"For example the story of Gauss and the sum 1+2+3+...+100 shows a
mathematical point and also demonstrates to the student that such
apparently difficult problems yield to alternative approaches and lie
within the grasp of quite young students. This is important. But it is
worth saying that the story may not be quite as true as Bell makes it."
In this instance, that would be unfair to Bell. His account there
is almost a word-for-word translation of the version given in
Sartorius von Waltershausen, _Gauss zum Gedaechtnis_ (1856). This is
a memorial volume written by a Goettingen friend and colleague
shortly after Gauss died. Of this specific anecdote, Sartorius says
that Gauss "told it repeatedly with great joy and liveliness". It is
of course possible that Gauss himself misremembered; but it seems
to me that this is exactly the sort of surprise triumph that a
9-year-old would remember vividly, so I think no doubts need be
expressed.
One thing that should be said is that the sum is not specified
as 1 + 2 + ... + 100; Sartorius wrote only "the sum of an
arithmetic progression". Bell has that right, adding a definition
and giving a more complicated example.
Turning to pure speculation, I note that the problem was supposed
to keep the students busy for an hour. If it was just
1+2+...+100, that means allowing almost 40 seconds for each addition
of a 2-digit number, which sounds very slow to me. So I would
guess that a more complicated progression was involved (where
computing the individual terms would also be part of the exercise).
William C. Waterhouse
Penn State