Combinatorics

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 5×5 array?

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

Given a 2 by 2 by 2 skeletal cube with one route 'down' the cube. How many routes are there from A to B?

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

Can you find squares within a number grid whose entries add up to an even total?

Is it possible to use all 28 dominoes arranging them in squares of four? What patterns can you see in the solution(s)?

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?