Conjecturing and generalising

problem
Favourite

Shape and Territory

Age

16 to 18

Challenge level

2 out of 3

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
problem

Absurdity Again

Age

16 to 18

Challenge level

1 out of 3

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
problem

Incircles

Age

16 to 18

Challenge level

1 out of 3

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
problem

Chocolate 2010

Age

14 to 16

Challenge level

1 out of 3

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
problem

Taking Steps

Age

7 to 11

Challenge level

1 out of 3

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
problem

Squares, Squares and More Squares

Age

11 to 14

Challenge level

1 out of 3

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
problem

Nim

Age

14 to 16

Challenge level

2 out of 3

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
problem

Card Trick 2

Age

11 to 14

Challenge level

2 out of 3

Can you explain how this card trick works?
problem

Repeaters

Age

11 to 14

Challenge level

2 out of 3

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
problem

Lower Bound

Age

14 to 16

Challenge level

2 out of 3

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =