How Do the 36 Officers Problem and Finite Fields Relate?

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The 36-officers problem, posed by Euler, involves arranging 36 officers into a 6x6 square such that each row and column contains one officer of each rank and regiment. This problem is related to orthogonal Latin squares and finite fields, as both require unique arrangements in rows and columns. Participants in the discussion expressed a desire for more detailed algebraic explanations and useful resources. A helpful link was shared for further exploration of the topic. The conversation highlights the mathematical significance of these arrangements and their connections to group theory.
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Can anyone explain me the problem of the 36-officers and the relation to finite fields ?

References to other explainatory website-links are also very usefull.
 
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Unfortunately, I (and, I suspect, others) have no idea what the "problem of the 36-officers" is! Could you give us more information?

Okay, I just googles on "36-officers" and "finite fields" and got this:

"Orthogonal latin squares have been considered by Euler probably for their entertaining value. He posed the problem of 36 officers: Is it possible to arrange 36 officers, each having one of six different ranks and belonging to one of six different regiments, in a square formation 6 by 6, so that each row and each file shall contain just one officer of each rank and just one from each regiment?"

Hmmm, I am reminded of the fact that the "operation table" for a group must have each member exactly once in each row and column, in order that each member have an inverse.
 
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HallsofIvy said:
Unfortunately, I (and, I suspect, others) have no idea what the "problem of the 36-officers" is! Could you give us more information?

Okay, I just googles on "36-officers" and "finite fields" and got this:

"Orthogonal latin squares have been considered by Euler probably for their entertaining value. He posed the problem of 36 officers: Is it possible to arrange 36 officers, each having one of six different ranks and belonging to one of six different regiments, in a square formation 6 by 6, so that each row and each file shall contain just one officer of each rank and just one from each regiment?"

Hmmm, I am reminded of the fact that the "operation table" for a group must have each member exactly once in each row and column, in order that each member have an inverse.


You are correct, this is exactly what I mean. Sorry for the bad description of mine. Do you have some more info concerning this problem. Don't mind if the explanaition is pure algebraic, i will try to understand. Any link to some nice sites will also be more then wellcome.


Thanks a lot
 
marlon said:
You are correct, this is exactly what I mean. Sorry for the bad description of mine. Do you have some more info concerning this problem. Don't mind if the explanaition is pure algebraic, i will try to understand. Any link to some nice sites will also be more then wellcome.

The answer to the question as Halls states is yes. What is your question?
 
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