### 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?

# Consecutive Seven

##### Stage: 3 Challenge Level:

Start with the set of the twenty-one numbers $0$ - $20$.

Can you arrange these numbers into seven subsets each of three numbers so that when the numbers in each are added together, they make seven consecutive numbers?

For example, one subset might be $\{2, 7, 16\}$

$2 + 7 + 16 = 25$

another might be $\{4, 5, 17\}$

$4 + 5 + 17 = 26$

As $25$ and $26$ are consecutive numbers these sets are the kind of thing that you need.

[Remember that consecutive numbers are numbers which follow each other when you are counting, for example, $4$, $5$, $6$, $7$ or $19$, $20$, $21$, $22$, $23$.]