## Non-Chess Retrograde Analysis

In most Retrograde Analysis puzzles, the solver is presented with an arrangement of pieces on a chess board and must deduce something about what has happened previously in the game based on the position in which the pieces are currently found. Lots of good examples can be found here: http://www.janko.at/Retros/index.htm

But there’s no reason that this “thinking backward” sort of problem can’t be applied to other things besides chess…

## Tic-Tac-Toe

by Les Marvin

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?

## Dots & Boxes

by J. Kisenwether

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?

## Othello

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?

## Checkers (International)

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.

By Jean Bertin

How did the man on e7 get there?

## Bridge

By Jonathan Mestel

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?

## 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.

## Life

(click here is you need to know the rules for Life)

Are there positions in John Conway’s famous cellular automata Life which cannot be reached from any previous position?

Prove it.

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## SOLUTIONS

## 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!**

*2009 Update – This page was originally written in the mid 1990’s and hasn’t really been changed much since (apart from moving it to WordPress from the now defunct Geocities). And yet, it has managed to inspire others to look into non-chess retros. Notably Alain Brobecker, who published two articles inspired by this page:*

http://abrobecker.free.fr/text/othello.pdf (In French, but you can puzzle out what the problems are from the diagrams.)

http://abrobecker.free.fr/text/tictactoe.pdf

*As of Sept 15th, 2009 – the Geocities site from which I copied this is still the #4 result when you search for Retrograde Analysis on Google. I’m amazed…*

2011 update – Retrograde Othello has turned up in the most recent MIT Puzzle Hunt! –

http://ihavetofindpeach.com/puzzles/mega_man/you_shall_understand_what_hath_befallen/

February 12, 2015 at 7:35 pm |

Howdy. I am a checkers (8×8 British/American checkers) problem composer. I usually don’t actually compose my problems, but rather find flaws in published games, and sometimes my corrections are delightful (and sometimes award-winning) problems. I rarely modify these problems to make them more artistic, because they have plenty of value as theoretical novelties at the checker board. If I begin actually composing, I would like to compose retrograde analysis checkers problems. I suppose the first step is to start investigating the various clues to earlier moves. And also: what can be deduced about preceding moves, and under what conditions can a move be deducible, and when can we deduce that no move could have led to the position in question.

December 13, 2015 at 3:23 pm |

[…] https://joekisenwether.wordpress.com/non-chess-retrograde-analysis/ […]

February 27, 2017 at 8:45 pm |

[…] the great trove for them is maintained by Angela and Otto Janko here. Joe Kisenwether has some examples from games other than […]

October 22, 2018 at 11:25 pm |

Retrograde analysis is important in both computer science and medicine, reasoning backward from symptom to root cause.

November 13, 2018 at 2:08 am |

You might be interested to know that there was a retrograde analysis puzzle in the 2013 MIT Mystery Hunt: http://www.mit.edu/~puzzle/2013/coinheist.com/oceans_11/turnary_reasoning/index.html It was written by me (chess), Tanya Khovanova (checkers), and Alan Deckelbaum (Magic). Also, just recently (2018), Jerry Butters and Jim Henle published an article in the Mathematical Intelligencer on retrograde analysis in baseball!