September 15, 2011

Mathematical Morsels I

The American Mathematical Monthly has a long tradition of publishing problems, going all the way back to its first issue in 1894.


In a letter that appeared in the debut issue, Monthly coeditors B.F. Finkel and J.M. Colaw argued the value of posing and solving mathematical problems.

"While realizing that the solution of problems is one of the lowest forms of Mathematical research . . . its educational value cannot be over estimated," they wrote. "It is the ladder by which the mind ascends into the higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem."

Readers of the Monthly continue to look forward to fresh doses of perplexity and ingenuity with the arrival of each new issue, and the problems sections of past issues remain a treasure house of mathematical gems to revisit and ponder anew.

Several decades ago, Ross Honsberger (University of Waterloo) chose scores of "elementary" problems, originally posed in the Monthly, to appear in a volume titled Mathematical Morsels (Mathematical Association of America, 1978). He wanted to illustrate that "all kinds of simple notions are full of ingenuity."

"Mathematics abounds in bright ideas," Honsberger wrote. "No matter how long and hard one pursues her, mathematics never seems to run out of exciting surprises. And by no means are these gems to be found only in difficult work at an advanced level."

Here are four classic problems from this selection for you to try.

THE FERRY BOATS

Two ferry boats ply back and forth across a river with constant speeds, turning at the banks without loss of time. They leave opposite shores at the same instant, meet for the first rime some 700 feet from one shore, continue on their way to the banks, return and meet for the second time 400 feet from the opposite shore. [Without using pencil and paper] determine the width of the river.

Source: American Mathematical Monthly, 1940, p. 111, Problem E366, proposed by C.O. Oakley, Haverford College, solved by W.C. Rufus, University of Michigan.

ROLLING A DIE

A normal die bearing the numbers 1, 2, 3, 4, 5, 6 on its faces is thrown repeatedly until the running total first exceeds 12. What is the most likely total that will be obtained?

Source: American Mathematical Monthly, 1948, p. 98, Problem E771, proposed by C.C. Carter, Bluffs, Illinois, solved by N.J. Fine, University of Pennsylvania.

RED AND BLUE DOTS

Consider a square array of dots, colored red or blue, with 20 rows and 20 columns. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a segment of their common color; adjacent dots of unlike colors are joined by a black segment. There are 219 red dots, 39 of them on the border of the array, none at the corners. There are 237 black segments. How many blue segments are there?

Source: American Mathematical Monthly, 1972, p. 303, Problem E2344, proposed by Jordi Dou, Barcelona, Spain.

A PERFECT 4TH POWER

Prove that the product of 8 consecutive natural numbers is never a perfect fourth power.

Source: American Mathematical Monthly, 1936, Problem 3703, proposed by Victor Th├ębault, Le Mans, France, solved by the Mathematics Club of the New Jersey College for Women, New Brunswick, New Jersey.

Reference:

Honsberger, R. 1978. Mathematical Morsels. Mathematical Association of America.

SOLUTIONS

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