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Geometry Level 5

Without cutting or bending the unit square, what is the side length (to 3 decimal places) of the largest square that can be covered completely with 7 unit squares?

Inspiration .


The answer is 2.207.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Mark Hennings
Feb 2, 2017

I don't know whether this is proved. 1 2 ( 3 + 2 ) = 2.207 \tfrac12(3+\sqrt{2}) = 2.207 is described here as the best-known, and proposed as the best.

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Mark, this is actually isn't difficult to compute. And I believe not that difficult to prove either.

Let S S be the side length of the largest possible coverable square. Then we can almost immediately see that

( S − 1 ) + ( S − 2 ) = 2 (S-1)+(S-2)=\sqrt{2}

and there you have it.

Michael Mendrin - 4 years, 4 months ago

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I agree that the answer is eminently plausible. Your calculation assumes a particular orientation of the squares. Given that the paper I reference discusses problems much more complex than 7 7 squares, and yet describes the 7 7 case as unproved, it can't be that easy... Not my area, though, so I don't know.

Mark Hennings - 4 years, 4 months ago

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Oh, best known! How about that. Maybe I'll have another look at trying to prove that it is the best. Amazing that this isn't even known for certain.

Michael Mendrin - 4 years, 4 months ago

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