Combinatorics

problem
Favourite

In a Box

Age

14 to 16

Challenge level

2 out of 3

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

1 out of 3

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

1 out of 3

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

Plum Tree

Age

14 to 18

Challenge level

1 out of 3

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

W Mates

Age

16 to 18

Challenge level

1 out of 3

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

1 out of 3

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

2 out of 3

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

2 out of 3

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

2 out of 3

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

2 out of 3

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?