly the square numbers, the diagram below might help you to understand why.


 


So, for three copies of a tile to form an equilateral triangle, the number of triangles which comprise the tile must be one third of a square number.

Of the four given tiles, only the third and fourth satisfy this condition, and both of these tiles can make equilateral triangles:

                        
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.