### 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:

We have received correct solutions from Sarah Dunn, Ben Falconer, Fern Smith and Ian Downie, all from Madras College. Well done to you all.

They all used a similar argument:

Conservative $38\%$ $38\% - 1654$ votes $14\% + 50$ votes $14%$

Since all the votes add up to 100%

$38\% + 38\% -1654 +14\% + 50 + 14\% = 100\%$

$104\% - 1604 = 100%$

therefore $4\% = 1604$ votes

$1\% = 401$ votes

and $100\% = 40100$ total votes.

Therefore

Conservative Labour $38\%$ of $40100 =$ $15238$ votes $15238 - 1654 =$ $13584$ votes $5614 + 50 =$ $5664$ votes $14% of 40100 =$ $5614$ votes $40100$ votes