The diagram (not to scale) shows a smaller rectangle (EFGH) inside a larger rectangle (ABCD), with the corners of the smaller rectangle touching all four sides of the larger rectangle.

The smaller rectangle measures 106 x 53. The width of the larger rectangle is 118. What is the height of the larger rectangle?

In summary, AB = 118; FG = 53; EF = 106; find the length BC.


I have drawn the answer diagrams to a scale of two pixels per unit.

I felt quite guilty at setting this tough puzzle as it was by far the most difficult that I have ever placed on this web site. Nevertheless, to my surprise, it attracted a record entry of 6, (although one of these was wrong). Only one competitor (John Stafford) gave both answers. There are various methods of solution. The one I have shown is based on the entry from Christopher Broughton and uses trigonometry but other methods exist.

Entries were from John Stafford, Richard Birkhill, Christopher Broughton, Andy DeBauch and Matt Bomber who all found the answer 101. John Stafford won the book token prize. Well done all! It seems that the level of difficulty may be inversely proportional to the enthusiasm of solvers to have a go.

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