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?
Topics: Starter  Area  Mensuration  Puzzles  Ratio  Shape
How did you use this starter? Can you suggest
how teachers could present or develop this resource? Do you have any comments? It is always useful to receive
feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
Click here to enter your comments.
If you don't have the time to provide feedback we'd really appreciate it if you could give this page a score! We are constantly improving and adding to these starters so it would be really helpful to know which ones are most useful. Simply click on a button below:
This starter has scored a mean of 3.9 out of 5 based on 168 votes.
Previous Day  This starter is for 26 July  Next Day
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.
Teacher, do your students have
access to computers? 

Here is the URL for a concise version of this page without comments or answers.
Here is the URL which will take them to a student version of this activity.
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
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
rightangled 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.