6. EQUAL SQUARES

A simple shape like the square tessellates the plane in a regular way, and we can obtain nice dissections by placing one such tessellation on top of another. In a tessellation of squares, the squares may be all the same size, or may occur in different sizes.





Problem S1. Dissect a square into 2 equal squares.

This problem is trivial (try it!), but we can use it to illustrate a new technique. At left is a portion of a tessellation of green 1-squares. Superimposed is a portion of a tessellation of red 2-squares. You can drag the red tessellation to various positions ((double) click on the red square, hold mouse button down and drag). Any position solves the above problem, but some positions give more simple or economical solutions. Here are two solutions showing the two resulting green squares: one a 4-piece dissection, the other a 5-piece. You might experiment to find a nice 6-piece solution.

In terms of area, this is a problem with 1-squares and 2-squares. However, the grids are nicely related too. This suggests the existence of other problems of this type.

Problem S2. (a) Dissect a square into 10 equal squares. (Abu'l-Wafa)
                        (b) Dissect 5 equal squares into 2 equal squares.

To carry out Part (a) we superimpose on the tessellation of green 1-squares, a red square grid of side length 10. A ‘nice’ solution is when the vertices of the red square lie on green lattice grid points. We then get a 16 piece dissection of the red square into 10 green squares, including 4 unbroken central squares.

Part (b) is more tricky. We use a red grid of half the size, but with the same orientation. If the green squares have area 1, what is the area of a red square? Place the red grid so that the central point lies at the centre of a green square. Now compare the 5-square green cross with two adjacent red squares. You should get a 12 piece dissection of one to the other. In this solution figure, notice that the 6 red pieces outside the green cross fit exactly onto the portion of the green cross outside the two shaded red squares.