Does Going First Matter in Dots and Boxes? Count the Dots

Going first in dots and boxes matters, but whether it helps or hurts depends on the board, and you can tell which by counting the dots. On all 14 small boards we solved exactly, the first player won only when the board had an odd number of dots; with an even number, the best the first player could do was a tie. The 3x3 board (16 dots) is a second-player win, 6 boxes to 3.

The same count shows up in real games. Between two copies of our Easy or Medium computer, the first player scored 62.7% and 66.6% on 4x4 (25 dots), but only 29.5% and 20.9% on 3x3 and 41.8% and 41.5% on 5x5 (16 and 36 dots), where 50% is an even game. Between two Hard computers, which fight over the long chains, the first player won only 86 of 10,000 games on 3x3, and scored 51.6% on 4x4 and 52.5% on 5x5, close to an even game.

Every number below comes from Poozee's own solver and from seeded games between the computer players behind Poozee's Dots and Boxes, 144,000 games in all. A tie counts as half a win for each player. New to the game? Start with how to play dots and boxes.

With perfect play, count the dots

A position in dots and boxes is just the set of lines drawn so far, so small boards can be solved outright. For boards with up to 24 lines (3x3 boxes and smaller), our program works out the value of every possible set of lines, from the full board back to the empty one. For three bigger boards, the exact search behind the game's Hard level settled the empty board instead, taking up to a few minutes each. Here's every board we solved:

On all 14 boards, the first player wins exactly when the number of dots is odd: 2x2 (9 dots) and 2x4 (15 dots). That's a pattern we found, not a proof, but it also fits a much bigger board. Joseph Barker and Richard Korf solved 4x5 in 2012, and their paper Solving Dots-And-Boxes (AAAI 2012) reports that the 4x5 board, with 30 dots, "is a tie given optimal play". Our solver can't settle Poozee's 4x4 and 5x5 boards (40 and 60 lines), so for those the computer games below are the evidence.

The tables also say where to start:

• On a board with an odd number of dots, stay out of the middle. On 2x2, all 8 outer lines win 3‑1 and the 4 inner ones only tie. On 2x4, 18 of the 22 first lines win 5‑3, and the 4 lines at the very center (the two middle pieces of the center line across, and both pieces of the center line down) only tie 4‑4.
• On a strip one box high and 2, 4 or 6 boxes long, only one first line holds the tie: the line across the middle that cuts the strip into two equal halves. Every other first line loses by two boxes.
• With an even number of dots, no first line wins. On 3x3 all 24 lose 3‑6, as our 3x3 solution shows in detail; on 2x3 all 17 lose 2‑4; and on the strips of 1, 3, 5 and 7 boxes every first line loses by one box.

Why the dots decide: the last turn

Every turn ends with a line that closes nothing, except the last turn of the game, which always ends by closing a box. Count it through and the number of turns in a game always equals the number of dots plus the number of double-crosses (lines that close two boxes at once). It held in all 144,000 of our games.

The player who takes the last turn usually wins, because the last turn takes the last chain of boxes. With no double-crosses, the number of turns is the number of dots, so:

1. With an odd number of dots (4x4 has 25), the odd-numbered turns belong to the first player, and so does the last one.
2. With an even number (3x3 has 16, 5x5 has 36), the last turn belongs to the second player.
3. Each double-cross hands the last turn to the other player.

Our Medium computer never hands boxes back on purpose, so its games show this plainly. In 10,000 games between two Medium computers on each board:

On 3x3, every one of the first player's 2,085 wins came in a game where it had the last turn. On 4x4 the odd dots hand the first player the last turn in most games. On 5x5, more games have double-crosses, so the last turn is more evenly shared, and the second player's edge is smaller than on 3x3.

How much going first is worth at each level

We had two copies of each computer level play 10,000 games on each board. Easy takes any box it's offered but otherwise draws a random line 40% of the time; Medium never draws a box's third side while it has a choice; Hard plays like Medium until 18 or fewer safe lines are left, then searches ahead and plays the endgame perfectly. All three pick at random among the lines they like best, so the games vary from seed to seed. The table shows the first player's share of the points:

Each score's 95% margin of error is under one point either way. What stands out:

• At Easy and Medium, the dots decide. The first player is favored on 4x4 and the second player on 3x3 and 5x5, at both levels. The first 1,000 games of each are the ones in our rules guide, which found the same flip.
• At Hard, 3x3 is a near-certain loss for the first player, as perfect play says it should be. The first player won 86 of 10,000 games.
• At Hard on 4x4 and 5x5, the edge almost disappears. The first player scored 51.6% on 4x4, where 2,012 games were ties, and 52.5% on 5x5: small edges, but bigger than the margin of error. On 5x5 that's the opposite of Easy and Medium. Hard hands boxes back to keep control, so its games almost always have double-crosses (all but 18 of the 10,000 on 5x5), and the dots alone no longer decide who gets the last turn.

Does your first line matter?

On the small boards it can: the exact results above include first lines that throw away a win or a tie. On Poozee's bigger boards, we played 2,000 games between two Hard computers from one line of each kind (lines that a rotation or reflection of the board turns into each other are the same kind), and counted the first player's score.

Hard's games from the same first line still differ, because its next lines are random safe ones until it starts searching. On 4x4, a first line on the outer edge scored about 4 points more than one inside the board, more than the margin of error. On 5x5 the gap was under 2 points, within the margin of error, and every kind of first line scored between 49.1% and 52.9%. So the first line is worth a few points at most, and only on 4x4 could we tell it apart from chance: start on the edge.

Going first or being the better player?

To compare the first move with skill, we matched each level against the next one up, 2,000 games each way on each board, with the stronger player moving first in one match and second in the other. The table shows the stronger player's score.

• Against a much stronger player, going first won't save you. Hard scored at least 95.7% against Medium on every board, whichever side it played. The first move changed its score by 4.3 points at most, on 3x3, where moving first is the losing side.
• Against a player one step better, the right side can beat the skill gap. On 3x3, Easy moving second beat Medium moving first: Easy won 1,043 of the 2,000 games, and Medium scored just 47.9%. Moving second instead, Medium scored 89.2%. On 4x4 the swing went the other way, from 87.9% for Medium moving first to 63.2% moving second.

Between Medium and Easy, the side that helps is the one the dots favor: second on 3x3 and 5x5, first on 4x4, the same as in the equal games above.

What to aim for when you go first

Once the safe lines run out, every line gives boxes away, and the open boxes form chains and loops. Whoever has to open the first long chain (three boxes or more) usually loses: the other player takes all but two boxes and hands the last two back, which keeps them in control. Our dots and boxes strategy guide explains this double-dealing in detail. Which player gets control is decided earlier, by the long chain rule: add the dots to the number of long chains (loops don't count). An even total gives control to the first player, an odd total to the second.

So when you go first, aim for:

Here's what that looks like on Poozee's default 4x4 board. This position came up in one of our Medium games after 20 lines. It's the first player's move, no box has been taken, and three safe lines are left (highlighted). The tinted boxes are a chain of eight (green), six boxes that aren't settled yet (blue) and a pair in the bottom-right corner.

• The top edge of the top-right box, or the right edge of the box under it, wins. The second player's best reply is the last safe line, the other one of those two, and the six boxes close into a loop.
• The line under the top-right box loses. It leaves no safe line, and the six boxes become a chain running from the top edge to the right edge.

Count it the first player's way. After the winning line, the board has one long chain (the eight) and a loop, which doesn't count: 25 + 1 = 26, even, so the first player gets control. After the losing line there are two long chains, the eight and the six: 25 + 2 = 27, odd, and control goes to the second player.

With best play from here, the first player gives the pair away, and the second player takes it and has to open the loop. The first player takes two of its boxes and hands the other four back, so the second player then has to open the chain of eight, and the first player takes all eight: 10‑6. After the losing line, the second player gets control instead and wins 12‑4.

How often the count decides a game

Real endgames aren't always this tidy. When the safe lines ran out in our Hard games, the board had split into chains and loops, with at least one long chain, in 2,040 of the 10,000 games on 4x4 and 529 on 5x5. In the rest, some group of boxes wasn't a simple chain or loop yet (it branched at a box, for example), so the count wasn't settled. In the tidy ones, the count picked the winner almost every time:

• On 4x4, the player the long chain rule gave control to won 1,511 of the 1,549 games that weren't ties. When the count favored the first player, it won 816 of 1,049 games and lost 33; when it didn't, the first player won 5 of 991.
• On 5x5, the rule's player won 511 of 529 games. When the count favored the first player, it won 277 of 285; when it didn't, 10 of 244.

On 3x3, the last turn is worth much less. In the Hard games there, the first player had the last turn in 850 games and still won only 68 of them.

What to do with this

1. Count the dots before you start. With an odd number (4x4 has 25), the first player gets the last turn unless double-crosses change it; with an even number (3x3 and 5x5), the second player does. Against casual players that decides a lot. Against strong ones, who hand boxes back on purpose, the long chain count takes over.
2. Take turns going first. Poozee does it for you: you draw first in your first game, and every new game switches.
3. Going first on 3x3, expect to lose against good play. The second player can always win 6‑3, so your hope is a mistake. The 3x3 solution shows which first line sets the most traps.
4. On 4x4 and 5x5, play for the long chain count. Going first, you want an odd number of long chains on 4x4 and an even number on 5x5. Closing a group of boxes into a loop instead of a chain, as above, changes the count by one.

Pro tip: When only a few safe lines are left, count the long chains each one would leave, add the dots, and check who that gives control to. Winning the count is only half the job: when your opponent opens a long chain, take all but two boxes and hand the last two back.

Keep playing

• Play Dots and Boxes on 3x3, 4x4 or 5x5, against the computer or a friend
• Dots and boxes strategy: safe lines, chains, control and double-dealing
• Dots and boxes 3x3 solution: who wins the smallest Poozee board, line by line
• Does going first matter in checkers?, the same question for a game where perfect play is a draw