Visualising and representing

Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

What's the largest volume of box you can make from a square of paper?

Can you do a little mathematical detective work to figure out which number has been wiped out?

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

Can you make sense of these three proofs of Pythagoras' Theorem?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?