You are here

The College Mathematics Journal Contents—May 2009

Vol. 40, No. 3, pp. 158-236


Interview With Martin Gardner

L-Tromino Tiling of Multilated Chessboards
By Martin Gardner
An n x n chessboard is called deficient if one square is missing from any spot on the board. Can all deficient boards with a number of cells divisible by 3 be tiled by bent (or L-shaped) trominoes? The answer is yes, with exception of the order-5 board. This paper deals with the general problem plus numerous related puzzles and proofs involving bent tromino tilings.

Polyomino Problems to Confuse Computers
By Stewart Coffin

Computers are very good at solving certain types combinatorial problems, such as fitting sets of polyomino pieces into square or rectangular trays of a given size. However, most puzzle-solving programs now in use assume orthogonal arrangements. When one departs from the usual square grid layout, complications arise. The author - using a computer, of course - exploits this limitation to devise puzzles that cannot easily be solved by a computer.

Puzzling Mechanisms
By Dr. M. Oskar van Deventer

The basis of a good mechanical puzzle is often a puzzling mechanism. This article will introduce some new puzzling mechanisms, like two knots that engage like gears, a chain who's links can be interchanged, and flat gears that do not come apart. It illustrates how puzzling mechanisms can be transformed into real mechanical puzzles, e.g., by introducing a maze structure.

Sets of Mutually Orthogonal Sudoku Latin Squares
By Timothy Vis, Ryan M. Petersen

A Latin square of order n is an n x n array using n symbols, such that each symbol appears exactly once in each row and column. A set of Latin squares is c ordered pairs of symbols appearing in the cells of the array are distinct. The popular puzzle Sudoku involves Latin squares with n = 9, along with the added condition that each of the 9 symbols appears exactly once in each of the 3 by 3 blocks that together tile the main array. In response to a problem in the American Mathematical Monthly, we provide two constructions for mutually orthogonal Latin squares (MOLS) that are also solution to Sudoku puzzles. We generalize this notion from n = 9 to n = k2 and construct sets of mutually orthogonal Sudoku Latin squares (MOSLS) for any integer k>1 with a lower bound on the attainable size of such a set.

Jeeps Penetrating a Hostile Desert
By Herb Bailey

Several jeeps are poised at base camp on the edge of a desert aiming to escort one of them as far as possible into the desert, while the others return to camp. They all have full tanks of gas and share their fuel to maximize penetration. In a friendly desert it is best to leave caches of fuel along the way to help returning jeeps. We solve the problem in a hostile desert where it is not prudent to leave unattended fuel deposits. A surprising result is that deeper penetration can be achieved if the jeeps stagger their departure times rather than all leaving camp together.

Three Poems
By Caleb Emmons

Flipping Triangles!
By Marc Zucker

We introduce a simple game made up of a board of coins on a triangular lattice. We then study the possibility of turning the board from one pattern of heads and tails to some other pattern. Given that a solution exists we find a precise answer to the number of solutions possible. We then generalize this to more complex boards with coins of many sides (sic) or colors.

n-Card Tricks
By Hang Chen and Curtis Cooper

Motivated by Fitch Cheney's 5-card trick, we present and analyze several n-card tricks. We determine the minimum n to perform the trick, with and without timing.

Reflections on the N + k Queens Problem
By Doug Chatham

The N queens problem is a classic puzzle. It asks for an arrangement of N mutually non-attacking queens on an N x N chessboard. We discuss a recent variation called the N + k queens problem, where pawns are added to the chessboard to allow a greater number of non-attacking queens to be placed on it. We describe some of what is known about the problem, prove a few new results, and propose some open questions.

By Gary Kennedy and Stephen Kennedy


An Interesting Property of x/π (x)
By Robert T. Harger and William L. Hightower