##### Document Text Contents

Page 1

A N D O T H E R

MATHEMATICAL

n I V E R S I O N S

OLLECTION OF

GI ES FRON

A E R I C "

Page 2

The

Unexpected

Hanging

Page 131

Peg Solitaire 133

sible to start with a corner vacancy on the standard board

and conclude with a nine-ball sweep. He gave no solution.

As fa r as I know, a solution to this difficult problem was first

rediscovered by Harry 0. Davis who gives an elegant eight-

een-move solution in his 1967 article, cited in my list of

references. Davis also shows in this article that no solution

to the standard game, regardless of what cell is vacant a t

the start, can contain a chain longer than nine moves.

Davis, who has been mentioned many times in this chapter,

first became interested in peg solitaire when he read about

i t in my column in 1962. Since then he has made enough

fresh discoveries-extending possibility tests, developing

techniques for obtaining minimum-move solutions and prov-

ing them minimal, creating and solving new problems, and

even extending solitaire to three dimensions (which he calls

"so1idaire")-to make a sizable book. So far he has published

only the one article listed. In recent years he has been col-

laborating with Wade E. Philpott, Lima, Ohio, who has done

important work on the theory of peg solitaire both in its

traditional orthogonal form and also on isometric (triangu-

lar) fields. (On isometric peg solitaire see my Scientific

American columns for February and May, 1966.)

A N S W E R S

For the first five problems, readers found shorter solu-

tions than the ones I gave in Scientific American. I have here

substituted minimal solutions, giving the names of those

who sen.t such solutions.

Greek cross in six moves: 54-74, 34-54, 42-44-64, 46-44,

74-54-34, 24-44. (R. L. Potyok, H. 0. Davis.)

Fireplace in eight moves : 45-25, 37-35, 34-36, 57-37-35,

25-45, 46-44-64, 56-54, 64-44. (W. Leo Johnson, H. 0. Davis,

R. L. Potyok.)

Pyramid in eight moves: 54-74, 45-65, 44-42, 34-32-52-54,

13-33, 73-75-55-53, 63-43-23-25-45, 46-44. (H. 0. Davis.)

Lamp in ten moves: 36-34, 56-54, 51-53-33-35-55, 65-45,

41-43, 31-33-53-55-35, 47-45, 44-46, 25-45, 46-44. (Hugh W.

Thompson, H. 0. Davis.)

Page 132

134 The Unexpected Hanging

Inclined square in eight moves: 55-75, 35-55, 42-44, 63-43-

45-65, 33-35-37-57-55-53-51-31-33-13-15-35, 75-55, 74-54-56-

36-34, 24-44. (H. 0. Davis.) Note the remarkable chain of

eleven jumps.

Wall: 46-44, 43-45, 41-43, 64-44-42, 24-44, 45-43-41.

This solves the problem. By continuing to play it is easy to

reduce the figure to four pieces on the corners of the central

3 by 3 square.

Square: 46-44, 25-45, 37-35, 34-36, 57-37-35, 45-25, 43-45,

64-44, 56-54, 44-64, 23-43, 31-33, 43-23, 63-43, 51-53, 43-63,

41-43. The finish is apparent: 15-35, 14-34, 13-33 on the left,

and the corresponding moves on the right, 75-55, 74-54, 73-

53. The puzzle is now solved. Four more jumps will leave

counters on the corners (36, 65, 52, 23) of an inclined square

-an unusually difficult pattern to achieve if one does not

know earlier positions.

Pinwheel: 42-44, 23-43, 44-42, 24-44, 36-34, 44-24, 46-44,

65-45, 44-46, 64-44, 52-54, 44-64. The position now has four-

fold symmetry. It is completed: 31-33, 51-31, 15-35, 13-15,

57-55, 37-57, 73-53, 75-73. The final figure is a stalemate.

The shortest stalemate, starting with a full board and a va-

cant center cell, is reached in these six moves: 46-44, 43-45,

41-43, 24-44, 54-34, 74-54. The next-shortest stalemate is a

ten-move game.

Robin Merson, who works on satellite orbit determinations

a t the Royal Aircraft Establishment in Farnborough, Eng-

land, sent a simple proof that a t least sixteen moves (a chain

of jumps counts as one move) are necessary in solving the

problem on the 6 x 6 square. The first move is 3-1, or its

symmetrical equivalent. This places a counter on each corner

cell. I t is impossible for a corner piece to be jumped, there-

fore each corner piece must move (including the counter a t

1, which must move out to allow a final jump into the cor-

ner). These four moves, added to the first, bring the total to

five. Consider now the side pieces on the borders between

corners. Two such pieces, side by side, cannot be jumped;

therefore for every such pair at least one counter must move.

On the left and right sides, and on the bottom, at least two

pieces must move to break up contiguous pairs. On the top

edge (assuming a 3-1 first move) one piece will suffice. This

Page 261

Bibliography 263

J. H. Cadwell, Topics in Recreational Mathematics. Cambridge, England: Cam-

bridge University Press, 1966. See pages 96-99.

A. S. Besicovitch, "The Kakeya Problem," American Mathematical Monthly,

Vol. 70, August-September 1963, pages 697-706.

A. A. Blank, "A Remark on the Kakeya Problem," American Mathematical

Monthly, Vol. 70, August-September 1963, pages 706-11.

F. Cunningham, Jr., "The Kakeya Problem for Simply Connected and Star-

shaped Sets," American Mathematical Monthly, Vol. 78, February 1971,

pages 114-29.

19. REP-TILES: REPLICATING FIGURES ON THE PLANE

C. Dudley Langford, "Uses of a Geometrical Puzzle," Mathematical Gazette,

Vol. 24, July 1940, pages 209- 11.

R. Sibson, "Comments on Note 1464," Mathematical Gazette, Vol. 24, De-

cember 1940, page 343.

Howard D. Grossman, "Fun with Lattice Points," Scripta Mathematics, Vol.

14, June 1948, pages 157-59.

Solomon W. Golomb, "Replicating Figures in the Plane," Mathematical Ga-

zette, Vol. 48, December 1964, pages 403- 12.

M. Goldberg and B. M. Stewart, "A Dissection Problem for Sets of Poly-

gons," American Mathematical Monthly, Vol. 71, December 1964, pages

1077-95.

Roy 0. Davies, "Replicating Boots," Mathematical Gazette, Vol. 50, May

1966, page 157.

G. Valette and T. Zamfirescu, "Les Partages d'un Polygone Convexe en 4

Polygones Semblables au Premier," Journal of Combinatorial Theory, B ,

Vol. 16, 1974, pages 1-16.

Jack Gies, Jr., "Infinite-level Replication Dissections of Plane Figures," "Con-

struction of Replicating Superfigures," and "Superfigures Replicating with

Polar Symmetry," Journal of Combinatorial Theory, 26A, 1979, pages

319-27, 328-34, 335-37.

Jack Gies, Jr., "The Gypsy Method of Superfigure Construction, " Journal of

Recreational Mathematics, Vol. 13, 1980181, pages 97-101.

F. M. Dekkiig, "Replicating Superfigures and Endomorphisms of Free

Groups," Journal of Combinatorial Theory, Vol. 32A, 1982, pages 315-20.

J. Doyen and M. Lenduyt, "Dissection of Polygons," Annals of Discrete

Mathematics, Vol. 18, 1983, pages 315- 18.

Page 262

ABOUT THE AUTHOR

MARTIN GARDNER, who studied philosophy as an undergraduate and

never formally pursued advanced study in mathematics, was the Mathe-

matical Games columnist of Scientij'ic American for 25 years, until his

retirement in 1981. Eleven book collections of his Scientz3c Amem'can

columns have been published. Mr. Gardner is the author of some fifty

books about mathematics, science, philosophy, and literary criticism.

Among his best known titles are The Annotated Alice, The Relativity

Explosion, The Ambidextrous Universe, The Whys of a Philosophical

Scrivener, Order and Surprise, and Science: Good, Bad and Bogus. He

has written one novel, The Flight of Peter Fromm, as well as many

books for children. A magician, he also writes books for the conjuring

trade. He lives in Hendersonville, North Carolina.

A N D O T H E R

MATHEMATICAL

n I V E R S I O N S

OLLECTION OF

GI ES FRON

A E R I C "

Page 2

The

Unexpected

Hanging

Page 131

Peg Solitaire 133

sible to start with a corner vacancy on the standard board

and conclude with a nine-ball sweep. He gave no solution.

As fa r as I know, a solution to this difficult problem was first

rediscovered by Harry 0. Davis who gives an elegant eight-

een-move solution in his 1967 article, cited in my list of

references. Davis also shows in this article that no solution

to the standard game, regardless of what cell is vacant a t

the start, can contain a chain longer than nine moves.

Davis, who has been mentioned many times in this chapter,

first became interested in peg solitaire when he read about

i t in my column in 1962. Since then he has made enough

fresh discoveries-extending possibility tests, developing

techniques for obtaining minimum-move solutions and prov-

ing them minimal, creating and solving new problems, and

even extending solitaire to three dimensions (which he calls

"so1idaire")-to make a sizable book. So far he has published

only the one article listed. In recent years he has been col-

laborating with Wade E. Philpott, Lima, Ohio, who has done

important work on the theory of peg solitaire both in its

traditional orthogonal form and also on isometric (triangu-

lar) fields. (On isometric peg solitaire see my Scientific

American columns for February and May, 1966.)

A N S W E R S

For the first five problems, readers found shorter solu-

tions than the ones I gave in Scientific American. I have here

substituted minimal solutions, giving the names of those

who sen.t such solutions.

Greek cross in six moves: 54-74, 34-54, 42-44-64, 46-44,

74-54-34, 24-44. (R. L. Potyok, H. 0. Davis.)

Fireplace in eight moves : 45-25, 37-35, 34-36, 57-37-35,

25-45, 46-44-64, 56-54, 64-44. (W. Leo Johnson, H. 0. Davis,

R. L. Potyok.)

Pyramid in eight moves: 54-74, 45-65, 44-42, 34-32-52-54,

13-33, 73-75-55-53, 63-43-23-25-45, 46-44. (H. 0. Davis.)

Lamp in ten moves: 36-34, 56-54, 51-53-33-35-55, 65-45,

41-43, 31-33-53-55-35, 47-45, 44-46, 25-45, 46-44. (Hugh W.

Thompson, H. 0. Davis.)

Page 132

134 The Unexpected Hanging

Inclined square in eight moves: 55-75, 35-55, 42-44, 63-43-

45-65, 33-35-37-57-55-53-51-31-33-13-15-35, 75-55, 74-54-56-

36-34, 24-44. (H. 0. Davis.) Note the remarkable chain of

eleven jumps.

Wall: 46-44, 43-45, 41-43, 64-44-42, 24-44, 45-43-41.

This solves the problem. By continuing to play it is easy to

reduce the figure to four pieces on the corners of the central

3 by 3 square.

Square: 46-44, 25-45, 37-35, 34-36, 57-37-35, 45-25, 43-45,

64-44, 56-54, 44-64, 23-43, 31-33, 43-23, 63-43, 51-53, 43-63,

41-43. The finish is apparent: 15-35, 14-34, 13-33 on the left,

and the corresponding moves on the right, 75-55, 74-54, 73-

53. The puzzle is now solved. Four more jumps will leave

counters on the corners (36, 65, 52, 23) of an inclined square

-an unusually difficult pattern to achieve if one does not

know earlier positions.

Pinwheel: 42-44, 23-43, 44-42, 24-44, 36-34, 44-24, 46-44,

65-45, 44-46, 64-44, 52-54, 44-64. The position now has four-

fold symmetry. It is completed: 31-33, 51-31, 15-35, 13-15,

57-55, 37-57, 73-53, 75-73. The final figure is a stalemate.

The shortest stalemate, starting with a full board and a va-

cant center cell, is reached in these six moves: 46-44, 43-45,

41-43, 24-44, 54-34, 74-54. The next-shortest stalemate is a

ten-move game.

Robin Merson, who works on satellite orbit determinations

a t the Royal Aircraft Establishment in Farnborough, Eng-

land, sent a simple proof that a t least sixteen moves (a chain

of jumps counts as one move) are necessary in solving the

problem on the 6 x 6 square. The first move is 3-1, or its

symmetrical equivalent. This places a counter on each corner

cell. I t is impossible for a corner piece to be jumped, there-

fore each corner piece must move (including the counter a t

1, which must move out to allow a final jump into the cor-

ner). These four moves, added to the first, bring the total to

five. Consider now the side pieces on the borders between

corners. Two such pieces, side by side, cannot be jumped;

therefore for every such pair at least one counter must move.

On the left and right sides, and on the bottom, at least two

pieces must move to break up contiguous pairs. On the top

edge (assuming a 3-1 first move) one piece will suffice. This

Page 261

Bibliography 263

J. H. Cadwell, Topics in Recreational Mathematics. Cambridge, England: Cam-

bridge University Press, 1966. See pages 96-99.

A. S. Besicovitch, "The Kakeya Problem," American Mathematical Monthly,

Vol. 70, August-September 1963, pages 697-706.

A. A. Blank, "A Remark on the Kakeya Problem," American Mathematical

Monthly, Vol. 70, August-September 1963, pages 706-11.

F. Cunningham, Jr., "The Kakeya Problem for Simply Connected and Star-

shaped Sets," American Mathematical Monthly, Vol. 78, February 1971,

pages 114-29.

19. REP-TILES: REPLICATING FIGURES ON THE PLANE

C. Dudley Langford, "Uses of a Geometrical Puzzle," Mathematical Gazette,

Vol. 24, July 1940, pages 209- 11.

R. Sibson, "Comments on Note 1464," Mathematical Gazette, Vol. 24, De-

cember 1940, page 343.

Howard D. Grossman, "Fun with Lattice Points," Scripta Mathematics, Vol.

14, June 1948, pages 157-59.

Solomon W. Golomb, "Replicating Figures in the Plane," Mathematical Ga-

zette, Vol. 48, December 1964, pages 403- 12.

M. Goldberg and B. M. Stewart, "A Dissection Problem for Sets of Poly-

gons," American Mathematical Monthly, Vol. 71, December 1964, pages

1077-95.

Roy 0. Davies, "Replicating Boots," Mathematical Gazette, Vol. 50, May

1966, page 157.

G. Valette and T. Zamfirescu, "Les Partages d'un Polygone Convexe en 4

Polygones Semblables au Premier," Journal of Combinatorial Theory, B ,

Vol. 16, 1974, pages 1-16.

Jack Gies, Jr., "Infinite-level Replication Dissections of Plane Figures," "Con-

struction of Replicating Superfigures," and "Superfigures Replicating with

Polar Symmetry," Journal of Combinatorial Theory, 26A, 1979, pages

319-27, 328-34, 335-37.

Jack Gies, Jr., "The Gypsy Method of Superfigure Construction, " Journal of

Recreational Mathematics, Vol. 13, 1980181, pages 97-101.

F. M. Dekkiig, "Replicating Superfigures and Endomorphisms of Free

Groups," Journal of Combinatorial Theory, Vol. 32A, 1982, pages 315-20.

J. Doyen and M. Lenduyt, "Dissection of Polygons," Annals of Discrete

Mathematics, Vol. 18, 1983, pages 315- 18.

Page 262

ABOUT THE AUTHOR

MARTIN GARDNER, who studied philosophy as an undergraduate and

never formally pursued advanced study in mathematics, was the Mathe-

matical Games columnist of Scientij'ic American for 25 years, until his

retirement in 1981. Eleven book collections of his Scientz3c Amem'can

columns have been published. Mr. Gardner is the author of some fifty

books about mathematics, science, philosophy, and literary criticism.

Among his best known titles are The Annotated Alice, The Relativity

Explosion, The Ambidextrous Universe, The Whys of a Philosophical

Scrivener, Order and Surprise, and Science: Good, Bad and Bogus. He

has written one novel, The Flight of Peter Fromm, as well as many

books for children. A magician, he also writes books for the conjuring

trade. He lives in Hendersonville, North Carolina.