[Maths-Education] RE: Group Theory and Soduko

[Ernest, Paul]

Dear Colleagues

Thank you those who responded about my Group theory Soduko question. (The question was: Does a completed Soduko (9X9) grid embody a group (of order 9)? It certainly looks a bit like one).

I have answered the question to my satisfaction in the negative. Apologies, because my reasoning below is a bit tedious.

A completed Soduko (9X9) grid has the nos 1-9 in each row and column once and only once. Thus it defines an operation (a function) on D X D --> D (where D is set of digits 1,2,…,9).

This operation is invariant under permutation of rows or of columns, so these can be moved in the search for patterns. If this represents a group, one of the elements (digits) is the identity I. Every element d must have an inverse d’ so that dd’ = d’d = I.

What I did was to take a random Soduko table, and test each digit to see if it was the identity. I tried each digit by moving it to the top left corner (transposing rows) and then copied the order of elements in the top row so the first column matched it. If the element was indeed the identity, the location of the identity throughout the grid must be symmetric about the leading diagonal (because dd’ = d’d = I). None of the 9 digits gave this pattern.

Therefore the tested Soduko grid does not represent a group, and hence not all such grids do.

Here is the grid I tested
X
 2 5 3 1 9 4 8 7 6
 7 9 4 6 3 8 5 2 1
 8 1 6 7 2 5 4 9 3
 5 8 1 9 4 6 7 3 2
 3 2 9 8 1 7 6 5 4
 6 4 7 2 5 3 1 8 9
 1 7 8 3 6 9 2 4 5
 9 6 5 4 8 2 3 1 7
 4 3 2 5 7 1 9 6 8

Here it is transposed to test 2 as the identity element

X
 2 5 3 1 9 4 8 7 6
 5 8 1 9 4 6 7 3 2
 3 2 9 8 1 7 6 5 4
 1 7 8 3 6 9 2 4 5
 9 6 5 4 8 2 3 1 7
 4 3 2 5 7 1 9 6 8
 8 1 6 7 2 5 4 9 3
 7 9 4 6 3 8 5 2 1
 6 4 7 2 5 3 1 8 9

2 is not positioned symmetrically about the leading diagonal so 2 is not I. etc.


The remaining unanswered question is does any Soduko grid represent a group? I assume the answer is trivially yes. Just choose a digit as I and position it symmetrically within the grid. Then add more digits.


Best wishes

Paul
__________________
Paul Ernest
Emeritus Professor
University of Exeter
SELL, St Lukes, Heavitree Road
Exeter  EX1 2LU, UK

Visiting Professor, HiST-ALT, Norway
Visiting Professor, UiO, Norway

http://www.people.ex.ac.uk/PErnest/
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