# Latin & Euler squares

By **Susan Milner**

I love these puzzles! They are visually appealing, with bright colours and distinct shapes rather than numbers or letters; they require puzzlers to be very clear about which rules they are using; there are thousands of solutions for each level; and solutions can be found via many approaches.

Children are often quicker than adults at finding solutions,which makes them feel justifiably proud of their abilities. Even kindergarten students pick up on the idea, although you might want to start them with 3x3 grids. Once they understand the goal, they often progress quite quickly through several levels.

Also, many other puzzles become accessible once one has internalised the idea of a Latin square. Examples include Futoshiki, Kenken, Towers, Neighbours, and Kakurasu. Sudoku fans will find the idea familiar, as Sudoku is a combination of Latin squares.

Latin square puzzles are best done with pieces of coloured paper or craft foam, as everyone seems to find colour attractive and non-threatening. They can be done with just numbers or letters, but I don’t recommend it to start with.

Euler squares are Latin squares with two attributes, say colour and shape. They are also known as Graeco-Latin squares because Euler, who seems to have invented them, used two types of letters, Greek and Roman, in each square. That is really hard to see! It was once a popular game to construct Euler squares with regular playing cards, using four cards (say Ace, Joker, Queen, King) in each of the four suits.

### For teachers: Introducing Latin and Euler square puzzles

I give each student a baggie with 16 pieces: a square, circle, triangle and rhombus in each colour. Then I ask the students to sort out the pieces, without giving any suggestions about how to do that. If someone seems uncertain what to do in the face of the deliberately vague instructions, I ask them to figure out if any pieces are missing. That seems to provide enough direction to get them going. It’s always interesting to see how many sort by colour, how many by shape, and how many use both at the same time, which is essentially level 1 of the Euler puzzles.

Unlike the way I in which introduce other puzzles to classes, I usually put the rules for these – one level at a time – on the board, as the first students complete each level. That way each student can double-check the rules that s/he is currently using. I think it is useful for students to see the progression in complexity, and for them to realise that the rules are what makes the game. If I don’t put up the next rule quickly enough, students will often suggest it.

Getting the idea across for the lowest level you start at often requires several re-statements, as some people oversimplify and others make it more complicated than necessary. I have discovered that in any age group there are people who can hear a rule and proceed to use it, others who can read it and proceed to use it, others who read a rule and need to re-state it in their own words before using it, and some who need to hear the re-statement from their peers.