In this partially completed tic-tac-toe game, both players were experts (neither one ever afforded the other an opportunity to force a win.) What were the first and the last moves played?
In this Dots & Boxes game between Leon and Noel, Leon made both the first and the last move. If both players always took the largest possible number of boxes available to them on their turn, what did the board look like before the last 16 lines were drawn?
In this short othello game, it is possible to determine every move that was made. Thanks to Erich for pointing out the possibilities in retro othello. There is a lot to be done here...
Scrabble
In Retrograde Scrabble, we assume that no player has played any word which would have failed if challenged (unless we can prove otherwise of course.)
By J. Kisenwether
In this two player game your opponent, who always makes the highest possible scoring word that she can, has just played. What is your best move?
International checkers is played on a 10 by 10 board. Uncrowned men are allowed to jump both forward and backward, but non-jumping moves may only be made forward.
You are walking past a bridge table when one of the players is called away to an urgent phone call.
Five cards are thrust into your hand, and you are told "You need all the remaininbg tricks, there is no trump."
You cash the three aces and both opponents follow suit. You then lead the club deuce and the next hand plays small. Do you play the Ace or the queen?
Author Unknown
In one hand of bridge, all four deuces took tricks. Game was bid and made. How many overtricks? Solutions Page
Lines of Action
The central question in applying retro-analysis to anything is: Which positions are legally reachable? Here is a page that asks that question for the game Line of Action.
Are there positions in John Conway's famous cellular automata Life which cannot be reached from any previous position?
Prove it.
A Parting Thought
As the previous example illustrates, retrograde analysis need not examine only games. In fact, any system which consists of a number of different states and a set of rules for how to go from one state to another can be retro-analyzed. In particular, it can be applied to axiomatic mathematics:
The axioms are analogous to the starting position.
The rules if induction are analogous to the rules for moving pieces.
Positions on the board are analogous to formal sentences.
Legally reachable positions are analogous to theorems.
and proofs of theorems are analogous to the games that lead to the legally reachable positions.
In fact, a game that leads to a particular position is called a proof game for that position by retro-chess fans.
The implications fo this analogy can be pretty surprising. for example there is Godel's Theorem:
For sufficiently complicated games, there exist positions for which it is impossible to decide if they are legal or not!
If you have any comments or any other examples of non-chess retros, then please Email me
Solutions Page
Solutions Page
Solutions Page
Solutions Page
Solutions Page