### 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.

### Mathland Election

A political commentator summed up an election result. Given that there were just four candidates and that the figures quoted were exact find the number of votes polled for each candidate.

# Euromaths

##### Age 11 to 14 Challenge Level:

Sarah Dunn, Madras College, St Andrew's, Scotland and Soh Yong Sheng, Raffles Institution, Singapore both solved this in the same way. 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 this array.

 E 0
 U 1
 R 1
 O 1
 M 1
 U 1
 R 2
 O 3
 M 4
 A 5
 R 1
 O 3
 M 6
 A 10
 T 15
 O 1
 M 4
 A 10
 T 20
 H 35
 M 1
 A 5
 T 15
 H 35
 S 70

We draw a grid denoting the number of moves possible to reach each place. The number of possible routes are calculated by adding the number of the gridplace on the left and top, and if it is on the extreme left or top then there is only 1 route to get there. The number added will be correct as the square can only be accessed through these squares. There are altogether 70 possible ways.

Can you generalise this result to a 6 by 6 square, or a 7 by 7 square ... or an n by n square? Have you seen this pattern before? You may like to try a 6 by 6 array written in a slightly different formation.

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