### Consecutive Numbers

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

### 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.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

# Magic Error

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

The diagram shows the totals of the rows and columns, and the bracketed numbers are the total of the numbers in the two main diagonals. By considering the average values of the rows and columns, we can see that the each column, row and diagonal should total 34. Row 2 and column 2 are both 2 short, so their common entry, 13, needs to increase by 2. Also, row 4 and column 3 are both 2 high, so their common entry, 15, needs to increase by 2. So swapping entries 13 and 15 increases row 2 and column 2 by 2 and reduces row 4 and column 3 by 2, and also increases the total of the main diagonal from 32 to 34, as desired. So the sum of the two numbers is 13 + 15 = 28.

This problem is taken from the UKMT Mathematical Challenges.
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