## Leonhard Euler

Everything about Sudoku that doesn't fit in one of the other sections

### Leonhard Euler

I was browsing some Su Doku books the other day, and I came across one that in the intro said that
Su Doku was invented by Leonhard Euler in 1783

I have now updated my site - but I was wondering; as I hadn't seen this statement anywhere else; has anybody heard this "fact" too?
heebiejeebieclub

Posts: 6
Joined: 15 August 2005

This is rubbish, I'm afraid. Euler discussed things called "Latin Squares", see

http://www.cut-the-knot.org/arithmetic/latin.shtml

If you drop the boxes requirement in sudoku, then you have a Latin square.
Any sudoku grid is a Latin square, but not vice-versa.

In other words, a 9x9 Latin square has each digit 1-9 in each row and column.

I don't think there is any evidence that Euler discussed sudoku.
Moschopulus

Posts: 256
Joined: 16 July 2005

Yes. It is ridiculous of course.

a) The name "Sudoku" was coined by Nikoli publications.

b) Sudoku is a variation of Number Place puzzles, invented by Dell Magazine.

c) Number Place is one of many different logic puzzles in which the *solutions* are Latin Squares. For example, the logic puzzle called "Skyscrapers" is actually much closer to a Latin Square, as unlike Sudoku, it doesn't use BOXES, just ROWS and COLUMNS.

d) What Euler invented *wasn't even a puzzle*! There is no evidence he ever presented partly filled in Latin Squares to the public with the challenge to complete the square.

e) The solution to Sudoku is a Latin Square, but NOT a Euler Square. It is highly unlikely that Euler originated the Latin Square, only that he created the Graeco-Latin square (aka Euler Square) from it.

He wrote about "A New Kind of Magic Square" refering to his Graeco-Latin squares, NOT Latin Squares.

Two order 3 Latin Squares:

Code: Select all
`a b cc a bb c a1 2 32 3 13 1 2`

... combined form a graeco-latin square or Euler square:

Code: Select all
`a1 b2 c3c2 a3 b1b3 c1 a2`

Each letter and digit appear once in each row and column. Further, each digit is paired with each letter exactly once.

f) It strains credulity that the relatively obvious -- and useful -- construction of a Latin Square did not predate the discovery of Magic Squares -- which go back many centuries. It would be as if the discovery of multiplication preceded addition. It's simply impossible to study Magic Squares in any depth without using Latin Squares!

For example, this order 4 magic square (known prior to Euler's birth):
Code: Select all
`  1  6  11  16  7  4  13  10 12 15   2   5 14  9   8   3`

... is actually a Graeco-Latin Square created by combining these two Latin Squares:
Code: Select all
`  1   2   3   4  3   4   1   2  4   3   2   1  2   1   4   3`

and
Code: Select all
`  0   4   8  12  4   0  12   8  8  12   0   4 12   8   4   0`

a=0, b=4, c=8, d=12

a1 b2 c3 d4
b3 a4 d1 c2
c4 d3 a2 b1
d2 c1 b4 a3

What does that have to do with Sudoku?

A better question is, why isn't anyone jumping on the graeco-Latin squares and make Sudoku variants with them?

http://mathworld.wolfram.com/LatinSquare.html
http://mathworld.wolfram.com/EulerSquare.html
tso

Posts: 798
Joined: 22 June 2005

If you made a Graeco-Latin sudoku variant, would it be any different to just doing two sudokus at once? Would you be able to utilise the fact that every letter was paired once with every number?
PaulIQ164

Posts: 533
Joined: 16 July 2005

Thanks guys, I will update my site!
heebiejeebieclub

Posts: 6
Joined: 15 August 2005

I believe it is possible to construct a 9x9 Graeco-Latin square. For any size that can be made, there will be only a tiny fraction of how many Latin squares there are. I'd guess that far fewer clues would be needed for a unique solution.
tso

Posts: 798
Joined: 22 June 2005

tso wrote:I believe it is possible to construct a 9x9 Graeco-Latin square. For any size that can be made, there will be only a tiny fraction of how many Latin squares there are. I'd guess that far fewer clues would be needed for a unique solution.

that could also be almost impossible to solve, when you just have a
few clues.
dukuso

Posts: 479
Joined: 25 June 2005

tso wrote:e) The solution to Sudoku is a Latin Square, but NOT a Euler Square. It is highly unlikely that Euler originated the Latin Square, only that he created the Graeco-Latin square (aka Euler Square) from it.

He wrote about "A New Kind of Magic Square" refering to his Graeco-Latin squares, NOT Latin Squares.

I think Euler is usally given as the first one having examined
Latin Squares, not only the Graeco-Latin ones.
You can't do Graeco-Latin without Latin.
And why is it called "Latin" ? I think it's because Euler
used Latin characters for it. Or was the name given before Euler ?
Is there any paper about Latin-Squares before Euler ?
dukuso

Posts: 479
Joined: 25 June 2005

dukuso wrote:
tso wrote:I believe it is possible to construct a 9x9 Graeco-Latin square. For any size that can be made, there will be only a tiny fraction of how many Latin squares there are. I'd guess that far fewer clues would be needed for a unique solution.

that could also be almost impossible to solve, when you just have a
few clues. You'll have to answer the question:
is there a an ortogonal pair with this property ?
Many such questions are unsolved AFAIK
dukuso

Posts: 479
Joined: 25 June 2005

dukuso wrote:
tso wrote:e) The solution to Sudoku is a Latin Square, but NOT a Euler Square. It is highly

unlikely that Euler originated the Latin Square, only that he created the Graeco-Latin square (aka Euler Square)

from it.

He wrote about "A New Kind of Magic Square" refering to his Graeco-Latin squares, NOT Latin Squares.

I think Euler is usally given as the first one having examined
Latin Squares, not only the Graeco-Latin ones.
You can't do Graeco-Latin without Latin.
And why is it called "Latin" ? I think it's because Euler
used Latin characters for it. Or was the name given before Euler ?
Is there any paper about Latin-Squares before Euler ?

Accoding to CHRONOLOGY OF RECREATIONAL MATHEMATICS

by David Singmaster

1) Latin squares were used on amulets circa 1200 in the medieval Islamic world.

2) Suveur finds first magic cube and invents(?) Latin squares in 1710.

3) 1782 Euler on Latin Squares.

I've seen no evidence that Euler either invented or discovered "latin" squares, nor did he claim to. His research *used* the already known constructions now known as latin squares. The term "latin square" is most certainly to be attributed to him, whether or not he used the term himself. His inventions in this area were mutually orthoganal latin squares aka Graeco-Latin or Euler Squares. The title of his book on the subject, "research on a new species of magic square" implies this to me. A magic square is made up of cells *all of which are different*. A Latin square does fit this description, but a Graeco-Latin square does. A brilliant, prolific mathematician like Euler wouldn't refer to a Latin Square as "new" or "magic square".

In other words, Nikoli's Sudoku and and Euler's Graeco-Latin squares share a common ancestor -- the Latin square.

But they have nothing to do with each other. The press has it wrong. Euler never should have been mentioned in connection with this puzzle.
tso

Posts: 798
Joined: 22 June 2005

tso wrote:In other words, Nikoli's Sudoku and and Euler's Graeco-Latin squares share a common ancestor -- the Latin square.

But they have nothing to do with each other. The press has it wrong. Euler never should have been mentioned in connection with this puzzle.

well, even if Euler didn't "invent" latin squares, he contributed to them
considerably so it's not so far fetched to mention him
when it comes to latin squares.
I would like however to mention the QCP and QWH -problems
in relation to sudoku

QCP: quasigroup completion problem
QWH: quasigroup with holes (-problem)

searching google for "quasigroup" and "sudoku" gave e.g.:

http://www.icparc.ic.ac.uk/~hs/sudoku.pdf

http://216.239.59.104/search?q=cache:d-Jq0edGsZEJ:www.cs.cornell.edu/gomes/TALKS/gomes-aaas-2005.pdf+quasigroup+sudoku&hl=de&ie=UTF-8
dukuso

Posts: 479
Joined: 25 June 2005

basic conclusion: write to the publishing company of this book and let 'em have it!
lobby__boy

Posts: 29
Joined: 08 June 2005

dukuso wrote:well, even if Euler didn't "invent" latin squares, he contributed to themconsiderably so it's not so far fetched to mention him
when it comes to latin squares.

Agreed. But not when talking about Sudoku or Number Place. Latin Squares lead to both Graeco-Latin Squares aka Euler's Squares as well as Euler's other work with Latin and Magic Squares. They also lead to Number Place and Sudoku. But NOTHING that Euler created or wrote lead to Sudoku. They simply have a common ancestor. They are sisters, separated at birth, not mother-daughter. If my mother did not exist, I would not.
tso

Posts: 798
Joined: 22 June 2005

One last thing about Euler. Some of his works are online here:

http://www.eulerarchive.org/

His complete "On Magic Squares" text is available in original language and English here: http://math.dartmouth.edu/~euler/pages/E795.html

"Investigations on a new type of magic square" isn't online, but a translation is being prepared now and should be available "soon". A summary is here: http://math.dartmouth.edu/~euler/pages/E530.html

"Summary:

Euler takes the concept of Latin square (an n by n square containing the numbers 1 through n, each of which appears exactly once in each row and in each column of the square) and generalizes it to a Greco-Latin square (essentially 2 Latin squares laid over each other in a special way). The primary question the paper addresses is: what sizes of Greco-Latin squares are possible to construct?

Euler gives hundreds of examples of Latin and Greco-Latin squares and takes many lengthy detours through this paper, asking questions about Latin squares in which the diagonals also satisfy the "Latin square" property. In the end, he argues, but fails to prove rigorously, that no Greco-Latin square of size 4k + 2 can ever be contructed. "

Take a look what SwissInfo says here.

They claim that Euler invented Sudoku!

"Sudoku grids have a long, international and not always certain history, but one thing is definite: they are not Japanese.

The puzzles were in fact invented 222 years ago by a Swiss maths genius, Leonhard Euler, who dominated 18th-century mathematics and whose collected works fill 75 volumes."

Do they not have access to the library? Do they even have internet access?

http://www.daviddarling.info/encyclopedia/L/Latin_square.html

http://anduin.eldar.org/~problemi/singmast/recchron.html
tso

Posts: 798
Joined: 22 June 2005

agreed that he didn't invent sudoku. OK ?
nor did he invent Latin Squares (though maybe was the first to
use that name ?)

but he has worked on similar things, so there is some connection. OK ?

who cares about Swiss info ...
Well, Euler was a Swiss guy, so they are maybe a bit "proud" of him.

And, did you notice, they claim copyright for that story.
So you can even copyright lies. Where is this going ....?
dukuso

Posts: 479
Joined: 25 June 2005

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