Conjecturing and generalising

problem

Starting to Explore Four Consecutive Numbers

Age

11 to 16

Challenge level

1 out of 3
problem

Sums of Pairs

Age

11 to 16

Challenge level

2 out of 3

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
problem

Circumference Angles

Age

11 to 16

Challenge level

2 out of 3

Can you prove the angle properties described by some of the circle theorems?
game

One, Three, Five, Seven

Age

11 to 16

Challenge level

3 out of 3

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
problem

The Bridges of Konigsberg

Age

11 to 18

Challenge level

1 out of 3

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.
problem

What Does It All Add Up To?

Age

11 to 18

Challenge level

2 out of 3

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
problem

Loopy

Age

14 to 16

Challenge level

1 out of 3

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
problem

Counting Fish

Age

14 to 16

Challenge level

1 out of 3

I need a figure for the fish population in a lake. How does it help to catch and mark 40 fish?
problem

Building Gnomons

Age

14 to 16

Challenge level

1 out of 3

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
problem

Equilateral Areas

Age

14 to 16

Challenge level

2 out of 3

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.