1. Can you cut the shape below into two identical parts using one single line? The line does not have to be straight.

2. Below are two figures, each divided into four parts that are all identical in shape. Can you divide the square into five parts that are also all exactly alike?

3. The Yin and Yang symbol below is a large circle divided into two equally-sized regions, one black and one white. The outline of each region is made from three semi-circles, a large one, and two smaller ones of equal diameter. The diameter of the small ones is exactly half the diameter of the big one.

Solution:

The simplest solution with a non-straight line, is to add in two semi-circles as illustrated here on the left.

The solution that is a straight line cut, is the line through the centre at a 45 degree angle, illustrated below.

We can see why by considering the “cake slice” of the black region enclosed by that line, and the horizontal line through the centres of the circles. Since the angle at the centre is 45 degrees, this slice has an area that is an eighth of the total area of the large circle. Now let’s say that the large circle has a radius of R, and thus an area of πR^{2}. The smaller circle (marked) has a radius of R/2 and thus an area of π(R/2)^{2} = πR^{2}/4, in other words a quarter of the area of the big circle. So the black semicircle underneath the horizontal line has an area of an eighth of the big circle. The area of the semi circle plus the 45 degree slice is therefore a quarter of the total area, and therefore exactly half of the black area. So it cuts the black region in half and by symmetry it must cut the white area in half too.