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Published in India by Prowess Publishing, YRK Towers, Thadikara Swamy Koil St, Alandur, Chennai, Tamil Nadu 600016

ISBN: 978-1-5457-5330-9

Library of Congress Cataloging in Publication

Games, Puzzles and Math Excursions_Interior.indd 4 15/10/20 4:51 PM Contents

Foreword...... xiii Preface...... xv Introduction...... xvii About The Author...... xix History...... xxi What’s in This Book?...... xxiii Games Unlisted...... xxv Paper and Pencil game with Templates...... xxix

GAMES...... 1

1. Alignment...... 3 Achi...... 3 Tic Tac Toe or Noughts and Crosses...... 4 Nine Men’s Morris...... 6 Picaria...... 8 Dara...... 9 Connect Four...... 10 Quarto...... 11 Go-Moku...... 13 Arni’s 3 in a row...... 14

2. Connection (Joining Dots or Lines)...... 15 Dots and Boxes Game...... 15 Sprouts...... 17

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Sim...... 19 Cram...... 20 Bridg-It...... 21

3. Territory (Linking or Occupying the Other Side)...... 22 Hex...... 22 Pentomino Space...... 24 Twixt...... 26 Five Field Kono...... 28 Reversi (Othello)...... 29 Arni’s Knights...... 30 Draughts...... 31 Go...... 32

4. Deduction Games...... 34 Battleships...... 34 Mastermind...... 36 Bulls and Cows...... 38 Wallpaper...... 39

5. Counting Games...... 41 Nim...... 41 Game of 31...... 46

Moore’s Nim (Nimk)...... 48 Wythoff’s Nim...... 51 Fibonacci Nim...... 52 Monopile...... 56 Bulo or TacTix...... 58 Shut the Box...... 59

6. Non Specific Games...... 61 L Game Edward Bono...... 61 Arni Sodoko Game...... 62 Arni’s Move and Rotate Game...... 64 Wythoff’s Game...... 66

Games, Puzzles and Math Excursions_Interior.indd 6 12/10/20 10:25 AM PUZZLES PRESENTATION...... 69

1. Jars...... 71 Decantation...... 71 Target Measure...... 72 2. Alignment Problems...... 73 Hearts and Coins...... 73 Jumping Coins...... 73 16 Discs in 4 Piles...... 73 Heart Pairs...... 74 Hearts Swap...... 74 Invert a Triangle...... 75 5 - Coin Puzzle...... 75 6 - Coin Geometry...... 75 Square Coin Geometry...... 76 3. Weight Problems...... 77 Problem # 1...... 77 Problem # 2...... 77 Problem # 3...... 77 Problem # 4...... 78 Problem # 5...... 78 Problem # 6...... 78 Problem # 7 – The Treasure Coup...... 79 Problem # 8...... 79 4. Expressing Numbers...... 80 The Four Fours Problem (4 4s)...... 80 The 1234 Problem...... 81 5. Matchstick Puzzles...... 82 17 Puzzles...... 82 6. Fairy Chess Puzzles...... 87 Sam Loyd’s Classic Puzzle...... 87 Chess in a Pistol...... 88

vii

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Swap 4 Knights...... 89 Swap 6 Knights...... 89 Knight’s Tour 30 Cells...... 89 Another Knight’s Tour 12 cells...... 90 The Queens Problem...... 90 Can I be the #2 World Chess Champion?...... 90

7. Cryptarithms...... 91 Restorations...... 91

8. Patterns and Series...... 93 Numerical Patterns ...... 93 What goes next – 19 Puzzles...... 93 Figure Patterns...... 94

9. Logic and Lateral Thinking Puzzles...... 97 Decision Matrix...... 97 Three Box Lables...... 98 Commuter...... 98 Time and Distance...... 99 The Shortlived Drone...... 99 How Wide is River...... 100 Simple Arthmetic and Fermat’s Last Theorem...... 100 The Electrician...... 101 Two Black Stones...... 101 Slicing a Horse...... 101 Driving Me Nuts!...... 101

EXCURSIONS...... 103

1. Distribution and Rearrangement Puzzles...... 105 Josephus Problem...... 105 Nine Men in a Trench...... 106 DUDENEYs Car Garage Puzzle 6 Cars...... 107 Dudeneys Garage Puzzle 8 Cars...... 107

viii

Games, Puzzles and Math Excursions_Interior.indd 8 12/10/20 10:25 AM River Crossing Problem...... 107 Shunting Problem...... 108

2. Fallacies and Paradoxes...... 109 Division by Zero...... 109 Taking One Value of Square Root...... 109 Anomalous Cancellation...... 110 All Triangles are Isosceles...... 110 The Duplication of the Cube...... 111 Trisection of an Angle...... 112 Squaring a Circle...... 112 Paradoxes...... 112 Achilles and the Tortoise Paradox...... 112 The Sum of all Positive Integers Equals -(1/12)...... 113 Uninteresting Numbers...... 114 Infinite Hotel Hilbert Paradox...... 115 The Missing ₹100...... 115 The Fibonacci Geometry Paradox...... 116 The Curry Geometrical Paradox...... 117

3. Dissection Puzzles...... 118 Archimedes Stomachion...... 118 Dudeney’s Dissection...... 119 Sam Loyd’s Dissection...... 120 Tangrams...... 120 Dots and Lines...... 122 Solution...... 124

4. The 7 Bridges of Königsberg...... 127 The 7 Bridges Puzzle...... 127

5. Tower of Hanoi, Pascal Triangle and Sierpinski Fractal...... 131 Tower of Hanoi Puzzle...... 131 The Pascal Triangle...... 134

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The Sierpinski Fractal...... 135 Puzzle Graph...... 137

6. Magic Squares...... 138 History of Magic Squares – In Pictures...... 138 Magic Square...... 140 Construction of Odd Magic Square...... 140 Construction of Even Magic Square, if Divisible by 4...... 141 The Durer Square...... 142 Big Magic Squares, Benjamin Franklin Squares...... 142 The Latin Square...... 145 Euler’s Square and 36 Officers Problem...... 145 More Magic Squares...... 147 Magic Night’s Tour...... 150

7. Findings, Hypothesis and Conjectures...... 154 Fermat’s Theorem...... 154 Goldbach’s Conjecture...... 155 Twin Prime Conjecture...... 155 Four Color Theorem...... 156 Collatz Conjecture...... 156 Odd Perfect Number...... 158

8. Puzzles Behind Art...... 159 A Persian Horse Puzzle...... 159 4 Deer with a Common Head...... 161

9. Major Mathematicians of India...... 162

SOLUTIONS FOR PUZZLES...... 167

1. Jars...... 169 2. Alignment...... 171 3. Weighments...... 174

Games, Puzzles and Math Excursions_Interior.indd 10 12/10/20 10:25 AM 4. Expressing Numbers...... 182 5. Matchsticks...... 188 6. Fairy Chess Puzzles - Solutions...... 193 7. Cryptarithms...... 197 8. Series and Patterns...... 199 9. Logic and Lateral Thinking...... 200

SOLUTIONS FOR EXCURSIONS...... 205

1. Distribution and Rearrangements...... 207 2. Dots and Dissections...... 213 3. Puzzles in Art...... 214 4. Four Deer with Common Head...... 217

TEMPLATES...... 219

1. Alignment...... 221 Achi...... 221 Tic Tac Toe/Quarto Markers Specs...... 222 Nine Men’s Morris...... 223 Picaria...... 224 Dara...... 225 Connect 4...... 226 Quarto...... 227 Go Maku...... 228 Arni 3 in a Row...... 229

2. Connection...... 230 Dots and Boxes...... 230 Sim...... 231 Cram...... 232 Bridg-It...... 233

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3. Territory...... 234 Hex...... 234 Pentominoes...... 235 Twixt...... 237 5 Field Kono...... 238 Riversi...... 239 Arni Knights...... 240 Draughts...... 241 Go...... 242

4. Deduction...... 243 Battle Ships/Wallpaper...... 243 Mastermind...... 244 Bulls and Cows...... 245

5. Counting...... 246 Nim/Moore’s Nim...... 246 Game 31...... 247 Wythoff’s/Fibonacci Nim...... 248 Bulo Tac Tix/Monopile...... 249 Shut the Box...... 250

6. Non Specific Games...... 251 L-Game...... 251 Arni Sodoko Game...... 252 Arni’s Move and Rotate...... 253 Wythoff’s Game...... 254

7. Dissection Puzzle...... 255 Circular Tangram...... 255

8. Puzzles in Art...... 257 One Head for Four Deer...... 257 A Racing Horse From a Mule...... 258

Index...... 259

xii

Games, Puzzles and Math Excursions_Interior.indd 12 12/10/20 10:25 AM Alignment

Achi

Achi is a two-player alignment game played in Ghana and similar to tic-tac- toe. Each player has four pieces to play on the board. A 3 × 3 board with 9 intersection points, as shown, is used.

Each player plays with different coloured pieces; it can also be black and white.

The players take turns to place one of their counters on a point where lines join, trying to create a 3-in-a-row either horizontally, vertically, or diagonally. When all eight counters have been placed, each player can move along a line to an adjacent empty point. The winner is the first player to create the afrementioned 3-in-a-row of one’s counters.

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Tic Tac Toe or Noughts and Crosses

It is the most widely known game and is found in almost all countries in one form or the other and is known to have been played in ancient times. It is the fore runner of most games thereafter like GoMaku, 9-man Morris, 3-man Morris, Connect 4, Achi, Dara, Pente, PICARIA and the modern Quarto. It is a 2 player pencil game. The players take turns in placing their marks, traditionally with a X and an O on a 3 X 3 grid. Whoever gets three of his markings in a row, wins. Even school children discover very rapidly that the best play from both players will lead to a draw and the very first opening moves by them will decide the winner.

For example X being the player who starts has only 3 openings. To that, the replies by O has to be very accurate or else X playing correctly will get a fork and win. A variant of the game is played with the numbers from 1 to 9 to make the sum of row, column or diagonal equal to 15. Player who goes first (A) has all the odd numbers like 1, 3, 5, 7, 9 whereas his opponent (B) has the even numbers 2, 4, 6, 8. A goes first playing 9 (Numbers 1, 3, 5, 7 are only left). B plays 8, closing any chance of A getting a diagonal 15. A plays 5 as he only has a 1 to get 15 on the diagonal and so on.

Games, Puzzles and Math Excursions_Interior.indd 4 12/10/20 10:26 AM Pigpen Cipher Did you know that this formation was used by the Freemasons and also in the American Civil war? And, it is a part of the Da Vinci code in Dan Brown’s book?

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Nine Men’s Morris

Nine Men’s Morris is the grown-up version of 3 in a row, played by 2 persons. It is a famous board game and is at least 2000 years old. Picture below is from a Spanish games book.

The game is played on a board consisting of three concentric squares connected by lines from the middle of each of the inner square’s sides to the middle of the corresponding outer square’s side. Holes are located in the 24 intersections and corners pointing where pegs can be inserted. Both players play with 9 pegs each of a distinguishing colour.

The basic aim of Nine Men’s Morris is to make “mills” — vertical or horizontal lines of three in a row. Every time this is achieved, an opponent’s piece is removed, the overall objective being to reduce the number of opponent’s pegs to less than three or to render the opponent unable to play.

To begin with the board is empty. Play is in two phases. To begin with, players take turns to play – inserting a peg of their own colour on any unoccupied point until all eighteen pegs have been played. After that, play continues alternately but each turn consists of a player moving one peg along a line to an adjacent point. During both of these phases, whenever a player achieves a mill, that player proceeds to remove one peg belonging to the opponent (that does not form

Games, Puzzles and Math Excursions_Interior.indd 6 12/10/20 10:26 AM part of a mill) from the board. If all the opponent’s pieces form mills then an exception is made and the player is allowed to remove any peg. It is only upon the formation of a mill that a peg is captured but a player will often break a mill by moving a peg out of it and then, in a subsequent turn, play the peg back again, thus forming a new mill and capturing another peg. In the paper version we play with a drawn diagram and coloured discs cut out of cardboard.

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Picaria The game is also known as “Tapatan” and “3-Man Morris”. This game also has ancient roots. It is a 2-person game and is played on a square board being connected as shown. Each player has three pieces. The winner is the first player to align their three pieces on the lines drawn on the board. There are 3 horizontal lines, 3 vertical lines and 6 diagonal lines, as in the diagram.

The board is empty to begin the game, and players take turns placing their pieces on empty intersections. Once all pieces are placed (assuming there is no winner by then), play proceeds with each player moving one of their pieces per turn. A piece may move to any adjacent vacant linked point on the board.

This game is ideally suited for paper and pencil. There is a Template for this game at the back of the Book.

There is an alternative version which does not allow the players to occupy the central spot on the first move.

Games, Puzzles and Math Excursions_Interior.indd 8 12/10/20 10:26 AM Dara

It is a 2-person game somewhat like 9 men’s Morris. The objective is to form many “3-in-a-row”s, to eliminate so many of the opponents pieces that he can’t play further.

It is played on a squared 5 × 6 board. Each player has 12 pieces. One player plays Black and the other plays White. However, any two colors will do. The board is empty in the beginning. Players take turn placing their pieces onto the empty cells of the square board with their strategy. Three in a row is not allowed for placement. This is known as Phase 1 of the game or the Drop Phase. After all 24 stones have been placed, Phase 2 or the Move Phase begins. Players will then take turns moving their pieces orthogonally into an adjacent empty cell. Players attempt to make a three-in-a-row with their own pieces. The three-in-a-row must be orthogonal and not diagonal. Furthermore, it must be strictly three consecutive pieces in a row. When a three-in-a-row is made by a player, he or she can remove one opponent’ piece from the board which is not part of a three-in-a-row itself. If a player has only 2 pieces left he or she is the loser. Three-in-a-rows made during the Drop Phase do not count. Therefore, a player cannot remove another player’s stone during the Drop phase even if one were to make a three-in-a-row. Moreover, the rule that four or more pieces in a row are illegal to form also applies in the Drop phase. If a player were to successfully form two three-in-a-rows in one move during the Move phase, only one of his opponent’s pieces can be removed.

In the paper version, the simple grid is drawn on a paper and 12 coloured circles or squares in 2 colours cut out.

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Connect Four

Also known as Four in a Row, this is a two-player board game in which the players first choose their colours and then take turns dropping one of their coloured disc from the top into a seven- column, six-row vertically suspended 7 × 6 grid, usually on a proprietary gadget. The pieces fall straight down.

The objective of the game is to be the first to form a horizontal, vertical, or diagonal line of four of one’s own discs. The first player can always win by playing the right moves.

In the paper version, a 7 × 6 grid is drawn and marked (columns A to G and rows 1 to 6). The players either choose pen marks (X and 0) or small colored (blue and red) circles to distinguish between them. On the grid the columns are identified by letters of the alphabet from A to G starting from the left, and rows identified by the digits 1 to 6 with 1 being at the bottom.

Since there is no gravity here, the rows 1 to 6 will have to be filled in ascending order. In other words, you can’t put your disc in C3 until there is a disc in C2.

Games, Puzzles and Math Excursions_Interior.indd 10 12/10/20 10:26 AM Quarto Quarto is a board game for two players. It was invented by Blaise Müller in 1991. The board has 16 squares (4 × 4), and the 16 different pieces that can be constructed combinating the following four characteristics:

Size (big/small) Colour (e.g. red/blue) Shape (circle/square) Hole (piece with hole/piece without hole) or Marking (marked or not marked)

Objective The aim of the game is to complete a line with four pieces that are similar at least about one of the four described characteristics (four big pieces, four little, four red, four blue, four circle, four square, four with hole or four without hole). The line may be vertical, horizontal or diagonal. The winner is the player who places the fourth piece of the line. Players move alternatively, one selecting a piece and giving his opponent to place on the board. After placement he in turn picks up a piece for the opponent to place. Once inserted, pieces cannot be moved. So, each turn consists of two actions: 1. Place on the board the piece given by the opponent. 2. Give to the opponent the piece to be placed in the next move. In the first turn of the game, the player who starts has only to choose one piece for the opponent. The game finishes in a draw when nobody reaches the objective after placing the 16 pieces

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Paper-Pencil Version Since we are not using proprietary hardware, the most important part is making the 16 pieces which are ALL different. I would advise making them as follows – In the paper version you have 16 cardboard cut outs as shown in the Glossary. Easier is take 2 cardboards in blue and red colours. Cut 4 big squares (2 cm × 2 cm) in each color. Cut 4 small squares (half inch × half inch) in each color. Trim 2 of the blue big squares into 2 “big” blue circles. Trim 2 of the red big squares into 2 “big” red circles. Do the same for the small squares in both colors. Now you will have 16 pieces but there will 8 duplicates. Mark all the duplicates with a full X on them. Now you will have the 16 different pieces for the game – based on size, colour, shape and marking. Draw a 4 inch × 4 inch grid on a sheet and you are ready to play.

Games, Puzzles and Math Excursions_Interior.indd 12 12/10/20 10:26 AM Go-Moku The game is of Japanese origin but played mostly in Korea. It is also known as: Five in a row, Go Bang, Pegit.

It is a 2-person game played on a regular Go board with unlimited white and black pieces, placing them by turn on the point of intersections of the grid. Whoever gets 5 in a row wins. (White will win in the next move in this diagram.)

In the paper version the players take turns in marking squares on the spaces of a grid. The players take turns in marking a square with their symbol (e.g. Blue O and Red X). The first player to get five squares in a row, then horizontally, vertically, or diagonally, win.

The example on the left shows a typical game won by the first player, Blue.

In the final example, all the 17 moves are shown. With A (white circles) starting first, B is just defending other than his first move.

On his 9th move A has opened 2 forks ‘963’ and ‘948’.

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Arni’s 3 in a Row (Chandru Arni is the developer of this game. It has the ingredients of “placing, moving and properties of the pieces”.)

This is a strategy game for two players. The board has 16 squares (4 × 4) and the 9 different pieces belong to a common pool. The pieces are of 3 different colours and each colour has 3 different shapes.

Colour (red/blue/green) Shape (circle/square/triangle)

Objective The aim of the game is to complete a line with three pieces that are similar at least about one of the 2 described characteristics. The line may be vertical, horizontal or diagonal. The winner is the player who has 3 in a row first. It can be 3 triangles in a row or 3 green pieces in a row, etc. Players move alternatively, placing one piece on the board. Advanced players may not have won after the placement of the 9 pieces. In that case they can move or jump a piece. To move a piece it can move only 1 cell and that too orthogonally. To jump a piece, the piece should be to an adjacent one and the cell adjacent to that must be empty. The move or jump cannot be diagonal. There is a template at the back.

Games, Puzzles and Math Excursions_Interior.indd 14 12/10/20 10:26 AM Connection (Joining Dots or Lines)

Dots and Boxes Game Dots and Boxes is a pencil-and-paper game for two players. It was first published in the 19th century by French mathematician Édouard Lucas.

The full game starts with an empty grid of dots (10 × 10 grid). The 2 players play with 2 different colour pencils. The players take turns adding a single horizontal or vertical line between two unjoined adjacent dots. A player who completes the fourth side of a 1 × 1 box earns one point and takes another turn. (A point is typically recorded by placing a mark that identifies the player in the box, such as an initial.) The game ends when no more lines can be placed. The winner is the player with the most points. The board may be of any size grid. This game can also be played by more than two players on a larger grid.

The diagram on the left shows a game being played on a 2 × 2 board (3 × 3 dots). The second player (“B”) plays a rotated mirror image of the first player’s moves, hoping to divide the board into two pieces and tie the game. But the first player (“A”) makes a sacrifice at move 7 and B accepts the

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sacrifice, getting one box. However, B must now add another line, and so B connects the center dot to the center-right dot, causing the remaining unscored boxes to be joined together in a chain (shown at the end of move 8). With A’s next move, A gets all three of them and ends the game, winning 3-1.

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