**From New Scientist #2760, 15th May 2010** [link]

My house appears on each of three overlapping local maps. Each map is half a metre square and all are to the same scale. To ease map handling, I have produced a composite version by pasting them on top of one another to form a single sheet which correctly aligns all the roads and other features. I have now trimmed off those parts of the map which I do not need, thereby creating a complete square composite map, almost exactly a whole number of millimetres along each side. As it happens, I could not have produced a larger square even if, instead, the maps had been entirely blank sheets of paper.

To the nearest millimetre, what is the length of the side of the composite map?

[enigma1595]

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This is another one that I didn’t solve programatically.

I think the description of the maps is confusing as maps are (usually) aligned to a grid, whereas this problem wants to know what is the maximum square that can be covered with three equal sized squares in any orientation. It may be better to think of it as creating a table cloth to cover a square table out of three equally sized square tablecloths.

I found a website from an Associate Professor of Mathematics at a U.S. university, and he collects solutions to Geometrical Packing problems.

This tells us that the dimension of the maximal square that can be covered with 3 unit squares is √φ, where φ is the Golden Ratio, (1 + √5) / 2. (The result is due to Henry Dudeney, and was published posthumously in 1931 – Dudeney died in 1930).

Hence the solution is 500mm × √φ, which is 636.0098247570345…mm, close to 636mm.

Solution:The composite map is 636mm along each side.