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sci.math > Alcuin's sequence
Schrift mit festem Zeichenabstand - Proportionalschrift
Hermann Kremer   18 Jun. 2004 20:49     Optionen anzeigen
Newsgroups: sci.math, alt.math.recreational
Von: "Hermann Kremer" <>Nachrichten von diesem Autor suchen
Datum: Fri, 18 Jun 2004 20:49:32 +0200
Lokal: Fr 18 Jun. 2004 20:49
Betreff: Re: Alcuin's sequence
Antwort an Autor | Weiterleiten | Drucken | Thread anzeigen | Original anzeigen | Missbrauch melden

Tony King schrieb in Nachricht

Hi Tony,
   ... late comes it, but it comes ...

>I am trying to do some research on alcuin's sequence, but all the
>references that I can find quote the terms of the sequence as the
>coefficients in the Maclaurin expansion of [(1-x^2)(1-x^3)(1-x^4)]^-
>1. I find this surprising, as I believe the sequence to have been
>named after Alcuin of York (735-804) who lived many years before
>Maclaurin expansions were discovered.

>This sequence also gives the number of different triangles that have
>integral sides and perimeter n. I would think that it is much more
>likely that this is what Alcuin discovered, and years later someone
>found that the above expansion acted as a generating function.

>That is, of course, if the Alcuin who the sequence is named after is
>Alcuin of York and not someone else with the same name.

>I would be most grateful if you could either confirm or refute this,
>or point me in the direction of any references.

It is in fact named after Alcuin of York [1].

Alcuin's sequence [2] does not only give the number of incongruent triangles
of given perimeter N with integer sides, but also the number of ways in which N
empty casks, N casks half-full of wine and N full casks can be distributed to
3 persons in such a way that each one gets the same number of casks and the
same amount of wine, and that's the link to Alcuin.

Alcuin is, more or less reluctantly, considered as the author of the Latin
manuscript "Propositiones ad acuendos juvenes", i.e. "Problems for the mind-
sharpening of youngsters", written around 800 AD in France. Roughly a dozen
copies have survived the 1200 years till now and are preserved in libraries in
Rome, Vienna, Munich, Karlsruhe, London, Leiden and Montpellier.
The manuscript contains a collection of 53 mathematical problems without any
theory, but with the solutions, some of which are incorrect. Both the complete
Latin Text and translations may be found at several websites [3].

In Problem No. 12 of the "Propositiones", 10 empty bottles, 10 bottles half-full
of oil and 10 full bottles are to be fairly shared among three sons. Counting
only incongruent solutions, i.e. solutions that are no mutual permutations of
others, there are 5 ones:

:  E1  H1  F1  |  E2  H2  F2  |  E3  H3  F3
: -------------+--------------+-------------
:   0  10   0  |   5   0   5  |   5   0   5   <-- Alcuin's solution
: -------------+--------------+-------------
:   1   8   1  |   4   2   4  |   5   0   5
: -------------+--------------+-------------
:   2   6   2  |   3   4   3  |   5   0   5
: -------------+--------------+-------------
:   3   4   3  |   3   4   3  |   4   2   4
: -------------+--------------+-------------
:   4   2   4  |   2   6   2  |   4   2   4
: -------------+--------------+-------------

where  E_i, H_i, F_i  means the number of empty, half-full and full bottles for
the i-th son.
Alcuin gave only one of the solutions, namely the first one above.

In 1201, Leonardo da Fibonacci published the similar problem of distributing
7 empty, 7 half-full and 7 full barrels of wine fairly to three persons, and he
also gave only one of the two solutions.

Likely the first author who noticed the existence of more than one solution for
problems of this kind was Claude Gaspard Bachet de Meziriac, best known as
the earliest translator of Diophant's works from Greek into Latin, who published
a book on recreational mathematics in 1612.
More on the history of such problems may be found in David Singmaster's gigantic
bibliography on the history of recreational mathematics [4].

The relations between fair sharing of barrels among three persons and triangles
with integral sides was established by David Singmaster in the paper:

  D. Singmaster: Triangles with Integer Sides and Sharing Barrels.
  The College Mathematics Journal 21, No. 4 (1990), p. 278 - 285 ,

a copy of which may be downloaded as a PDF file from [5].

The equivalence of the two problems can easily be seen by observing that
for any triangle with sides  a, b, c  and circumference  N  the relation

  a + b + c = N

and the inequalities

  a  <=  b + c
  b  <=  c + a
  c  <=  a + b

hold true, and by adding  a  to the first inequality etc., one gets the

  a  <=  N/2
  b  <=  N/2
  c  <=  N/2 .

For the fair barrel sharing with 3 persons, on the other hand, the

  E1 + E2 + E3 = H1 + H2 + H3 = F1 + F2 + F3 = N
  E1 + H1 + F1 = E2 + H2 + F2 = E3 + H3 + F3 = N

trivially hold, and, as Singmaster proved, the relations

  E1   =  F1
  E2   =  F2
  E3   =  F3

  E1  <=  N/2
  E2  <=  N/2
  E3  <=  N/2

must additionally be valid, and the latter three are just the triangle
inequalities from above.

In 1993, Dominic Olivastro published the book

  D. Olivastro: Ancient Puzzles. Classic Brainteasers and Other
  Timeless Mathematical Games of the Last 10 Centuries.
  New York: Bantam Books, 1993

and in the chapter on fair sharing problems he introduced the sequence
of the number of solutions for different values of  N,  which he called A_q,
 "... in honour of Alcuin ...".

Following Olivastro's hommage to Alcuin, Neil Sloane seems to have coined
the name "Alcuin's sequence" for the OEIS, and from there it found its
way into Eric Weissteins "Mathworld".

Thanks to Jutta Gut and Francisco Salinas for their help.




Latin text with English translation:
Latin text with German translation:   -->  Skript
Latin text with Italian and Spanish translations:
Portuguese translation:

[4]   -->  Paragraph 7.H.1

[5] contains a link to Singmaster's paper:
  -->  teacher handbook  -->  Getting Information --> READING ...

Best regards

>Many thanks

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