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A game that is between Noughts and Crosses and Sprouts for complexity?

I love the game of Sprouts but unfortunately at the moment it seems to be a bit too complex for the young student I have introduced to it. She already knows noughts and crosses (also called Tictactoe). I need a game that is

more complex than noughts and crosses,
less complex than Sprouts, and
similar to both in having a small number of possible moves at each turn.

Either a boardgame or a pencil-and-paper game will be fine. The game of Dots and Boxes using a small grid of squares is a possibility, but I'm hoping to find something that is better at getting a student into the frame of mind of "if I go there, he has three possibilities; now if he chooses the first, then..."

Answer*

"Connect 4" and "Gomoku" are similar to tic-tac-toe, but more interesting. Gomuko is simply tic-tac-toe on a larger board, where one has to connect stones into lines of length 5 with a few extra rules to reduce the first player advantage.

In connect 4, again the goal is to connect lines of a certain length, 4 this time, but now you can only place your disks at the lowest unoccupied position in a column.

You can also play the games above on a hexagonal grid instead of a square grid, if you want some more variation.

Something different and a bit more complex, but also nice is the game Hive. In Hive, each player either moves or places a hexagonal tile with an insect on it adjacent to the already placed tiles, according to specific placement and movement rules. The goal is to encircle the enemy Queen (bee). Thinking up to 4 moves deep is often required for good strategy, although good intuition can also get you pretty far. If you think Hive has too many rules, you can try removing some of the pieces from the game.

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