### Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

### Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

# Latin Numbers

##### Age 14 to 16Challenge Level

This problem is available as a printable worksheet: Latin Numbers

### Why do this problem?

This problem offers an engaging context in which students are challenged to solve a problem that requires systematic working and strategic thinking, while applying their knowledge of place value and divisibility.

### Possible approach

If students have not met Latin Squares before you may wish to show them this image and ask them to say what they notice.

Introduce the problem:

A six by six grid needs to be filled in so that the first row is a six digit number N, and the rows beneath are 2N, 3N, 4N, 5N and 6N.

The completed grid has to be a Latin Square, that is, it must have the same six digits in every row and every column.

Give students some time on their own to think about the problem, then invite them to discuss in pairs any ideas they have. Then share any suggestions about where to get started in a class discussion.

In the problem, there is a grid shaded in sections to indicate one possible order in which cell values can be deduced; the grid, together with some prompts, is available on this worksheet

### Key questions

If the bottom row is 6N, what can you deduce about the first digit of N?
If the fifth row is 5N, what can you deduce about the last digit in that row?
What can you say about the last digit of 2N, 4N and 6N?

### Possible extension

Students may wish to read more about Latin Squares and Cyclic Numbers.

### Possible Support

Two and Two requires similar systematic working and would be a good activity to work on before trying Latin Numbers.