## Lo Shu Magic Squares Puzzle

A 4,000 year old Chinese magic squares puzzle is both fun and a way to learn basic problem solving skills. Plus turtles.

Magic squares are a simple math puzzle with a grid of equal numbers of columns and rows. Each cell in the grid is assigned a number. The columns, rows, and diagonal directions total to the same amount. For older kids and adults, this amount is called a magic constant.

Legend says an ancient Chinese emperor name Yu happened to be down by the Yellow River one evening when he noticed a turtle swim at his feet. The turtle had unusual marks on his shell which formed a magic square where all columns, rows, and diagonals summed to the same amount. Here’s the pattern of dots Yu saw on the turtle’s shell:

If you love numbers, this set of dots has some amazing relationships. First, notice the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 are made of dots organized into lines and boxes. Count from 1 to 9 on the dot pattern above for yourself. Then notice the corners, starting from top right, are multiples of 2. The numbers 1, 3, 5, 7, and 9 are on the edges or center of this pattern.

The 3 cell by 3 cell square grid pattern is called a Lo Shu magic square after a Chinese story about a turtle star pattern and the Lo river.

### The Math Squares Puzzle

To solve a math squares puzzle, the sum of all rows, columns, and diagonals must be the same. You can only use each number once. Here’s a blank 3 cell by 3 cell grid you can print out to try and solve:

Because this is a 3 cell by 3 cell grid, and numbers can be used only once, the 9 cells tell you your available numbers are 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Can you solve the puzzle with these basic rules? How long does it take you?

### How to Solve a Math Squares Puzzle

The easiest solution is in the pattern of dots Emperor Yu saw on the back of the turtle. In the pattern, notice how each row, column, and diagonals of dots sum to 15:

Of course, this is cheating to simply refer to the dot pattern. It’s more fun to see if there is a mathematical way to solve any math squares puzzle.

As it turns out, there is a way to calculate the sum you need to find for each row, column, and diagonal in your grid. Add all your available numbers and divide by the number of rows in your grid:

`1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 / 3 = 15`

A three cell square grid, therefore, must sum to 15 for all its rows, columns, and diagonals. The dot pattern Yu saw on the turtle’s back sum to 15.

How you sum to 15 turns out to be somewhat flexible. Here are eight solutions:

Note Yu’s dot pattern is the upper left solution. The other seven solutions, however, also sum to 15 for their rows, columns, and diagonals.

### Bigger Square Grids and Math Squares Puzzles

If you’re ambitious, or nuts, you may wonder if this mathematical pattern holds for larger square grids. It does. While legend says Emperor Yu of China discovered magic squares, they appear in many cultures through history. The question of how to expand magic square grids from 3 cells by 3 cells was solved by Thakkura Pheru in India in 1315 AD. You can find his solution and other ways to do and teach magic squares in the links below.

### Learn More

#### Lo Shu Magic Squares

http://en.wikipedia.org/wiki/Lo_Shu_Square

#### Magic Squares (Nationellt centrum för matematikutbildning)

http://ncm.gu.se/media/maut/india/8_magic_square.pdf

http://ncm.gu.se/

#### Suzanne and Richard Alejandre’s Lo Shu Activity

http://mathforum.org/alejandre/magic.square/loshu1.html

http://mathforum.org/alejandre/magic.square/loshu2.html

#### National Council of Teachers of Mathematics (NCTM)

Includes Thakkura Pheru’s solution for larger magic square puzzles.

http://illuminations.nctm.org/Lesson.aspx?id=655

#### Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey

Some skeptical background for the Emperor Yu legend with alternate origins.

http://books.google.com/books?id=UqugpLLyZUEC&pg=PA20&lpg=PA20&dq=emperor+yu,+mystic+symbol,+turtle,+magic+squares&source=bl&ots=NGVwCmLQtS&sig=zsTWHb_9NyT5GdRo8NXLf21nozM&hl=en&sa=X&ei=itMmVPiKIcWZsQSijIHYDw&ved=0CB0Q6AEwADgK#v=onepage&q=emperor%20yu%2C%20mystic%20symbol%2C%20turtle%2C%20magic%20squares&f=false

#### Magic Square Group on Flickr

### Also In The October 2014 Issue

### Andrew Mills Talks about Bits & Bytes and How to Design Games

Andrew created Bits & Bytes, a fun card game to teach kids computing skills: logic, problem solving, and critical thinking.

### Tim Kropp Talks About Creating Games for Kids

Tim describes how he created his game company, Glide Games, and with his young son created two video games, Elevator Adventures and Subway Adventures.

### Lo Shu Magic Squares Puzzle

A 4,000 year old Chinese magic squares puzzle is both fun and a way to learn basic problem solving skills. Plus turtles.

### History of Video Games

Can you name the first video game? The first game likely was Tennis for Two in 1958 but it could be Space Wars! in 1962 or other games. It's complicated.

### Playing Harvest Moon in Japanese

The story of an English-speaking person learning a little Japanese by playing the latest Harvest Moon game, Connect to a New World, in original Japanese.

### Wendy Norman and Skype in the classroom

Wendy Norman, the Director of Social Good at Microsoft, talks about the history and features of the Skype in the classroom service for teachers.

### Game Play

You can use the concepts of game play to turn almost any task or information into a game. Assuming you can define game play.

### Maker Faire NY 2014

The Greatest Show and Tell on Earth turned out to be tons of fun for kids and parents, plus a place to wander and find new technology. Links and video.

### We need storytelling. Otherwise, life just goes on and on like the number Pi.

### October 2014 Learn More Links

Links from the bottom of all the October 2014 articles, collected in one place for you to print, share, or bookmark.

### Papyrus

This language lets you modify the Skyrim game to learn game coding plus have fun adding objects and functionality to the game.

### October 2014 News Wire

Interesting stories about computer science, software programming, and technology for the month of September 2014.

### Unity, PyGame, GamePress

Three game creation software tools you can use to create games. Includes a brief description and lots of links to these and other game creation tools.