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The classic pencil game Sim on six points. You and the AI take turns drawing lines between the dots of a hexagon, and whoever closes a triangle in their own colour loses. No luck, no draws — just nerve and counting.
Six dots sit on a hexagon, and every pair of dots can be joined by one line — fifteen lines in all. Tap one dot and then another to draw the line between them in orange, or slide your finger from one dot to the other. If three of your own lines ever form a triangle with its corners on the dots, you lose.
The AI draws its lines in dashed blue and has the same rule: a triangle of three blue lines loses for the AI. Its most recent line glows white. Lines of different colours never count together, and a triangle only counts when its three corners are dots, not where lines cross in the middle. You always move first, and a line can never be erased or taken by the other side.
After you pick your first dot, the other dots show what each line would do. A dot with a red dashed ring means that line would close one of your triangles and lose the game on the spot; a dot with an orange ring is safe. Tap the same dot again to cancel your choice, or tap a different dot that is already joined to it to start from there instead.
The numbers next to "You" and "AI" count the safe lines each side has left — lines still free that would not close a triangle in that side's colour. A game can never end in a draw: before all fifteen lines are drawn, someone must close a triangle. If you run out of safe lines, every line you can still draw loses.
Beat the AI and the next game begins with your win streak one higher; your score is that streak. The AI plays more carefully the longer the streak runs, making fewer loose moves each game. One loss ends the run, and you can save it to the Hall of Fame from the result card.
Your first three lines can be completely safe. A triangle needs two of your lines meeting at a dot, so as long as no two of your lines share an end, nothing you have drawn can ever be closed. Six dots can take three such lines at once — one from each opposite pair of the hexagon is an easy pattern — and every line after that starts to create danger. The moment two of your lines meet at a dot, the line joining their far ends becomes forbidden for you, and the red rings will show it.
The game is decided by the safe-line counters, not by the picture. Every free line is in one of four states: safe for both sides, safe only for you, safe only for the AI, or dead for both. A line that is safe only for you is a private reserve: the AI cannot draw it without losing, so it will still be there later. Spend your turns on lines that are safe for both sides — each one you take is also a move the AI no longer has — and keep your private lines for the end, when the side that runs out first loses.
Watch what your move does to both counters. A line that the AI could also draw takes one option away from it. A line joining the far ends of two AI lines that meet at a dot costs the AI nothing — that line was already forbidden for the AI, so if it is safe for you, it is one of your reserves; save it. And a line that touches several of your own lines can cost you two or three safe lines at once, because every pair of your lines that now meets at a dot forbids the line across their far ends. Before you lift your finger, glance at both numbers and ask which one this line will lower.
Sim was invented by the mathematician Gustavus Simmons in 1969. It cannot end in a draw because of a result from Ramsey theory: however fifteen lines on six dots are coloured with two colours, a one-colour triangle appears. Computer searches in the 1970s showed that the second player always wins with perfect play. This AI does solve the game, so as the first player you win by exploiting its loose moves — and those grow rarer with every win in your streak.