Tag Archives: Games

Test Your Intuition (17): What does it Take to Win Tic-Tac-Toe


(A few more quantum posts are coming. But let’s have a quick break for games.)

Tic Tac Toe is played since anciant times. For the common version, where the two players X and O take turns in marking the empty squares in a 3 by 3 board – each player has a strategy that can guarantee a draw.

Now suppose that the X player has a budget of Y dollars, the O playare has a budget of 100 dollars and before each round the two players bid who makes the next move. The highest bidder makes the move and his budget is reduced by his bid.

For example, if, to start with, the X player have 120 dollars, and if for the first move X bids 40 dollars and O bids 30 dollars, then X will make the first move and will be left with 80 dollars while O will be left with his 100 dollars.

What would you expect the minimal value of Y is so the X player has a winning strategy? Of course if Y>300, X can make 3 uninterrupted moves and win, but perhaps much less is enough?

(Thanks to Reshef Meir and Moshe Tennenholtz)

Update: Another variant of this game is when the player winning the turn pays his bid to the other player. (This version is called “Richman game,” while the variant above is called “Poorman game”. In this case if Y> 800 the X player can play three moves and win. And again the question is what is the infimum value of Y for which the X player can force a win…

The Retaliation Game


We have two players playing in turns. Each player can decide to stop in which case the game is stopped and the two players can go on with their lives, or to act.

The player that acts gains and the other player loose twice as much. 

So in the first round: player I can stop the game, if he acts he gets 1 and the other player II gets -2.
If the games continues in the second turn player II can stop the game, if he acts he gets 1 and player I gets -2.
If the games continues in the third turn player I can stop the game, if he acts he gets +1 and player II gets -2.

How to play this game?

Amir Ban on Deep Junior

Kasparov and Junior

Ladies and Gentelmen: Amir Ban (right, in the picture above) the guest blogger, was an Israeli Olympiad math champion in the early 70s, with Shay Bushinsky he wrote Deep Junior, and he is also one of the inventors of the “disc on key”. This post is about computer chess. 

Let me introduce myself: I’m Amir Ban, and I wrote the computer chess program Junior, also known as Deep Junior. When Gil invited me to guest-blog for him on the subject of computer chess, I was honored and pleased, but what can I write to introduce the subject to the uninitiated? Well, as luck has it, I participated last week in a unique event in Barcelona, Spain: a man with machine vs. machine match! The “man” was veteran grandmaster and Spanish champion Miguel Illescas. The “machine” he assisted himself by was my own Junior 10 (a commercially available product) on his Toshiba notebook. On the other side of the board was my latest Deep Junior 11, due to be released later this summer, running on an ordinary core-duo Dell desktop.

Susan Polgar, a former women’s world chess champion, and the eldest of the three Hungarian-Jewish Polgar sisters, describes the event on her popular blog. The comments to the blog entry show some difference of opinion:

Comment 1: “That’s not fair to the computer at all.”

Comment 2: “Big deal, Junior 10 vs. Junior 11 with a grandmaster moving the mouse….”

Visualization of Deep Junior Bxh2 sacrifice in the 5th game, New York, 2003.
Hmm… We need some perspective here. For that, let us take a few quick flashbacks, starting ca. 60 years ago with the pioneering efforts of Alan Turing, and especially Claude Shannon, who in 1950 wrote “Programming a Computer For Playing Chess“. In the article, Shannon lays out the foundations of computer chess, still practiced to this day: Given the current position where the computer must play a move, it will generate the tree of all hypothetical continuations: All moves playable at the position, then all possible replies to each of these moves, then all possible replies to the replies, and so on. Theoretically this process may continue indefinitely until a terminal position is reached, i.e. a checkmate, or a draw by the rules of chess. Unfortunately (or rather, fortunately) this possibility is merely theoretical, as the typical number of legal moves per position is around 40, and a chess game may last a hundred or so moves, the exponential explosion of possibilities forces a limit to the practical depth of the moves tree.

The Shannon trophy

Shannon proposed stopping the tree generation at some practical depth, and attaching an evaluation to the position at the leaf.

Continue reading