[FOM] history of the 'sum and product' riddle

Hans van Ditmarsch hans at cs.otago.ac.nz
Thu Jun 9 01:20:26 EDT 2005

Dear reader,

The 'Sum and Product' riddle has been known to the academic community
since the late 1960s. It is as follows:

Let x and y be two integers with 1 < x < y and x+y =< 100.
S is given their sum x+y and P is given their product x*y. All this is
commonly known to S and P. The following conversation now takes place:

- P:  I do not know the two numbers.
- S:  I knew that you didn't know the two numbers.
- P:  Now I know the two numbers.
- S:  Now I know the two numbers.

Rineke Verbrugge, University of Groningen, and Hans van Ditmarsch,
University of Otago, are trying to uncover part of the origin of this
riddle. The original publication is by Hans Freudenthal in the (Dutch)
"Nieuw Archief voor Wiskunde" (New Archive for Mathematics), and is from
1969. It was brought to the attention of the AI community by John
McCarthy's "Formalization of Two Puzzles Involving Knowledge" which is
dated 1978-1981 and published in a 1990 collection of his papers (V.
Lifschitz (editor), Formalizing Common Sense: Papers by John McCarthy,
Ablex). Later publications on this riddle include work by Lee Sallows,
Giovanni Panti, Martin Gardner, Jan Plaza, and various others.

Given recent developments in the logics of knowledge, such as the
development of logics for knowledge change, and the availability of tools
for model checking knowledge, there is renewed interest in analyses of
this problem. There are two questions about the history of the Sum and
Product problem to which we do not know the answer:

- Did the problem originate with Hans Freudenthal in the NAW 1969
formulation, or did he pass it on from an older source?

- How did the problem travel from this Dutch original source to its
"alleged appearance on a bulletin board at Xerox Parc" in the early 1970s?
This was the source of the problem for John McCarthy, and at that time he
was unaware of the Dutch origin of the problem (personal communication).

Concerning an answer to the first question: this is most likely a fruit of
Hans Freudenthal's fertile brain. Or otherwise shrouded in mystery. We did
not find any clue to the origin of this problem in any of Hans
Freudenthal's writings, nor was this available at the Dutch Freudenthal
Institute for Mathematics Education.

Concerning an answer to the second question: We have two hunches on how
the jump might have been made from Europe to America, in the early 1970s.
Either a Dutch academic, mathematician or computer scientist, visited
Xerox Parc at that time, or, vice versa, a US academic visited Holland in
the early 1970s and brought the riddle back home.

But we would much appreciate any individual coming forward with more
information than just that, and the main reason for our public request is
our hope to get an answer to the second question.

It may be relevant to observe that we have been in contact with the
researchers mentioned above (with the exception of Hans Freudenthal -
although Hans van Ditmarsch, who was at that time an undergraduate math
student in Utrecht, remembers him well).

Yours sincerely,
Rineke Verbrugge (rineke at ai.rug.nl) and Hans van Ditmarsch
(hans at cs.otago.ac.nz)

* Hans van Ditmarsch                  Department of Computer Science *
* hans at cs.otago.ac.nz                 University of Otago            *
* +64 3 479 8475   fax 479 8529       PO Box 56                      *
* www.cs.otago.ac.nz/staffpriv/hans/  Dunedin            New Zealand *

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