$page_title       = "Exhibit: latin squares";
$page_description = "Exhibit on a problem of latin squares in Math 21b, Linear Algebra and Applications";
${base}           = "../../";
$robots = "all";

sub main_page {
print OUTPUT<<"EOF"



<center><table class="exhibits"><tr valign=top><td>
<h2> Latin squares</h2>
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 
<pre>
 A =   | 1   2 |
       | 2   1 |
</pre>
An example of a 3x3 latin square, you have encountered in the first midterm. 

Here is an example of a 5x5 latin square:

<pre>
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 |
</pre>
</td></tr></table></center><br><br>

<h2> Suduku squares </h2>

<center><table class="exhibits"><tr valign=top><td>
A 9x9 matrix is a <b>Suduku square</b>, 
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:

<pre>
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 |
</pre>
</td></tr></table></center><br><br>

<h2> Magic squares </h2>

<center><table class="exhibits"><tr valign=top><td>
A nxn matrix is called a <b>Magic square</b> if it contains all
integers 1,...,n<sup>2</sup> exactly once and each row each column
and each diagonal column has the property that the sum is constant.

An example: 

<pre>
| 4 9 2 |
| 3 5 7 | 
| 8 1 6 |
</pre>

Note that this square appears in the center of the above Suduku matrix
which has been found by 
<a href="http://www.muljadi.org/MagicSudoku.htm">Paul Muljadi</a>.

<center>
<a href="melancolia.jpg"><img src="melancolia-s.jpg"></a>
</center>

The magic square which appears in Durers Melancolia I is

<pre>
A = | 16  3  2 13 |
    |  5 10 11  8 |
    |  9  6  7 12 |
    |  4 15 14  1 |
</pre>

</td></tr></table></center>

<h2> Questions </h2>

Something to think about: 

<ul>
 <li> Which of the three classes of matrices has the property that its transpose A<sup>T</sup> has the same property too?
 <li> Can you find an eigenvalue and eigenvector in each of the above three classes of matrices? 
 <li> Can you see why every Suduku matrix has a nontrivial kernel and is therefore not invertible? 
</ul>

<h2> Links </h2>

<center><table class="exhibits"><tr valign=top><td>
<ul>
 <li><a href ="http://en.wikipedia.org/wiki/Magic_Square">Magic square entry in Wikipedia</a>
</ul>
</td></tr></table></center><br><br>

EOF
}

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