Friday, 25 March 2016

Variations on Magic Squares - Interesting Party Games

Blog Contents; Who am I?

An nxn magic square is an arrangement of the numbers from 1 to n^2 (n-squared) in an nxmatrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same and is equal to n(n^2+1)/2.


Magic squares are fascinating objects.  I have always found them intriguing - how do they construct magic squares was a question that I could never find a sensible answer to. Wiki has all the historic and mystical significance of magic squares - we shall miss this out here - you can have a look for background reading. 

Recently, I happened to stumble on a book about magic squares and decided to find out how to make one.  3x3 is easy and straightforward.  4x4 recipes on the internet, Youtube and the rest all looked very complicated.  Thankfully after a few hours of struggle, I could find my own way of easily making 4x4 pseudo-magic squares - a simple method that is trivial to remember. (Strictly - this is not a true magic square as the numbers are repeated but we shall ignore this as it still makes an excellent party game - actually more fun as some of the complexities of magic squares is taken out)
This exercise also allowed me to dig out some very interesting party games constructed around the 4x4 grid and I shall describe a couple of games here at the end of the blog.

3x3 Magic Square:   In the strict definition of a magic square, a 3x3 square must add to 15 (see the formula at the end of first para)
A general recipe to construct a 3x3 number square is as follows (I call it number square as the sum is not equal to 15 and numbers can be in any range and not necessarily from 1 to 9 as per definition of a magic square)

As you might have guessed, N=5 for a true magic square. But you can make it for any other number and choose a and b to get a good spread of numbers.  Remember 2a + b must be less than N otherwise you will get negative numbers but that is still okay - great to have a magic square with negative numbers!
4x4 Magic Square:  
A 4x4 magic square has numbers from 1 to 16 and the sum of rows, columns and diagonals should be individually equal to 34.  In the following, I show how to construct a 4x4 magic square:

This is the pure magic square as per definition. 
Now we look at some generalizations which make interesting party games:  
Party Games:  Variations on 4x4 Magic Square:  
Game 1:  I call it the A-Square  (some numbers are repeated etc.) as it is not strictly a magic square. The best way to describe the game is to play it.  Follow the instructions in the slide:  

One does not have to limit to sum being 38.  The sum can be any number - birthday at the party, for example or the wedding anniversary.  

Game 2:  This is a simpler way to make a 4x4 number square that I concocted up.  Easy to remember and only uses four numbers instead of eight so there is more repetition of numbers in the grid.  Follow the slides:






Hope you have enjoyed reading this.  

Post Script:  The purist will not like this blog.  nxn Magic square by definition is unique - so what do you do with it?  By taking liberty to go beyond the strict definition, we have got some interesting games which are quite manageable even for youngish children and of course adults will enjoy these as well.  
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