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# Latin Numbers

*In a Latin Square each symbol or colour occurs exactly once in each row and exactly once in each column. *

Work out the number in the pale blue cell first.

Then the numbers in the**column** marked "2nd", then "3rd" and so on...

There is some more advice in the Getting Started section.

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Age 14 to 16

Challenge Level

* Latin Numbers printable worksheet*

Here is an example of a 4 by 4 Latin square:

Can you see what is going on?

**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

1st | |||||

3rd | 4th | 5th | 2nd | ||

There is some more advice in the Getting Started section.

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?