Place four pebbles on the sand in the form of a square. Keep adding
as few pebbles as necessary to double the area. How many extra
pebbles are added each time?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you work out the area of the inner square and give an
explanation of how you did it?
Chua Zhi Yu, from River Valley High School, Singapore and Andrei
Lazanu, from 12, School No. 205, Bucharest, Romania both saw
through the deceptiveness of these diagrams and sent explanations
of why one square unit of area seems to disappear when the pieces
Look at the triangle with arms of lengths 5 and 13 units. We
shall call this triangle T. Although the figure at the top looks
like this triangle it is not a triangle at all. What appears to be
the longest side is not a straight line but actually two lines
along the hypotenuses of the red and blue triangles. The gradient
or slope of the hypotenuse of the red triangle is 3/8, while the
gradient of the hypotenuse of the blue triangle is 2/5. We know 3/8
< 2/5 so the figure at the top has two edges that 'dip inwards'
making it into a concave quadrilateral. This shows that the four
pieces of this jigsaw fit inside the right angled triangle T
leaving a small space uncovered. Exchanging the positions of the
red and blue triangles makes these two hypotenuses project outwards
enclosing extra area in the shape of a long thin parallelogram.
Adding up the areas, the red triangle has area 12 square units,
the blue triangle 5 square units and the other two pieces together
15 square units making a total of 32 square units. The right angled
triangle T with arms of length 5 and 13 units has area 32.5 square
units, an extra half a square unit. By rearranging the pieces the
extra area included along the hypotenuse of triangle T is twice
this, namely one square unit, and this accounts for the indentation
on the bottom of the lower figure.