Combinatorics

How many noughts are at the end of these giant numbers?

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

Label this plum tree graph to make it totally magic!

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board for any value of n. How many ways can a knight do this on a 3 by 4 board?

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.

All the words in the Snowman language consist of exactly seven letters formed from the letters {s, no, wm, an). How many words are there in the Snowman language?

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How many such students are there?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find there are 2 and only 2 different standard dice. Can you prove this ?

How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?