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Carl Gauss is one of the
smartest mathematicians who every lived. There is a famous story about
him listed
here.
It says "in
elementary school his teacher tried to occupy pupils by making
them add up the
integers from 1 to 100. The young Gauss produced the correct
answer within seconds by a flash of mathematical insight, to the
astonishment of all. Gauss had realized that pairwise addition of
terms from opposite ends of the list yielded identical intermediate
sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a
total sum of 50 × 101 = 5050 (see
arithmetic series and
summation)".
It is
amazing that Gauss did this. Still, someone who played a lot
with blocks could probably do this also. Consider the
following example: suppose you were a boy in elementary school, and asked to add numbers from 1 to
5. Now, instead of thinking about a series of numbers as a series
of digits, a little boy might think about them as a staircase of blocks.
For example, in the staircase below, the number "1" is
represented by the single block on the end of the staircase. the number "2"
is represented by the two vertically stacked blocks next to it.
"3" is represented by the 3 vertically stacked blocks, and so on
all the way through 5. Note that for staircases of this type,
a series of "n" steps ("n" can be any number) will always be "n"
wide and "n" tall.
Now, suppose
you copied the above staircase and flipped one over, like we do
below.
Now,
fit the staircases together. This makes a rectangle that is 5
tall and 6 (not 5!) wide.
The
number of blocks in a rectangle is easy to figure, it's just the
width times the height or 6 x 5 = 30. But to get the
number of blocks in just one staircase, you have to divide the area by 2. So
the sum of the digits from 1 to 5 is
5 * (5 + 1) / 2 = 5 * 6
/ 2 = 15.
If we had "n" numbers
instead of "5", then the general formula for summing this series is:
n * (n + 1)/2
So long as "n" is any
integer larger than 0.
The key thing to note here is
that the "insight" to solving this problem is not to think about "pairwise
addition of terms from opposite ends of the list yielded identical
intermediate sums".
Instead, it is more helpful to think about shapes that fit together
than numbers in a series. Another example where thinking about
shapes is helpful is
Pythagoras' theorem.
Acknowledgements:
This problem was submitted by our
father.
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