### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Calendar Capers

Choose any three by three square of dates on a calendar page...

### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

# Nine in a Line

##### Stage: 3 and 4 Short Challenge Level:

If n is the smallest number, the nine consecutive numbers can be expressed as:

n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8

We are told that they all add up to 2007

so n + n+1 + n+2 + n+3 + n+4 + n+5 + n+6 + n+7 + n+8 = 2007

so 9n + 36 = 2007

9n = 1971

n = 219

If the smallest number is 219, the largest will be 219 + 8 = 227

Alternatively, the average of the 9 numbers is 223, so they are 219, 220,..., 226, 227.

So the last number is 227.

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
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