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Tilted Squares

Age 11 to 14 Challenge Level:

We received three excellent solutions from Jonathan, Bryn and Marissa of Madras College. Marissa's solution follows:
 


The results above suggested a relationship between the area of tilted squares and the distance travelled along and up to get from one vertex to an adjacent vertex.
 
Alex from Llandovery College was able to prove the relationship and sent us this proof that the formula will always work.