Explaining, convincing and proving

problem

Matter of scale

Age

14 to 16

Challenge level

Can you prove Pythagoras' Theorem using enlargements and scale factors?
problem

Square pair circles

Age

16 to 18

Challenge level

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
problem

Parallel universe

Age

14 to 16

Challenge level

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.
problem

Master minding

Age

11 to 14

Challenge level

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?
problem

Not necessarily in that order

Age

11 to 14

Challenge level

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?
problem

Pattern of islands

Age

11 to 14

Challenge level

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
problem

Flight of the Flibbins

Age

11 to 14

Challenge level

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to the new planet?
problem

Add 3 dice

Age

7 to 11

Challenge level

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do this?
problem

Eleven

Age

11 to 14

Challenge level

Replace each letter with a digit to make this addition correct.
problem

Cross-country race

Age

14 to 16

Challenge level

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?