An introduction to puzzles involving Latin Squares.

A Latin Square is an NxN square array of N unique symbols – each row has one of each symbol, each column has one of each symbol.

Examples with N=4:

Originally, a Latin Square was a square of *Latin letters*,
but any symbol will do. In fact, we can even use colors or numbers:

Latin Squares are different from *Magic Squares*

- (“Magic” = all rows and columns add up to the same total, numbers are unique across the entire square)
- (a Magic Square has all numbers from 1 up to N^2... for example, a 4x4 Magic Square will contain numbers from 1 to 16)

One of the most popular puzzles are based on Latin Squares: **Sudoku**

- You are given a partially-filled-in array (normally 9-by-9)
- Your goal is to fill in the rest of the Latin Square
- With a
*constraint*– no duplicate symbols within each of the nine 3-by-3 boxes - No mathematical operations required – it is just a
*logic puzzle*

We are not going to analyze Sudoku solving techniques today

- Some techniques are very simple
- But really
*hard*Sudokus require some more complex solution approaches - Check the
*rating*of the puzzle you are working on – or you might spend hours on one puzzle

**KenKen** is a puzzle invented by Tetsuya Miyamoto in 2004

- KenKen is another “enhanced Latin square” puzzle
- Usually a very small array (smaller than Sudoku) – 4-by-4 and 6-by-6 are common
- KenKen requires the solver to know some rules of arithmetic (add, subtract, multiply, and divide)

The 4-by-4 grid below is divided into special outlined boxes called *cages*

- Most cages have a number and an arithmetic operation
- It's an extra constraint on the numbers you can enter

- “6x” means that when we multiply the first two numbers in the first row, the answer must be 6
- “3-” means that when we subtract the last two numbers in the first row, the answer must be 3
- And... because it is a Latin Square, we can only use each digit (1, 2, 3, 4) once in the row

- “6x” boxes must contain
**2**and**3**)- It could be “2 3” or “3 2”

- “3-” boxes must contain
**1**and**4**)- It could be “1 4” or “4 1”

Note that we fill in some information based on our deductions... using small numbers near the edge to indicate that the numbers might switch

- We also have “3-” in the last row, so when we subtract first two numbers in the last row, the answer must be 3
- “1-” in the last row means that when we subtract the last two numbers in the last row, the answer must be 1

- Single box cages are “free squares” – we can fill them in immediately
- Two free squares in this puzzle... 1 and 3

- These two free squares give us more clues to solve the last row
- The “3-” cage *must* be “4 1” (not “1 4”)
- and the “1-” cage must be “2 3”

- We can‘t make any progress on the first row, but we can start to fill in the second row
- The “7+” cage *must* be “3 4”
- Now we have enough information to fill in the rest of the puzzle

- The first row must be “3 2 1 4” (to prevent clashing with the second row)
- and we can use Latin Square logic to fill in the remaining squares

Finally, we can check the answers:

- 3x2 = 6, 4–1 = 3
- 3+4 = 7
- 2÷1 = 2
- 4÷2 = 2
- 4–1 = 3, 3–2 = 1

KenKen puzzles are a good mental exercise:

- It is necessary to do some
*simple*arithmetic - as well as the usual Latin Square logic of a Sudoku puzzle

But... the puzzles are relatively small, and a 4-by-4 puzzle can be done in 2 to 5 minutes (compared with a hard 9-by-9 Sudoku, which might take an hour).

For more about larger KenKen puzzles (and examples of 6x6 and 7x7 KenKen puzzles), see More KenKen Puzzles.

The Swiss mathematician Leonhard Euler wrote about Latin Squares in the 18th century

Martin Gardner wrote about a more complicated version of Latin Squares – “Greco-Latin Squares” –

- See Gardner's “Mathematical Games” column,
*Scientific American*, November 1959. - also described in a Wikipedia article: https://en.wikipedia.org/wiki/Graeco-Latin_square

*Orthogonal Latin Squares* are used in software testing (to ensure good test coverage)

Online KenKen practice:

- KenKen puzzle website: https://www.kenkenpuzzle.com/
- New York Times has a daily 6-by-6 KenKen

This work is licensed under a Creative Commons Attribution 4.0 International License.

Last modified: Jan. 3, 2020

Dennis Mancl - http://manclswx.com