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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.
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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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B. Notation
Showing a position as a picture takes a lot of space, further
on a notation will be used to tell which moves are played
in order to skip showing every trivial turn. The notation that
has become somewhat established assigns each line, box and
dot a name from a1 to k11.
b d f h j
2
4
6
8
10
f3
i6
The notation for these two moves would be 1.f3 2.i6.
IV. ADVANCED STRATEGY
Knowing the long-chain-rule the next step to learning the
game is figuring out how to aim for the right number of
chains.
A. Preventing chains
We are taking a look at following position:
b d f h j
2
4
6
8
10
Turn: 29 Score: 0-0
There is one very long 19-chain and whoever gets it will
be the winner of the game. 29 moves have been played, it
is the turn of player 2 and according to the long-chain-rule
player 2 should aim for an odd number of chains. There is
a 2-chain at the top left which could extend into a 3-chain
with a move like a6. If that 2-chain becomes a 3-chain it
will be two long-chains in total, an even number. Not what
player 2 wants! The coming turn will prevent the short 2-
chain turning into a long 3-chain:
b d f h j
2
4
6
8
10
Turn: 30.b9 Score: 0-0
Player 2 played a sacrifice preventing the 2-chain from
growing into a 3-chain. Player 1 will now keep trying to build
another chain as he needs another one and plays 31.a4. Again
player 2 will play a sacrifice to prevent the second chain:
b d f h j
2
4
6
8
10
Turn: 32.b5 Score: 2-0
No second long-chain was created, there was two 2-chains
but those don’t lead to the creation of a domino thus don’t
affect the long-chain-rule. The only long-chain is the long
19-chain, the only one, odd. Player 2 wins the game by
sacrificing four boxes.
Sacrificing boxes is not the only way to prevent chains.
b d f h j
2
4
6
8
10
Turn: 32.a10 Score: 0-0
Player 2 plays 32.a10 creating a 6-loop at the top left
preventing the 6-chain that could otherwise be created by b9.
Note that sacrificing and building loops are both expensive
ways to prevent chains. Sacrificing gives boxes away while
loops will require the winner of the chain-battle to give
away two dominos which is four lost boxes.
The least expensive way to prevent an additional chain is
to merge the upcoming chain with an already existing one.
b d f h j
2
4
6
8
10
Turn: 32.d1 Score: 0-0
Player 2 has merged the 19 and the 4-chain, creating a long
25-chain. If he hadn’t done that but played 32.a2 instead then
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player 1 could have played 33.c2 in order to seperate the two
chains.
b d f h j
2
4
6
8
10
Turn: 33.c2 Score: 0-0
B. Preemptive Sacrifices
Given the position:
b d f h j
2
4
6
8
10
Turn: 27 Score: 0-7
No dominos have been created yet, it is the turn of player
2 who is seven boxes ahead but he has lost the long-chain- battle, the total number of long-chains is two, even. In order
to create a third chain player 2 could try to build space
for another chain at the bottom right thus playing 28.e2.
Player 1 would now merge the bottom chain with that space
preventing the additional chain by playing 29.k4 and then
player 2 would play 30.i2 to prevent the bottom chain from
growing even longer. Let’s try to predict the score resulting
from 28.e2 29.k4 30.i2.
b d f h j
2
4
6
8
10
Turn: 27+ Prediction: 14-11
Player 2 gets one of the two 2-chains and a domino from
the bottom 6-chain, the rest of the boxes goes to player 1.
The better move to get additional space for a third chain
is the loony move 28.e4.
b d f h j
2
4
6
8
10
Turn: 28.e4 Score: 0-7
Player 1 will take the d4 box and is then given a domino
decision. As he’s already seven boxes behind winning the
long-chain battle is his only chance of victory, he plays 29.ti4
(t for the taken d4 box). Player 2 now has lots of space at
the bottom to try building another chain with f1.
b d f h j
2
4
6
8
10
Turn: 30.tf1 Score: 1-9
Player 1 now has to sacrifice three more boxes at the
bottom. The next moves will be 31.e2 32.tk2 33.j3 34.tk4.
Aside from the three sacrificed boxes player 2 will also get
two boxes from one of the 2-chains and wins with a total of
14 boxes.
b d f h j
2
4
6
8
10
Turn: 34.tk4+ Prediction: 11-14
Worth noting: A preemptive sacrifice into a n-chain can
only be a winning move if the sacrificing player is ahead
by at least n − 1 boxes (e.g. if you’re one box ahead and
play into a 3-chain, your opponent will take one box and is
then given a domino-decision so he’ll definitely win if he
makes the right choice, hence an advantage by two boxes is
required). Likewise you need to be m−3 boxes ahead when
playing into a m-loop.
C. Dividing the board
Aiming for the right number of chains starts with aiming
for a number of regions that will later on contain chains.
Player 1 should start off trying to aim for two chains
(going for four chains doesn’t work out for reasons that will
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