MATH
21 B
Mathematics Math21b Spring 2008
Linear Algebra and Differential Equations
Exhibit: latin squares
Course Head: Oliver Knill
Office: SciCtr 434

Latin squares

A nxn matrix is called a latin square if the numbers 1,...,n occur exactly once in each row and exactly once in each column of the matrix. A simple example is
 A =   | 1   2 |
       | 2   1 |
An example of a 3x3 latin square, you have encountered in the first midterm. Here is an example of a 5x5 latin square:
A = | 1 2 3 4 5 |
    | 2 3 5 1 4 |
    | 3 5 4 2 1 |
    | 4 1 2 5 3 |
    | 5 4 1 3 2 |


Suduku squares

A 9x9 matrix is a Suduku square, if it is a latin square and if additionally in each of the 9 3x3 submatrices all numbers 1..,9 occur exactly once too. Here is an example:
A = | 3 4 2 9 7 8 1 5 6 |
    | 6 9 5 2 4 1 3 7 8 |
    | 1 7 8 6 3 5 2 4 9 |
    | 7 6 3 4 9 2 5 8 1 |
    | 8 1 9 3 5 7 4 6 2 |
    | 2 5 4 8 1 6 7 9 3 |
    | 9 3 7 1 6 4 8 2 5 |
    | 4 8 1 5 2 9 6 3 7 |
    | 5 2 6 7 8 3 9 1 4 |


Magic squares

A nxn matrix is called a Magic square if it contains all integers 1,...,n2 exactly once and each row each column and each diagonal column has the property that the sum is constant. An example:
| 4 9 2 |
| 3 5 7 | 
| 8 1 6 |
Note that this square appears in the center of the above Suduku matrix which has been found by Paul Muljadi.
The magic square which appears in Durers Melancolia I is
A = | 16  3  2 13 |
    |  5 10 11  8 |
    |  9  6  7 12 |
    |  4 15 14  1 |

Questions

Something to think about:

Links



Please send questions and comments to math21b@fas.harvard.edu
Math21b | Oliver Knill | Spring 2008 | Department of Mathematics | Faculty of Art and Sciences | Harvard University