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