### Consecutive Numbers

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

### Tea Cups

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

### 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?

# Distinct in a Line

##### Stage: 3 and 4 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.

Alternatively, consider the bottom-left corner. $2$ and $3$ are in the same corner, $4$ is in the same row and $5$ is in the same diagonal. Therefore, this must contain a $1$.

Then, the centre is in the same diagonal as each of the corners, so cannot be $1$, $3$, $4$ or $5$. This means the centre must contain a $2$.

This problem is taken from the UKMT Mathematical Challenges.
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