### Counting on Letters

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

### Pair Sums

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

### Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

# Weekly Problem 51 - 2008

##### Stage: 3 Short Challenge Level:
$3$, $4$ and $5$ are on diagonals, so can't go in the centre square (since each number must appear just once on each diagonal).

So the number in the centre sqaure must be a $1$ or a $2$. Suppose it was a $1$. Then we must put $1$s as shown in red:

and $2$s as shown in blue:

So there must be a $1$ in place of one of the crosses:

But the top and bottom cross couldn't be $1$, since they are on the diagonal, and the middle cross can't be $1$ since it's on the $3$rd row (where there's already a $1$). So $1$ can't go in the centre square after all!

$2$ goes in the centre square.

This problem is taken from the UKMT Mathematical Challenges.

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