### Greetings

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?

### 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.

### Euromaths

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?

# Mathland Election

##### Stage: 3 Challenge Level:

On Mathland TV a political commentator summed up an election result as follows.

A Labour majority of $1729$ last time has been turned into a Conservative majority of $1654$ in this election and the conservative candidate has obtained $38\%$ of the poll. Labour has taken second place. The Liberal Democrat has obtained only $14\%$ of the poll and has been beaten into fourth place by the SNP candidate who has $50$ more votes than the Liberal Democrat.

Given that there were just four candidates and that the figures quoted were exact find the number of votes polled for each candidate.
[This question comes from the book 'Mathematical Challenges' published in 1989 for the Scottish Mathematical Council by Blackie, ISBN 0216 92622X]