Latin numbers
Can you create a Latin Square from multiples of a six digit number?
Problem
Latin Numbers printable worksheet
Here is an example of a 4 by 4 Latin square:
Can you see what is going on?
In a Latin Square each symbol or colour occurs exactly once in each row and exactly once in each column.
In the grid below, N is a 6 digit number with a very special property:
if you double the number and write it in the second row,
treble the number and write it in the third row,
and so on...
you end up with a Latin Square!
| N: | ||||||
|---|---|---|---|---|---|---|
| 2N: | ||||||
| 3N: | ||||||
| 4N: | ||||||
| 5N: | ||||||
| 6N: |
Can you find the six digit number N?
If you're finding it difficult to get started, click below to see a diagram showing one possible order in which you can work out each value.
Work out the number in the pale blue cell first.
Then the numbers in the column marked "2nd", then "3rd" and so on...
| 1st | |||||
| 3rd | 4th | 5th | 2nd | ||
There is some more advice in the Getting Started section.
Getting Started
| 1st | |||||
| 3rd | 4th | 5th | 2nd | ||
Work out the number in the pale blue cell first:
If the bottom row is 6N, what can you deduce about the first digit of N?
Work out the numbers in the column marked "2nd":
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?
What can you deduce about the last digit of the first row?
Will it be even or odd?
The first digit of N will appear as the last digit in one of the other rows.
Multiply different possible values for the last digit of N and see which gives you the results you are looking for.
Once you have completed the right hand column you'll know which numbers will fill the column marked "3rd".
After completing the column marked "3rd" try to complete the column marked "4th"...
Student Solutions
Thank you to Isabelle from Maidstone Grammar School for Girls who sent a correct solution for the completed square. Andrew, Brian, Dylan and Maroun from Greenacre Public School in Australia and Lawrence from the UK sent in very similar solutions that were clearly explained. This is Lawrence's solution:
1. We started off by thinking about what digit could go in the top LH corner (A,N) and soon realised that it had to be 1 otherwise the multiple of 6 would not fit in the table.
2. We then looked at the RH column and realised that as 5N would have to end in 5 or 0, the only place a 1 could go in the RH column was the only other odd multiple left, which was 3 (F,3).
The only number to give a multiple ending in 1 is 3 x 7, so (F,N) was 7, and we were then able to fill in the rest of column F and carry the 10s over into column E.
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| N | 1 | 7 | ||||
| 2N | 4 | |||||
| 3N | 1 | |||||
| 4N | 8 | |||||
| 5N | 5 | |||||
| 6N | 2 |
3. Now that we had all the digits we went back to column A and put the digits in, in numerical order (checking that it was feasible), which allowed us to work out that (B, N) was 4.
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| N | 1 | 4 | 7 | |||
| 2N | 2 | 4 | ||||
| 3N | 4 | 1 | ||||
| 4N | 5 | 8 | ||||
| 5N | 7 | 5 | ||||
| 6N | 8 | 2 |
4. We then used trial and error to put the remaining 3 digits of N in starting at the RHS of the table and working down each column.
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| N | 1 | 4 | 2 | 8 | 5 | 7 |
| 2N | 2 | 8 | 5 | 7 | 1 | 4 |
| 3N | 4 | 2 | 8 | 5 | 7 | 1 |
| 4N | 5 | 7 | 1 | 4 | 2 | 8 |
| 5N | 7 | 1 | 4 | 2 | 8 | 5 |
| 6N | 8 | 5 | 7 | 1 | 4 | 2 |
We only realised when we had finished that there is a pattern in the rows, i.e. the digits are always in the same order: 1, 4, 2, 8, 5, 7.
Teachers' Resources
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
This printable worksheet may be useful: Latin Numbers
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.