Explaining, convincing and proving

problem
Sixty-seven squared

Age

16 to 18

Challenge level

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?
problem
Eyes down

Age

16 to 18

Challenge level

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?
problem
Dalmatians

Age

14 to 18

Challenge level

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.
problem
Make 37

Age

5 to 11

Challenge level

Four bags contain a large number of 1s, 3s, 5s and 7s. Can you pick any ten numbers from the bags so that their total is 37?
problem
Königsberg

Age

11 to 14

Challenge level

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
problem
Take three from five

Age

11 to 16

Challenge level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
problem
Top-heavy pyramids

Age

11 to 14

Challenge level

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
problem
Semi-detached

Age

14 to 16

Challenge level

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
problem
Terminology

Age

14 to 16

Challenge level

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
problem
Repetitiously

Age

14 to 16

Challenge level

Can you express every recurring decimal as a fraction?