October 4, 2001
OLD QUESTION. I remember noticing as a kid, contemplating cutting diagonally across a square lawn, that the diagonal was about one and a half times as long as the side. (The actual ratio is the square root of two, about 1.4.) Do you have an early mathematical memory?
ANSWER. The winning memory comes from Carl Eichenlaub:
When I was about 6 years old, I remember splitting a candy bar three ways with my younger sister and an older playmate named Diane. Diane was about 9 years old, and she responded to our objection that her share was larger than either of ours by explaining, "You can't make all the pieces the same size. After you break it in half the first time, you have to pick one of the two pieces to break in half to make three, and those last two pieces will be smaller."
We accepted this reasoning, being unable to answer it, but I was troubled and puzzled over the conclusion that night at home. Surely it was possible to make three equal pieces, I thought; yet her logic seemed ironclad. The answer burst upon me like a revelation: when making the first division, you make one piece twice as large as the other! Even though the goal was to produce all equal pieces, it wasn't necessary that all pieces be equal at every stage of the process, which was the assumption that had misled us.
Honorable mention goes to this account from Mark Thompson:
I think it was the summer after my 9th grade when I noticed that the product 12 x 14 was 168, which I recognized was one less than the square of 13. That made me curious, and I soon discovered that it worked in other instances, and I decided to try to prove it with algebra. Of course, it turned out to be an identity I already knew from introductory algebra, but somehow it hadn't occurred to me until then that the identity had a significance for actual numbers--and that it could be a useful aid for computation. So I memorized some more squares, and from then on when I had to multiply two even or two odd numbers whose average was below 20 or so, I did it in my head using the difference of squares.
Joshua Green recalls that after a baffling introduction to subtraction and borrowing in first or second grade, "when I woke up the next morning, suddenly everything made sense."
NEW CHALLENGE. How could you tell if the universe doubled back on itself?
September 22, 1996
It's a miracle the way the world fits together, lot against lot, road meeting road, one jagged property line meshing perfectly with the neighbor's. The whole land with its homes and factories and streets continues thousands of miles right up to the edge of the ocean, and stops right there. At hole in the ocean off the coast is filled perfectly by a small island. Under our feet the earth descends thousands of miles and stops precisely at the other side. The seas rise to where the atmosphere begins, itself then rising seamlessly, except for a few gaps perfectly filled by clouds and airplanes, to the very edge of outer space.
In like manner the moments of our lives continue seamlessly from birth to death. We sleep until the moment we wake. Our morning preparations, measured and totaled, add up precisely to our moment of departure. Our commute lasts precisely to the moment we arrive at work. Each project, conversation, break, or moment's rest, summed and totaled, tallies precisely with the extent of our working day. Every second of our free time is used for some task, recreation, or rest, and not a second more. The events of a lifetime, down to the smallest thought or chuckle, added up moment by moment, tally perfectly with its whole span. It's a miracle how it all fits.
Copyright 2001, Frank Morgan.
Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.