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Oswin Aichholzer Algorithms and Games1

Dots and Boxes

Algorithms & Games WS 2019/20

Oswin Aichholzer Algorithms and Games2 i

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

Oswin Aichholzer Algorithms and Games2 ii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

3× 4 grid

Oswin Aichholzer Algorithms and Games2 iii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

Oswin Aichholzer Algorithms and Games2 iv

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

A

Oswin Aichholzer Algorithms and Games2 v

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

Oswin Aichholzer Algorithms and Games2 vi

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

A

Oswin Aichholzer Algorithms and Games2 vii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

Oswin Aichholzer Algorithms and Games2 viii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

A

Oswin Aichholzer Algorithms and Games2 ix

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

Oswin Aichholzer Algorithms and Games2 x

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

A

Oswin Aichholzer Algorithms and Games2 xi

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

Oswin Aichholzer Algorithms and Games2 xii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

B

Oswin Aichholzer Algorithms and Games2 xiii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

B

Oswin Aichholzer Algorithms and Games2 xiv

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

AA

Oswin Aichholzer Algorithms and Games2 xv

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

AA

Oswin Aichholzer Algorithms and Games2 xvi

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

AA

Oswin Aichholzer Algorithms and Games2 xvii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

AB

Oswin Aichholzer Algorithms and Games2 xviii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

AA

Oswin Aichholzer Algorithms and Games2 xix

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

A

BB

Oswin Aichholzer Algorithms and Games2 xx

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

A

B

BB

Oswin Aichholzer Algorithms and Games2 xxi

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

A

B

BBB

Oswin Aichholzer Algorithms and Games2 xxii

Dots and Boxes

Dots and Boxes is played by two players (Alice and Bob)on and n×m grid.

In each turn a player connects two adjacent dots with ahorizontal or vertical unit segment.

B

If a player completes a unit box, he/she owns that box andmust then draw another segment (except in the last move).

A

A

B

BB

At the end of the game the players count their boxes. Inthe example B wins 4 : 2.

Oswin Aichholzer Algorithms and Games3 i

Move Generator: Pseudo Code

NEXT STATES(state, player)E = Set of unused segments in stateIF E = ∅ THEN

add state to list(state, NOT player)ELSE FOR all segments s ∈ E DO

IF s closes one or two boxes in state THENlabel new boxes with playernext states(state + s, player)

ELSEadd state to list(state + s, NOT player)

Main call:E = Set of unused segments in current stateIF E 6= ∅ THEN next states(current state, current player)

ELSE No moves possibleRemark: add state to list() removes duplicates

Oswin Aichholzer Algorithms and Games3 ii

Move Generator: Pseudo Code

NEXT STATES(state, player)E = Set of unused segments in stateIF E = ∅ THEN

add state to list(state, NOT player)ELSE FOR all segments s ∈ E DO

IF s closes one or two boxes in state THENlabel new boxes with playernext states(state + s, player)

ELSEadd state to list(state + s, NOT player)

Main call:E = Set of unused segments in current stateIF E 6= ∅ THEN next states(current state, current player)

ELSE No moves possibleRemark: add state to list() removes duplicates

Final segment closeda box, no move left(rule exception).

Oswin Aichholzer Algorithms and Games4

Small Boards

. . .

2× 2, 4, 6, 8, 10: B (4,8,12,16,20?) 2× 3, 5, 7, 9: draw

3× 3: A (9); 3× 4: B (12); 3× 5: A (15); 3× 6: B (18?)

4× 4: B (16)

Oswin Aichholzer Algorithms and Games5

Schedule:• Testrun 1: Tuesday 21.01.2020 14:00-16:00• Testrun 2: Thursday 23.01.2020 14:00-16:00• Competition 1: Thursday 30.01.2020 10:00-18:00• Competition 2: Friday 31.01.2020 10:00-18:00

Interviews:• By appointment with TAs, and on request of a group• January, February, first week of March.• All members of a group have to be present• Approach has to be explained• Source code has to be provided for the interview and

can be inspected

Oswin Aichholzer Algorithms and Games6

Evaluation: maximum of 34% for practical part• Minimum requirement: All valid moves (and only valid

moves) for all legal positions are provided; results in theminimum of 17%

• 17-34% depending on approach and level of play