Divide the below figure into $5$ equal pieces (same shape, same size, possibly reflected).
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I believe this is one of Martin Gardner's, but I could not find the source.
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asked Apr 13, 2015 at 14:59
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$\begingroup$ I think the dashed lines are misleading: if you consider them, we have a total area of 16 squares to divide into 5 parts $\endgroup$– leoll2Apr 13, 2015 at 15:12
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$\begingroup$ I actually find the dashed lines helpful, as they give me the relative sizes of each portion of the shape $\endgroup$– Brian JApr 13, 2015 at 18:58
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$\begingroup$ I think this might have been in one of the Aha! books? $\endgroup$– zebApr 14, 2015 at 0:43
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$\begingroup$ @zeb, thank you, you are right! This appears in aha! Insight. $\endgroup$– Mike EarnestApr 14, 2015 at 21:25
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1 Answer
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The first observation we make is that
the left and right edges are identical
the top and bottom edges are identical
If we then
mark four equidistant points along each horizontal line,
we have constructed the basis for making 5 shapes that are basically
thinner versions of the original shape.
Connecting the dots vertically now subdivides the original shape into 5 equal pieces of the same shape and size.
x-x-x-x-x-x \ \ \ \ \ \ / / / / / / | | | | | | x-x-x-x-x-x
Perhaps the most important observation to make is that
the lattice on which this shape is placed is a visual distraction. Once ignored and only the shape itself is considered, the solution is quite obvious.
answered Apr 13, 2015 at 15:21
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$\begingroup$ This means we could divide this into an infinite number of identical shapes, right? $\endgroup$ Apr 13, 2015 at 18:13
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$\begingroup$ @VictorHenry You sure could. It's not what the puzzle was asking for, though. $\endgroup$ Apr 13, 2015 at 18:17