Latin Squares

Players: 1
Ages: 5 and up
Cost: Free!
Math Ideas: Logic, spatial reasoning, operations
Questions to Ask:
How can you color in this square so that no color appears twice on the same row or column?
How many different ways can you color the same square?

My wife is so glad I started this newsletter.

There was a time when I'd spend all day on a journey through the Internet, discovering new mathematical concepts and patterns. Then I'd come home with a notebook full of drawings and a brain full of new ideas, but nobody to share them with!

Except her.

This week's newsletter is a perfect example: I started out by finding an interesting chess puzzle and ended up stumbling on a whole treasure trove of mathematical ideas.

So instead of giving you a single recommendation this week, I'll take you on a little mathematical journey, with little recommendations along the way. My wife thanks you.

(I don't even know if she reads these things, honestly. I guess I'll find out this week!)

The Eight Queens Puzzle

My son has started playing chess, so I've been looking around for some interesting chess puzzles to give him.

I found one that was quite intriguing: Queens can move in any direction on the board, vertically, horizontally, or diagonally. How many queens can you place on the board so that no two queens can capture each other?

The answer was clearly less than nine; chess boards only have eight rows, so the ninth queen would have no safe row to hide on. But what about eight queens?

If your child is a fan of chess, this is a wonderful little practice puzzle to pose to them. Or you can follow me to the next stop of my journey...

In one of the websites about the eight queens puzzle, I noticed a reference to Latin squares. I like Latin and I like squares, so I followed the link.

It turns out Latin squares are an ancient visual puzzle, where you color in a set of square tiles so that no color appears twice in the same column or in the same row. The stained glass window shown is an example of an 8x8 Latin square, where eight colors are used

This leads to a great game to play with kids of any age: Create Your Own Latin Square

Draw a square grid of any size (3x3 is a good place to start)
Pick some crayons or colored pencils whose number matches the height of your grid (5x5 squares require 5 colors)
Shade in each square in the grid! Make sure that no color appears twice on the same row or the same column.

Once your kids feel comfortable making Latin squares, you might want to give them a harder challenge: Fill in this 4x4 grid, but make sure your four colors appear in each of the four big, bold squares.

This adds an element of strategic thinking that makes the game a lot more fun and challenging.

If the 4x4 square is still too easy, you could give your kids a 9x9 Latin square, where each color must be used once in each of the nine big bold squares. Here is an example of a completed 9x9 Latin square under that restriction. Say, it sort of looks like a...

Sudoku

That's right! The Sudoku craze of the past 15 years is actually centuries old. Not only that, but there are tons of variations of these puzzles that are appropriate for learners of all ages.

People tend to call them Sudoku-style puzzles, but really they are variations on the Latin Square. Let's check out a few:

KenKen

In a KenKen puzzle, the goal is to fill in each square with a number. As with Latin squares, no number can appear twice in any row or column.

In addition, the numbers in each bolded section must be used to create the answer given. For example, in the bottom right corner, the two numbers must be able to divide to get a result of 2. So you have two pairs of numbers that could work: 4 and 2, or 2 and 1.

These KenKen problems, which can get quite large and complicated, are fantastic for any students who are familiar with the four operations. A book of KenKen puzzles would be the perfect item to bring in your purse to any restaurant or waiting room - it certainly requires more mathematical thinking than any Candy Crush game. Or you can print some free KenKens of various difficulties online.

Jigoku

Want to see the hardest puzzle I've ever tried to solve? Welcome to Jigoku.

Usually in a Sudoku, you begin with a few numbers. Without those hints, the puzzle would be impossible to solve!

But in Jigoku, you get no starting numbers. Instead, you get a set of inequality signs between the empty cells, indicating which number is greater than which.

So in the mini-puzzle shown, we need to place the numbers from 1 to 4. If you start in the top left corner, you will see that it is less than the number below it, which is in turn less than the number in the bottom right corner, which is less than the number in the top right corner. So that train of inequalities indicates that the answer should look like

1 4
2 3

But that's just a tiny 2x2 version of the game. The full Jigoku is a complete 9x9 Sudoku puzzle with absolutely no numbers provided.

I'm not kidding: this is the hardest puzzle I've ever tried. One time I spent twenty minutes working on a big Jigoku with a bunch of math teachers at this conference called Twitter Math Camp (Am I the nerdiest person you know?), and we got through about 10% of the puzzle.

Actually, on Tuesday I printed out a new one and started working on a new one, just to see if I could finally crack the code. I actually got almost to the finish and then I HAD TWO *%&$ SEVENS ON THE SAME %&*$ ROW AND I QUIT!!!

Anyway, I used to spend a ton of time playing these little logic games as a kid, and the mental structure for problem-solving has helped me a great deal in the years since. If your child likes solving logic puzzles, let them give Jigoku a try.

And thanks, again, for letting me share my random walks down the mathematical Internet. Whether your child ends up cracking a Jigoku or just coloring in a 3x3 Latin square, they'll have met me somewhere along my path.