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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?

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Writ Large

Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

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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?

Even Up

Age 11 to 14 Challenge Level:
Consider all of the five-digit numbers which we can form using only the digits 2, 4, 6 and 8.
If these numbers are arranged in ascending order, what is the 512th number?