### Pebbles

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?

### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

# Muggles Magic

##### Stage: 3 Challenge Level:

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 are rearranged.

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.