**Jate's Game**

We played a couple rounds of another game I got off NRICH. Here is the link: http://nrich.maths.org/6589. The basic rules:

- make a number line and label the whole numbers from 1 to 20 (when we started playing, we only went from 1 to 10)
- The first player chooses two numbers on the number line, crosses them out and circles either their sum or their difference and circles that number. The two crossed out numbers are now out of play.
- The next player works with the circled number and chooses another number that hasn't been crossed out. They then circle either the sum or the difference and cross out their numbers.
- Each player repeats step 3 until there are no legal moves.
- The last player to make a legal move is the winner!

It might help to study the picture below. It isn't especially clear or tidy, but you should be able to get the point:

Jate is working with 17 (on the right of the paper, circled and not yet crossed out)

He has to choose another number. Since so few options remain, he has two choices, either of which causes him to win the game. See if you can figure out his move. . .

He chose 12, 17-12 is 5, so he circled 5 and crossed out 12 and 17. That left me with no remaining legal moves.

Now, this hasn't come up yet, but I have seen the opportunity to argue about a special case: what if he had chosen 5 and circled 12 (= 17-5), crossed out the 17 and 5? Could I have chosen 6 and circled it, because 12-6 = 6? Seems to me that you have four options for rules to deal with this situation:

- 2n - n = n isn't allowed as a legal move
- 2n - n = n means that you cross out 2n, circle n and the next player uses n for their move
- 2n - n = n means that you cross out 2n and n, the next player chooses 2 new numbers
- 2n - n = n means that you circle n, then cross out 2n and n, and this player automatically wins (because the next player can't make a legal move with the circled n, it has already been killed)

In our case, Jate avoided this ambiguity (did he do it intentionally?) As always, I invite you to explore each variant.

We played 1 and a half games. The second game, it turned out Jate had set a booby-trap on one of the numbers, so I automatically lost when I circled that number. How do I know it was pre-planned and not an arbitrary, last minute rule change? Because he told Jin and they both started laughing as soon as I circled the trick number.

*Jin's Game*
Writing up the notes on the tidy-up game (see here) reminded me that I hadn't ever fully explored toetactic (inverse tictactoe.) Jin was amenable to playing, so we drew a couple of boards and played tictactoe as a warm-up. Then, I explained we were trying to force the other person to get 3 in a row and we played a couple of boards that way. No deep analysis tonight, but it was fun hearing him say "inverse tic-tac-toe" and seeing the plans go through his mind as he worked out his strategy.

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