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Overlapping Rectangles

Problem

Pairs of identical rectanglular strips, each measuring 3 by 1, are overlapped in a number of different ways to form three different shapes, shown in the diagram below.

Which shape has the greatest perimeter?


Solution

It should be clear the the L-shape on the left hand side has a perimeter of 12, and assuming symmetry of the cross-shape in the middle, we can see that the perimeter is also 12. But what if the cross-shape is not perfectly symmetrical, and what can we conclude about the shape on the right hand side?

Consider the following diagram.

If the vertical edges on the right hand side of the shape are moved to the right we can see that they add to 3. The same is true of the vertical edges on the left hand side, and the horizontal edges on the top and bottom.

In other words, as long as the rectangles completely overlap the resulting perimeter will always be 4 times 3 = 12.

Alternatively, it can be seen that the perimeter of the overlap shape is equal to the perimeter of two rectangles minus the lengths of the four concealed edges, and indeed, this value will be constant. Note that the diagram on the left hand side demonstrates an extreme case: there are two (internal) concealed edges, the other two edges coincide with the external edges and so it is necessary to subtract four concealed edges from the perimeter of the two rectangles.

What if the dimensions of the rectangles were 5 by 2? 9 by 7? $a$ by $b$?
What if the rectangles do not overlap at right angles?

Problem ID: 289 (22 Sep 2006)     Difficulty: 1 Star

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