Latin Phrasebook?

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Latin Phrasebook?

Postby Mathimagics » Thu Dec 27, 2018 4:28 am

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Monty Python made the "Hungarian Phrasebook" famous (I will not buy this record, it is scratched!), but I recently had serious need of a Latin version, or to be more specific, a Latin Square Phrasebook.

This is needed for decoding published papers on the theory of Latin Squares. As we all know, Sudoku is essentially a house-extension of the Latin Square. Two sets of houses becomes three. Or, conversely, 9x9 Latin Squares are Sudoku's without boxes.

These two examples should demonstrate the need for some sort of glossary:

Nicholas J. Cavenagh, Diane Donovan, G.H.J. van Rees, A note on the completion of partial latin squares (2004)

P. Adams, R. Bean and A. Khodkar, A census of critical sets in the latin squares of order at most six, Ars Combin., 68 (2003), 203–223.

In fact, just in the titles, there are some "Latin" phrases that will only be recognisable after translation:

  • partial latin square: a puzzle. If uniquely completable (UC) to a latin square, a valid puzzle.
  • critical set: a minimal puzzle (no clue is redundant)

Less obvious ones:

  • strong critical set: a minimal puzzle that is solvable entirely by singles.
  • totally weak critical set: a minimal puzzle that has no singles.

On a historical note:
The modern Sudoku was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from Connersville, Indiana, and first published in 1979 by Dell Magazines as Number Place (the earliest known examples of modern Sudoku).


Now this is indeed curious, because the idea of a critical set in Latin Squares dates from the same period. The first published paper on this topic was in 1978.

Did Howard get his idea from reading one these papers? :?:
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Re: Latin Phrasebook?

Postby Serg » Thu Dec 27, 2018 10:24 am

Hi, Mathimagics!
Mathimagics wrote:.
This is needed for decoding published papers on the theory of Latin Squares. As we all know, Sudoku is essentially a house-extension of the Latin Square. Two sets of houses becomes three. Or, conversely, 9x9 Latin Squares are Sudoku's without boxes.

These two examples should demonstrate the need for some sort of glossary:

Nicholas J. Cavenagh, Diane Donovan, G.H.J. van Rees, A note on the completion of partial latin squares (2004)

P. Adams, R. Bean and A. Khodkar, A census of critical sets in the latin squares of order at most six, Ars Combin., 68 (2003), 203–223.

In fact, just in the titles, there are some "Latin" phrases that will only be recognisable after translation:

  • partial latin square: a puzzle. If uniquely completable (UC) to a latin square, a valid puzzle.
  • critical set: a minimal puzzle (no clue is redundant)

Less obvious ones:

  • strong critical set: a minimal puzzle that is solvable entirely by singles.
  • totally weak critical set: a minimal puzzle that has no singles.

What about isotopism and parastrophy?
Mathimagics wrote:On a historical note:
The modern Sudoku was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from Connersville, Indiana, and first published in 1979 by Dell Magazines as Number Place (the earliest known examples of modern Sudoku).


Now this is indeed curious, because the idea of a critical set in Latin Squares dates from the same period. The first published paper on this topic was in 1978.

Did Howard get his idea from reading one these papers? :?:

I think there is no connection between sudoku invention and Latin Square critical sets. Otherwise Howard Garns would invent Latin Square like puzzle. But he introduced new "dimension" - boxes.

One more paper about Latin Squares (2010):
Douglas S. Stones, "The many formulae for the number of Latin rectangles".

The paper contains nice history overview. It's curious, but some formulae for the number of Latin rectangles (squares) are more difficult in calculation than direct "brute force" enumerations!

I don't understand why, but it looks like australian mathematicians dominate now in Latin Squares investigations :?:

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Re: Latin Phrasebook?

Postby Mathimagics » Thu Dec 27, 2018 12:05 pm

Hi Serg,
Serg wrote:I think there is no connection between sudoku invention and Latin Square critical sets. Otherwise Howard Garns would invent Latin Square like puzzle. But he introduced new "dimension" - boxes.

Really? We know that he had a knack for puzzle creation, so having got the idea that a critical set was in fact a puzzle, he could well have tried to give it a twist (so as not to be just copying the idea), and came up with the even better idea of adding that 3rd dimension.

Serg wrote:I don't understand why, but it looks like australian mathematicians dominate now in Latin Squares investigations :?:

We are just naturally gifted ? :roll:
Combinatorics is easier than pure math? :?
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