COMPUTING SCIENCE
Gauss's Day of Reckoning
A famous story about the boy wonder of mathematics has taken on a life of its own
Brian Hayes
Doing It the Hard Way
So much for how the prodigious Carl Friedrich Gauss solved the
problem. What about the rest of the students in the class? Let me
invite you to take a sheet of paper and actually try adding the
numbers from 1 to 100.
Finished? Already?
What I discovered when I tried this experiment is that it's really
hard to do it the hard way. You may set out to plod dutifully
through all the addition operations, but shortcuts present
themselves even when you're not looking for them. Suppose you adopt
the standard primary-school algorithm, writing down all 100 numbers
in a tall column and then starting work on the units digits. After
the first 10 digits, the partial sum is 45; the next 10 digits add
another increment of 45, bringing the partial sum to 90; then 45
more makes 135, and so on. How far would a student get in this
process before recognizing a repetitive pattern? On turning to the
tens digits, the pattern is even harder to miss: There are ten 1s
followed by ten 2s, then ten 3s, etc. Surely any student who has the
skills to complete this task at all would not add those repeated
numbers one by one. A more likely strategy would be the one
Sartorius implied when he wrote "count, multiply and
add"—the phrase that Helen Worthington Gauss reduced to
mere "adding."
On a small slate or a sheet of paper, it's difficult to write 100
numbers in a column, and so students would likely break the task
down into subproblems. Suppose you start by adding the numbers from
1 to 10, for a sum of 55. Then the sum of 11 through 20 is 155, and
21 through 30 yields 255. Again, how far would you continue before
spotting the trend?
Admittedly, these shortcuts can't match the elegance and ingenuity
of Gauss's method. They are tied to the decimal representation of
numbers, and they also don't generalize as well to arithmetic
progressions other than lists of consecutive integers. But they do
remind us that there's usually more than one good way to solve a problem.
I suspect that only one kind of student would ever be likely to add
the numbers from 1 through 100 by performing 99 successive
additions—namely, a student using a computer or a programmable
calculator. And for that student, the simplest strategy
might in fact be the best one.
We can hope that a modern Büttner—deprived of his whip,
of course, and teaching in a classroom where computers have replaced
slates—would not be drilling students on skills of such
dubious utility as adding up a long series of numbers by hand. But
the new Büttner just might ask his pupils to write a program to
calculate the sum of any arithmetic progression. A new Gauss, with
the same keen insight, could create a very efficient program based
on the pairing idea—and that feat still deserves the highest
admiration. But the modern Gauss might not be the first to fling his
or her laptop on the table and cry "There it lies!"
Writing that clever program—and testing and debugging it, and
proving its correctness—would be no quicker than writing the
straightforward step-by-step version. In this respect, technology
may be something of an equalizer.
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