Reflections

The number 746 can be written as the sum of two numbers, one of which is the reversal of the other.

For example: In how many other ways can 746 be written as the sum of two numbers, one of which is the reversal of the other?

What other numbers between 700 and 800 can be written as the sum of two numbers, one of which is the reversal of the other?

SOLUTION Reflections (ages 8 to 13) Two other ways: 746 = 472 + 274 = 373 + 373.

Nine other numbers between 700 and 800 can be similarly expressed: 706, 707, 726, 727, 747, 766, 767, 786 and 787.

Space problems

1 The height of a cone equals the radius of its base. Two such cones fit inside a sphere so that the circumference of the bases just touches the sphere.

What proportion of the volume of the sphere is taken up by the cones?

2 A sphere sits inside a cube so that it just touches the faces of the cube. Does the sphere take up more or less than half the volume of the cube?

SOLUTION Space problems (ages 14 to 16) 1 Exactly half. The volume of each cone is 1Z3 pr3 and the volume of the sphere is 4Z3 pr3. 2 Just over half.

If l is the length of the edge of the cube, the volume of the sphere is 4Z3 p (lZ2)3 which as a proportion of l3 is p6, and p6

12.