### 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 numbers from $0$ - $20$:

Can you arrange these numbers into seven sets of three numbers, so that the totals of the sets are consecutive?

For example, one set 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 might be part of your solution.

Once you've found a solution, here are some questions you might like to consider:

• Is there more than one possible set of seven consecutive totals? How do you know?
• Is there more than one way to make the seven totals?
• Could you make seven sets that all had the same total?
• Could you make seven sets whose totals went up in twos? Or threes? Or...