With some 8-year-olds, you know, you just don't know, you know?

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“At [a] party with 100 guests, everyone shakes hands [just once] with everyone else. How many handshakes take place?”
 — Norman Geras

The puzzle that Norm posed (via Marcus du Sautoy) happens to be equivalent to adding the sequence of numbers from 1 to 99. This, in turn, reminded me of a wonderful experience I had as a freshman at Cornell University in 1959.

I was in a large lecture hall attending an applied math class, where the professor was busily writing out some proof on the blackboard, accompanied by his spoken commentary. Suddenly, he cavalierly said, "And, since the sum of all integers from 1 to N is N(N+1)/2 ..." as he substituted that formula and continued writing out his proof. Of course this was a trap. And, of course, he snared one unsuspecting student, who raised the objection, "But professor, that isn't at all obvious to me."

The professor turned from the blackboard, and with a smile on his face he replied, "Why, that formula was derived by an 8-year-old boy!" The puzzled student was humiliated in front of about 150 of his peers, so the professor continued the story of that 8-year-old boy:

In a one-room schoolhouse, a teacher wanted to keep his younger students busy while he spent some time with a few of the oldest students. He told the youngsters to add up all the numbers from 1 to 99. When finished, each student was to stack his slate on the teacher's desk (the year was 1785). This would no doubt keep the youngsters busy for some time.

As the youngsters proceeded to add the numbers, the youngest student (the 8-year-old boy) just sat for a few minutes contemplating the assignment. Then he simply wrote the answer on his slate and placed it on the teacher's desk. After a longer period of time, all the other young students stacked their slates on top of the youngest student's slate.

When the teacher was finished with his older students, he examined the stack of slates on his desk. Of course, only the bottom slate had one number on it, and it was the only correct answer. It was then that the teacher realized he had a genius in his class. His name was Carl Friedrich Gauss, the Prince of Mathematicians.

Gauss supposedly visualized the problem as follows: He imagined writing the sequence 1 through 99 down one side of his slate. Then he imagined writing the sequence in reverse order down the other side. Next he observed that the addition of each pair of numbers in all 99 rows of numbers equaled 100. Hence, the two columns together yielded a grand total of 9900, which, however, was exactly twice the sum of one column. Thus, adding all the numbers 1 through 99 equaled 9900/2 equals 4950.

Another version of this story has Gauss deriving his formula when his mother needed to know how many candles to buy for Hanukkah. This story, however, is known to be apocryphal, since Gauss was not Jewish, and I just made it up. BTW, the answer is 44 candles (it's a trick question).

Post #806 With some 8-year-olds, you know, you just don't know, you know?