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Latin Numbers

Age 14 to 16
Challenge 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. 

N:                                    
2N:            
3N:            
4N:            
5N:            
6N:            

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.