The four coloured pieces can be put together in two different ways to make these
shapes with base 13 units and height 5 units. Why is there one
square missing in the second arrangement?
The four coloured pieces can be put together in two different ways to make these
shapes with base 13 units and height 5 units. Why is there one
square missing in the second arrangement?
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Neither of the 13×5 triangles has the same total area as its supposed component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area, but the triangles are 13 wide and 5 tall, so it seems, that the area should be 32.5 units. But the blue triangle has a ratio of 5:2, while the red triangle has the ratio 8:3, and these are not the same ratio. So the apparent combined hypotenuse in each figure is actually bent. A full explanation can be found on Mark Wieczorek's Weblog.
Sam Loyd presented this Chessboard Paradox at the American Chess congress in 1858. Notice the Fibonacci numbers which can be found in both of these diagrams.
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Students can create their own presentation of the Missing Square Puzzle to show to other classes or in an assembly. Here are some guidelines for using PowerPoint 2007:
On the Home tab, in the Drawing group, click Arrange,
point to Align, and then click Grid Settings.
Tick the Snap objects to
grid and the display grid on screen boxes. Select from the dropdown box a
spacing of 1cm.
The red and blue
right-angled triangles can be made using the "Right Triangle" tool which can
be found in the Home tab, in the Drawing group.
The green and yellow
shapes can be created by putting together a number of 1cm by 1cm squares.
Upon completion of the shape drag over the shape to select all of the
squares then select "Group" from the Format tab, Arrange group.
Turn the Snap To Grid option off an add custom animations to each of the shapes to make the first arrangement of shapes transform into the second.
The images on this page are from the Wikimedia Commons. The descriptions of the licences can be found on the following pages: Missing Square Puzzle and Sam Lloyd Image.
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