## Mathematical gems, Volume 2 |

### From inside the book

Results 1-3 of 46

Page 22

Math. Monthly, 51 (1944) 84. 2. L. F. Toth, New proof of a minimum property of the

regular n-gon, Amer. Math. Monthly, 54 (1947) 589. 3. For Thompson's solution,

see Amer. Math. Monthly, 58 (1951) 38,

theorem, Amer. Math. Monthly, 23 (1916) 161. 5. C. N. Schmall, Amer. Math.

Monthly, 32 (1925) 99 (

Sierpinski, Theory of Numbers, Warszawa, 1964. 2. 22 MATHEMATICAL GEMS II

.

Math. Monthly, 51 (1944) 84. 2. L. F. Toth, New proof of a minimum property of the

regular n-gon, Amer. Math. Monthly, 54 (1947) 589. 3. For Thompson's solution,

see Amer. Math. Monthly, 58 (1951) 38,

**Problem**E913. 4. R. A. Johnson, A circletheorem, Amer. Math. Monthly, 23 (1916) 161. 5. C. N. Schmall, Amer. Math.

Monthly, 32 (1925) 99 (

**Problem**3080, posed in 1924, p. 255). References 1. W.Sierpinski, Theory of Numbers, Warszawa, 1964. 2. 22 MATHEMATICAL GEMS II

.

Page 23

Everybody knows the leap-frog jump in the game of checkers. There is an

interesting

begins by arranging a number of men in the starting zone, which consists of the

half-plane of lattice points on and below the x-axis. The object is to get a man as

far as possible above the x-axis by checker-jumps in the directions of the lattice

lines (diagonal jumps are not allowed). The

number of men ...

Everybody knows the leap-frog jump in the game of checkers. There is an

interesting

**problem**about checker-jumping on the lattice points of a plane. Onebegins by arranging a number of men in the starting zone, which consists of the

half-plane of lattice points on and below the x-axis. The object is to get a man as

far as possible above the x-axis by checker-jumps in the directions of the lattice

lines (diagonal jumps are not allowed). The

**problem**is to determine the leastnumber of men ...

Page 136

The

ingenuity. In this essay we take up two solutions to the interesting fifth

the 1971 competition. In a game, one scores on a turn either a points or b points,

a and b positive integers with b < a. Given that there are 35 nonattainable

cumulative scores, and that one of them is 58, what are the values of a and bl

Cumulative scores have the form ax + by, x and y nonnegative integers, obtained

from scoring a on ...

The

**problems**posed are very challenging and often require considerableingenuity. In this essay we take up two solutions to the interesting fifth

**problem**ofthe 1971 competition. In a game, one scores on a turn either a points or b points,

a and b positive integers with b < a. Given that there are 35 nonattainable

cumulative scores, and that one of them is 58, what are the values of a and bl

Cumulative scores have the form ax + by, x and y nonnegative integers, obtained

from scoring a on ...

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### Contents

CHAPTER PAGE 1 Three Surprises from Combinatorics and Number Theory | 1 |

Four Minor Gems from Geometry | 10 |

A Problem in CheckerJumping | 23 |

Copyright | |

13 other sections not shown

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### Common terms and phrases

1-factor Accordingly Amer attainable score beads BICENTRIC POLYGONS black edges bricks checkerboard circumcenter circumcircle circumsphere collinear column complete graph composite solutions configuration congruent Consequently consider contains contradiction copies deleted determine diagonal digits distance can occur divides dominoes eight-point circle endpoints equal exceed Figure gives graph G harmonic bricks harmonic series hexominoes implying incircle integer intersection isosceles 6-point isosceles tetrahedron key multiple Klarner L-tromino lattice point layers least line ax Math monomino Monthly n-gon natural numbers necklaces number of vertices obtain occupy odd block odd components odd number odd prime pack the line packable pair Paul Erdos perimeter plane polyominoes prime number problem proof proved puzzle quadrilateral red edges relatively prime result segments sides Similarly Slothouber-Graatsma Puzzle spheres subset Suppose tangent tetrahedron tetrominoes theorem total number tromino unattainable unit cubes vertex Wilson's theorem yields