Triangles strategy: how to win the dot game

Every game has two phases: a quiet phase where both players draw lines that give nothing away, and then a phase where every move hands the opponent at least one triangle. On a board this small the second phase is often brutally one-sided, with one player taking almost all 36 triangles. Good strategy is mostly about reaching it in the right position.

New to the game? Start with the Triangles rules.

Every triangle on the board is half of a small square, made from one horizontal side, one vertical side and one diagonal. Two facts follow from that.

Two lines means danger

A triangle with two of its three lines drawn is a free point for whoever moves next. Before every move, look at the triangles your line belongs to. If any of them already has a line, you are giving it away.

Each line touches 2 or 4 triangles

A diagonal or a side on the edge of the board belongs to two triangles in one square. A side in the middle of the board belongs to four, two in each square it separates, so it can do twice the damage.

A safe line is one that does not give any triangle its second line. Inside a single square, only three pairs of lines can sit together safely: the top and bottom sides, the left and right sides, or the two diagonals. Any other pair, such as two sides meeting at a corner or a side and a diagonal, belongs to the same triangle. So during the quiet phase a square can hold at most two lines.

Diagonals are the most self-contained lines on the board, because they only affect their own square. A side in the middle of the board limits what can safely be drawn in two squares at once.

Nobody scores during the quiet phase, so turns simply alternate, and that makes it a counting game. If the quiet phase lasts an even number of lines, the player who moved first has to make the first giveaway. If it lasts an odd number, the second player does.

The length is not fixed. On this 4 × 4 board, if both players stick to safe lines, the quiet phase can end after as few as 9 lines or as many as 18, depending on which lines get drawn. The longest version fills every square with both diagonals and leaves every side empty. That position is a disaster for whoever has to move next: every remaining line is a side, any side hands over at least two triangles, and each triangle taken sets up the next, so the opponent can collect all 36.

When you close a triangle, the line you just drew is also part of other triangles: one more if it is a diagonal or an edge side, three more if it is a middle side. Any of those that already had a line now has two, and you can close it on your extra turn. That is how a single giveaway turns into a long run.

Count the whole run

Before you give something away, follow it through: which triangle can your opponent close, and what does that line set up next? The first triangle is rarely the whole cost.

Order doesn’t matter when you take

Closing a triangle never makes another one harder to close. If you plan to take everything on offer, you will end up with the same triangles whatever order you take them in.

Once the safe lines are gone and there is nothing to take, every line you draw gives your opponent at least one triangle. Now the goal is to give away as little as possible and leave your opponent in a position that is just as bad.

Look for one-triangle sacrifices

The cheapest giveaways are usually lines that touch only one square: a diagonal or a side on the edge of the board. Middle sides touch two squares and are much more likely to start a long run.

A sacrifice passes the problem back

After your opponent takes the triangle, they must draw another line. If they have no cheap giveaway of their own, they have to open the board for you. The endgame is a race to run your opponent out of small sacrifices first.

On Hard, the computer always takes a triangle when one is available, otherwise picks a random safe line, and once no safe line is left gives away as few triangles as it can. It never counts safe lines ahead. On Easy it also draws a random line about a third of the time. So:

Leave nothing open

A triangle with two lines drawn will almost always be taken on the computer’s next turn, and on Hard every time.

Own the count

The computer does not count safe moves, so you can steer which of you runs out first.

Cash in on its giveaways

Once it is out of safe lines, it has to give you something, and on Easy its random lines often give away a lot. Take everything.

Give the minimum

When it is your turn to give, hand over one triangle if you can, ideally with an edge side or a diagonal.

• Drawing the second line of a triangle while safe lines are still available.
• Forgetting that one middle side can hand over four triangles at once.
• Giving away “just one” triangle without checking where the run leads.
• Treating lines in your color as yours. Whoever draws the closing line scores the triangle.