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Dots and Boxes 5x5 guide

Philipp Quach

November 2, 2016

Abstract— A hopefully easy to understand guide on Dots

and Boxes strategy covering basics as well as more advanced

knowledge.

CONTENTS

I Introduction 1

I-A Rules . . . . . . . . . . . . . . . . . . 1

I-B The third wall . . . . . . . . . . . . . 1

I-C Multi-captures . . . . . . . . . . . . . 1

II Strategy basics 1

II-A Doublecross-strategy . . . . . . . . . . 1

II-B The long-chain-rule . . . . . . . . . . 2

III Formalities 3

III-A Technical terms . . . . . . . . . . . . 3

III-B Notation . . . . . . . . . . . . . . . . 4

IV Advanced strategy 4

IV-A Preventing chains . . . . . . . . . . . 4

IV-B Preemptive Sacrifices . . . . . . . . . 5

IV-C Dividing the board . . . . . . . . . . . 5

IV-D Indirect chains, wasting moves . . . . 6

IV-E Ladders and edge moves – force

loops/merge regions . . . . . . . . . . 7

V Expert strategy 8

V-A Balance – second player approach . . 8

V-B Openings . . . . . . . . . . . . . . . . 9

V-C Visual approach to nim . . . . . . . . 10

VI Extra 14

VI-A Counting chains in a ”maze” . . . . . 14

VI-B Mirroring . . . . . . . . . . . . . . . . 15

I. INTRODUCTION

Dots and Boxes is a game that many people probably

remember having played in school, yet very few know that

it involves any strategy at all. While there’s lots of math to

Dots and Boxes this guide shall focus merely on aspects that

are relevant for playing the game and deliver them short and

simple.

A. Rules

On a grid of dots, players take turns drawing lines from

one dot to another horizontally or vertically adjacent dot. In

the process of doing so they will draw the walls of boxes.

Drawing the fourth wall of a box will earn the player a point

and forces him to draw another line. The player with the most

points wins.

B. The third wall

An obvious conclusion derivated directly from the rules is

that you don’t want to draw the third wall of a box as your

opponent can thereafter draw the fourth wall earning him a

point. As it is often the case with interesting games, we will

see exceptions to this simple statement.

C. Multi-captures

It is oftenly possible to capture more than just one box

per turn.

The leftmost box of aboves position has three walls, hence

the box can be completed by drawing the fourth wall as seen

below.

The rules state that upon completing a box you’re forced

to draw another line, hence the second box from the left,

which now has three walls, can be completed in the same

turn.

As a box has been completed once again the turn is still

ongoing. Analougously the rest of the boxes can be taken.

II. STRATEGY BASICS

If you want to beat your clueless buddies understanding

the following two subsections should suffice in order to beat

them almost all the time.

A. Doublecross-strategy

Key to playing Dots and Boxes strategically is an “all but

2”-trick used during the endgame.

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Turn1

: 34 Score: 0-0

Given endgame position consists of two chains one of

length 5 the other of length 20, there’s no free move that

doesn’t give away a box. As you want to offer as few boxes

as possible O has just played the dashed line into the shorter

5-chain. The clueless reaction by X on the dashed move

would be following:

Turn: 35 Score: 5-0

He just took the five boxes and played into the 20-chain. O

will take those 20 boxes and win the game. However, there

is a better move!

Turn: 35 Score: 3-0

Instead of taking all five boxes he uses the ”all but 2”-trick

finishing his move early as the dashed line does not complete

a box. X loses two boxes by letting O doublecross two boxes

of the 5-chain but gets the entire 20-chain in return.

1Turn includes the last played, dashed move

Turn: 36 Score: 3-2

B. The long-chain-rule

X won previous endgame by using doublecross-strategy,

O lost as he had to play into either 5 or 20-chain not having

any chance to defend himself. We conclude that we don’t

want to be the player who has to play into a chain first,

because if we do then doublecross-strategy will be used

against us. This leads us to the question answered by the

long-chain-rule: How can we avoid having to play into a

long-chain first?

The long-chain-rule for the 5x5 game of Dots and

Boxes states that if the number of long-chains is even

player 2 will have to play the first move into a long chain

and that if the number of long-chains is odd it will be

player 1. In other words: Player 1 should aim to build an

even number of long-chains, while player 2 should aim

to build an odd number of long-chains. Our previous

endgame had two long-chains, an even number, hence it was

O, player 2, who had to play into one of the long-chains.

Not all regions count as long-chains, a chain is long if it’s

length is ≥ 3 otherwise it’s a short-chain and then there’s

loops. A more realistic endgame position could look like

this:

Turn: 30 Score: 0-0

The number of long chains is 3, there’s a 3-chain in the

top-left, a 5-chain in the top-right and an 8-chain in the

bottom right. And then we have two short-chains, a 1-chain

in the bottom-left corner and a 2-chain above the 1-chain,

as well as a 6-loop. As the number of long-chains is odd it

should be player 1 who has to play into a long-chain first.

Let’s play some moves and see what happens.

2

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Turn: 31 Score: 0-0

Turn: 32 Score: 0-1

Turn: 33 Score: 2-1

Player 1 has now played into the loop. Similarly to the

”all but 2”-trick for chains there’s an ”all but 4”-trick for

loops:

Turn: 34 Score: 2-3

Player 1 will now take his 4 boxes, but is the one who

has to play into a chain first which happens as according to

the long-chain-rule. A curiosity however is that if the loop

would’ve been taken completely, without applying the ”all

but 4”-trick to it, it would have been player 2 playing into

the first long-chain. Loops do not count as chains, but every

loop that gets taken completely without having the ”all

but 4”-trick applied to it inverts the long-chain-rule!

Another curiosity could occur in turns 32, 33:

Turn: 32 Score: 0-1

Turn: 33 Score: 0-1

Player 2 made a mistake in turn 32 very common among

beginners. Always play into the center of a 2-chain! The

reason that a 3-chain is long and a 2-chain short, is that

you can always apply the ”all but 2”-trick after a move into

a 3-chain is played while the center move into a 2-chain

prevents it. Decisive is in fact not how many long-chains

exist but rather the number of times the ”all but 2”-trick

will be applied which is why a long-chain that gets taken

completely doesn’t add to the long-chain count.

III. FORMALITIES

Quick linguistic input to make further writing a bit easier.

A. Technical terms

Using the ”all but 2” or ”all but 4”-trick is not mandatory,

it is oftenly better not to use it, we call the decision of

whether to use it or not a ”domino decision”. A domino

is the shape of 2 boxes created after the ”all but 2”-trick is

used:

Domino

Any move that gives away one or more boxes is called a

”sacrifice”, while a move that also creates a domino decision

is called a ”loony move”.

Sacrifice Loony move

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