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?

So Many Sums

Stage: 3 Short Challenge Level:

42

Consider the thousands column. The letters represent different digits so, as $S$ is 3, $M$ is 2 and there is a carry of 1 from the hundreds column. Therefore $A$ is 9, $U$ is 0 and there is also a carry of 1 from the tens column. In the units column, $O+Y$ produces a units digit of 3, so $O+Y=$3 or $O+Y=$13. However, $O+Y=$3 requires one of $O$, $Y$ to equal zero (impossible as $U=$0) or 2 (also impossible as $M=$2). So $O+Y=$13. We can also deduce that $N$ is 8, since, in the tens column, 1 $+$ 3 $+N=$12. The pairs of digits that produce a sum of 13 are 4 and 9, 5 and 8, 6 and 7. As $A$ is 9 and $N$ is 8, the only possible values for $O$ and $Y$ are 6 and 7. These are interchangeable, but in both cases $Y\times O=$42.

This problem is taken from the UKMT Mathematical Challenges.