Joseph's Squares is a game to be played on pen and paper. It goes like this:
1. draw 2 squares;
2. player A draws a line between one side of a square to a side of the other square;
3. once a side of a square has been connected it can't be used again.
4. lines can't be crossed ever.
5. player B draws another line again connecting one unused side of a square to an unused side of the other square.
6. and so on until a player can't draw any lines, that player loses.
In a game with 2 squares the second player can always easily win, but once you get that you can start playing with 3 squares, 4 squares and so on.
It isn't meant to be the best game of all times, just better than tic-tac-toe.
I haven't tried to play with triangles or other shapes but could be interesting too.
You have to be a reasonable human and not try to draw confusing lines. These confusing lines cannot change the game in theory but they can confuse everything and ruin the game for both players (it's ridiculous that I have to say that but many people to whom I tried to teach the game had this stupid idea, so I'm saying here).
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Interesting. Ever played Nim?
It's very mathematical. I used to teach it to my math and critical thinking students.
If you're playing me in Nim and you give me the option to go first or second, I will win 100% of the time.
Nim should obviously be implemented in Nim.
Yes, I did. I once said this was like a 2D version of Nim, but no one understood because no one else had ever heard of Nim.
Wait, now that I'm thinking, when I was a kid my grandfather once played a game with matches with me, he put a bunch of matches on the table then we took turns removing either 1, 2 or 3 matches until one couldn't.
Yep. That's the Nim I'm most familiar with. Simple game. You can use any tokens that are readily available like matches, sugar packets, coins, toothpicks, pebbles, etc.
I found that leveraging children's desire to be victorious was a good way to interest them in math. Go was another great game for teaching them math.
I forgot to attach an example game to the explanation: 

