Conjecturing and generalising

problem

Adding in Rows

Age

11 to 14

Challenge level

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
problem

One O Five

Age

11 to 14

Challenge level

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by 3, 5 and by 7...
game

Sliding Puzzle

Age

11 to 16

Challenge level

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
problem

Starting to Explore Four Consecutive Numbers

Age

11 to 16

Challenge level
game

One, Three, Five, Seven

Age

11 to 16

Challenge level

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

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

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

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

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

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