Conway's Soldiers

Conway's Soldiers

Conway's Soldiers or the checker-jumping problem is a one-person mathematical game or puzzle devised and analyzed by mathematician John Horton Conway in 1961. A variant of peg solitaire, it takes place on an infinite checkerboard. The board is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". As in peg solitaire, a move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible.

Arrangements of Conway's soldiers to reach rows 1, 2, 3 and 4. The men marked "B" represent an alternative to those marked "A".

Conway proved that, regardless of the strategy used, there is no finite series of moves that will allow a soldier to advance more than four rows above the horizontal line. His argument uses a carefully chosen weighting of cells (involving the golden ratio), and he proved that the total weight can only decrease or remain constant. This argument has been reproduced in a number of popular math books.

Simon Tatham and Gareth Taylor have shown that the fifth row can be reached via an infinite series of moves [1]; this result is also in a paper by Pieter Blue and Stephen Hartke [2]. If diagonal jumps are allowed, the 8th row can be reached but not the 9th row. It has also been shown that, in the n-dimensional version of the game, the highest row that can be reached is 3n-2. Conway's weighting argument demonstrates that the row 3n-1 cannot be reached. It is considerably harder to show that row 3n-2 can be reached (see the paper by Eriksson and Lindstrom).

Contents

Proof that the fifth row is inaccessible

Notation and definitions

Let the target square be labeled x0 = 1, and all other squares be labeled xn, where n is the number of squares away (horizontally and vertically, as in manhattan distance) from the target square. If we consider the starting configuration of soldiers, below the thick red line, we can assign a score based on the sum of the values under each soldier, (e.g., x2 + 2x3 etc.) When a soldier jumps over another soldier, there are three cases to consider:

  1. A Positive jump: this is when a soldier jumps towards the target square over another soldier. Let the value of the soldier's square be xn, then the total change in score after a positive jump is xn − 2xn − 1xn = xn − 2(1 − xx2).
  2. A Neutral jump; this is when a soldier jumps over another soldier but remains an equal distance from the target square after his jump (should this be necessary). In this case the change in score is -xn − 1.
  3. A Negative jump: this is when a soldiers jumps over another into an empty square away from the target square. Here the change in score is xn + 2xn + 1xn = xn(x2x − 1).

Choosing a value of x

Let us choose a value of x such that the change in score for a positive jump is 0. Thus, we require 1 − xx2 = 0 (choosing xn − 2 = 0 gives x = 0 which gets us nowhere). Therefore, x2 + x − 1 = 0, and solving this with the quadratic equation yields {x =\frac{\sqrt{5} -1}{2}, \frac{-\sqrt{5} -1}{2}}. We choose the positive root, {\frac{\sqrt{5}-1}{2} = \varphi \approx 0.61803\,39887\dots\,}, as its absolute value is less than 1, which becomes useful later in the proof. Rearranging φ2 + φ − 1 = 0, we can see that:

φ2 = 1 − φ [and multiplying by φ;]

φ3 = φ − φ2

φ4 = φ2 − φ3

etc...

Summing this to infinity causes all terms on the right hand side to cancel apart from the 1, i.e.,

{\sum_{n=2}^{\infty}\varphi^n = 1}

This can also be shown with the common ratio, where:

 \sum_{k=0}^\infty r^{k} = \frac{1}{1-r} (r < 1)

When r = φ:

 \sum_{k=2}^\infty \varphi^{k} = \frac{1}{1-\varphi} - \varphi - 1= 1

Solutions

Let us take the first example, where the target square is in the first row above the red line. We now consider the maximum possible initial score, that is when every square has a soldier on it. The sum of the squares on the first row below the red line, is {\varphi +2\varphi^2 +2\varphi^3 + \ldots = \varphi +2(\varphi^2 +\varphi^3 + \varphi^4 + \ldots)}. [Drawing a diagram helps to visualise this]. In the next line down, every square is one further away from the target square, and so has value φ times the square above it, and so on for all the rows below the line.

Therefore, the total value of all the squares below the line is equal to:

{{S}_1=(\varphi +2(\varphi^2 +\varphi^3 + \varphi^4 \ldots))(1 +\varphi + \varphi^2 + \varphi^3 + \ldots)}.

At this stage we observe that \sum_{n=2}^{\infty}\varphi^n = 1 from above, and thus the above expression simplifies to:

S1 = (φ + 2)(1 + φ + 1) = (φ + 2)2 = 4 + 4φ + φ2 = 5 + 3φ, using φ2 = 1 − φ for the last step.

Therefore the sum of all the squares below the line when the target square is immediately above the line is 5 + 3φ.

The next case we consider is when the target square is in the second row above the red line. In this case each square under the red line is one square further away from the target square than in the previous example, so the total score now is found by multiplying the total score we obtained by φ to give:

S2 = φ(5 + 3φ) = 5φ + 3φ2 = 5φ + 3(1 − φ) = 3 + 2φ.

Similarly:

S3 = 2 + φ

S4 = 1 + φ

S5 = 1

Thus, we have shown that when the target square is five rows above the red line, the maximum possible original total score of all the soldiers is 1. In reality, with a finite number of soldiers the total will be less than one. Therefore, since a positive jump towards the target square leaves the total score unaltered, and the final score on all the soldiers must be at least 1 (φ0 = 1 and the scores on any other soldiers left), the fifth row cannot be reached with a finite number of soldiers originally below the line.

This completes the proof.

QED.

References

  • E. Berlekamp, J. Conway and R. Guy, Winning Ways for Your Mathematical Plays, 2nd ed., Vol. 4, Chap. 23: 803—841, A K Peters, Wellesley, MA, 2004.
  • R. Honsberger, A problem in checker jumping, in Mathematical Gems II, Chap. 3: 23—28, MAA, 1976.
  • H. Eriksson and B. Lindstrom, Twin jumping checkers in Z_d, Europ. J. Combinatorics, 16 (1995), 153–157.

External links


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Conway — may refer to: Contents 1 People 1.1 Surname 1.2 Given name 1.3 …   Wikipedia

  • John Horton Conway — Infobox Scientist name = John Horton Conway |300px image width = 300px birth date = birth date and age|1937|12|26|mf=y birth place = Liverpool, Merseyside, England residence = U.S. nationality = English death date = death place = field =… …   Wikipedia

  • John Horton Conway — Pour les articles homonymes, voir Conway. John Horton Conway John Horton Conway en 2005 Naissance 26& …   Wikipédia en Français

  • John Horton Conway — (* 26. Dezember 1937 in Liverpool) ist ein englischer Mathematiker. Inhaltsverzeichnis 1 Leben und W …   Deutsch Wikipedia

  • James T. Conway — Infobox Military Person name= James Terry Conway born= died= placeofbirth= Walnut Ridge, Arkansas placeofdeath= placeofburial= caption= 34th Commandant of the Marine Corps (2006 present) nickname= allegiance= flagicon|United States United States… …   Wikipedia

  • Tom Conway — Infobox actor bgcolour = silver name = Tom Conway imagesize = 205px caption = from the trailer for Grand Central Murder (1942) birthname = Thomas Charles Sanders occupation = actor birthdate = September 15, fy|1904 location = St. Petersburg,… …   Wikipedia

  • Tom Conway — (* 15. September 1904 in Sankt Petersburg, Russland; † 22. April 1967 in Culver City, Los Angeles County) war ein britischer Schauspieler und Hörspielsprecher. Inhaltsverzeichnis 1 Leben 2 Filmografie …   Deutsch Wikipedia

  • Jack Conway — Nombre real Hugh Ryan Conway Nacimiento 17 de julio de 1887 …   Wikipedia Español

  • Jack Conway (filmmaker) — Infobox actor bgcolour = silver name = Jack Conway imagesize = 200px caption = birthdate = July 17 1887 location = Graceville, Minnesota, USA height = deathdate = October 11 1952 deathplace =Pacific Palisades, California yearsactive = 1909 1948… …   Wikipedia

  • Craig Conway (actor) — Craig Conway Born 1975 South Shields, England[1][2] Occupation Actor Years active …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”