Combinatorics

problem
Favourite

In a Box

Age

14 to 16

Challenge level

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?
problem
Favourite

Snooker Frames

Age

16 to 18

Challenge level

It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?
problem

Euler's Officers

Age

14 to 16

Challenge level

How many different ways can you arrange the officers in a square?
problem

Plum Tree

Age

14 to 18

Challenge level

Label this plum tree graph to make it totally magic!
problem

W Mates

Age

16 to 18

Challenge level

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.
problem

Lost in Space

Age

14 to 16

Challenge level

How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?
problem

Flagging

Age

11 to 14

Challenge level

How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?
problem

Paving Paths

Age

11 to 14

Challenge level

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?
problem

Counting Binary Ops

Age

14 to 16

Challenge level

How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.
problem

Bell Ringing

Age

11 to 14

Challenge level

Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?