Boardwalk Philosophy Page

by Mark G. Meyers

Of the Finite Versus the Infinite, or the Mind’s Rise to Infinity

I was a novice when I first created these games as a teenager. I made attempts at forging practical strategic concepts in the face of various random card arrangements, or “deals”. By way of those concepts, I began to orient my mind to the playing field.

With the passage of time, I felt that I had developed some degree of familiarity with the games, and I established a list of what I felt would be the most commonly useful concepts in strategy.

Over the years, I wondered how much my skill could improve. I experienced a number of plateaus in how many good options I could see. At each plateau, I felt like I was seeing all that could be seen, but over time, I would then be surpassed by new breakthroughs, producing even greater spheres of strategic options.

I have now come to see why my development as a player may ultimately be open ended. I have come to see that there may never be a best or most perfect play, such as what may or may not be possible by way of a perfect game-playing computer program.

For any real game situation, there is the chance of success in each specific strategic response, but there is also the variation in the amount of success gained. Both the odds and the amount of success must be considered together, where options range widely. Even the goal of the game itself is just not cut-and-dried, such as winning at solitaire versus outscoring one’s opponent in competition, or by way of what is called “tossing” versus attempting to score a perfect game. (Tossing is where a player chooses greater odds of success by targeting a lesser end point score.)

After 30 years, I have documented a sufficient series of strategic paradoxes to support the position that, in the end, the master of the game may be the player who sincerely sees more than any such thing as a best possible play. I now see so many interesting and provocative options that I find myself amazed by their diversity. In Boardwalk Solitaire, I have come to experience a love and appreciation as a player in the face of its expanding universe.

On Luck Versus Skill in Boardwalk Solitaire

A deal is defined as a specific arrangement of cards at the beginning of a game.  We will presume any arrangement we consider to be random. 

The number of possible deals is so large, that the odds are very strongly against any one player experiencing the same game twice in a lifetime (or of being able to recollect such an occurrence if it did).  Forming a precise, premeditated strategy is not regarded herein as humanly possible. 

In these games, the element of randomness also gives us the element of luck, and luck cannot be removed.  Because of this, there is no ultimate law by which the player may know precisely what to do. The only way to know what precisely to do is to develop a strategy by which one performs plays that are in perfect accordance with a random deal.  A random (or indiscernible) strategy appears to be needed in order to play perfectly.  Vertigo is used as a Boardwalk term to name the effect produced by the deal's capacity to disorient the player on strategy (all values are transient).

When playing identical deals in competition, a player who portends to know what to do may be beaten by another player who creatively draws upon greater possibilities.  But in order to beat the game itself, and not just another player, one must first obtain a position whereby they can accurately predict its outcome.  Where the game itself has been beaten, its outcome has been shown to be known, and it should no longer be regarded as a Boardwalk game. The most direct example of knowing the outcome in any attempt to beat the game would be shown by a player's ability to win (the given variation) on every attempt. 

Hysteresis in Boardwalk is a word for saying, "variability in outcome by luck alone". It may be possible to produce an argument for beating the game on this basis:  That when the player has exercised absolutely every last bit of skill that is possible, only the hysteresis will remain.  Where hysteresis produces the only variation from a totally predictable outcome, then the variation in outcome has been reduced to being purely the product of luck, and one might consider arguing the variation as no longer being a Boardwalk game.  Care needs to be taken when producing the statement that all non-hysteresis has been removed, and to say so surely.  Prior to this, the argument should not be honored.

This game offers the player an opportunity to apply great skill, and even to make variations requiring greater skill (such as with a larger deck of cards). My experience as inventor of the games, over the years, is now that the sense of strategy in Boardwalk teases the player into attempting to beat the game itself. Since by its definition it does not appear possible to achieve this result as a player and still call it a Boardwalk game, the game itself is offered as the equivalent of a modern, Buddhist koan, or as a question which cannot be answered.  The difference here is that there may be, but by definition, for a living variation it is never presently known if such a thing can be found.

If a computer player (or game-playing algorithm) were developed such that it could calculate everything, then that would produce a perfection standard. I think we have to bear in mind on this that such an algorithm would have to be able to quantify each possible arrangement of cards, in its mathematical/strategic value with absolute precision. There will be a diversity of play possibilities in tandem with a diversity of end goals, and each play possibility may have its own array of possible end goals, each with its own measurable value. This seems to present an astronomical number of calculations. A par algorithm is the idea of a computer program that would play the game with such absolute mathematical thoroughness and precision, that it would serve as a mathematical perfection standard. One would only be able to beat such a program, on average, less than half of the time. Where an opponent to par does better than that, then at that point the par algorithm would be found to be less than what it was held up to be (and in need of revision).

The Complexity of the Game

There are four existing variations in Boardwalk Solitaire; Hopscotch, Lucky Seven, Sqatsi and Cheshire. Hopscotch is the simplest and hardest to win, and Cheshire is the most complex and easiest to win (when played very carefully). In fact, the purpose of Cheshire is to illustrate a point. Where it can be won every time, it is intended to portray itself as mundane.  (Note: As it turns out, this statement may have been premature!)

Boardwalk Solitaire introduces a genre of gaming, pitting skill against random luck, each at its own height. From a design perspective, this is the intended nature of the game. Ideally, it is like golf. The range of all of human skill is to fill a boat that swims in a sea of random possibilities. Recall that in these games there is only one human player, and so a human opponent is not available to implement the level of skill for the game.

The problem that appears to arise with attempts at incorporating an opponent into a deck of cards is the narrowness of the opponent’s range in potential skill, from its floor to its ceiling. To make a simpler opponent for the human player, the skill floor is lower, but the skill ceiling also tends to be badly reduced. When making a variation that can match greater human skill, the skill floor also tends to rise. As an example, Lucky Seven seems about half as complicated to play as Sqatsi. Note that the rules of the two are not greatly different, but that strategy in Sqatsi is significantly more involving than in Lucky Seven.

In order to step up to Sqatsi, the player must face twice the complexity. Any reward for doing so comes in the form of a game with a higher skill ceiling. While it is worth noting that the skill floor for Sqatsi still serves one who plays with lesser thought, there is still the business of having to cope with the complexifying aspect of Sqatsi over Lucky Seven, which is to have to cope with two ascending walks in tandem.

Enter two categories of ideas for achieving greater complexity while remaining within the constraints of the three existing entities; upper walk, lower walk and hand, as well as within the three existing game types; Hopscotch, Lucky Seven and Sqatsi . 1) A larger deck of cards could be used, and 2) cards colored on both sides by suit could be used to thrust the player into a significantly greater world of “information”, or that which can be known by looking at the board.

In the first idea, 5 each of 20 face values could be dealt as "100-card Boardwalk" (such as with a 9-column upper walk, a 10-column lower walk and a 5-card hand). There would be 14 cards face-up on the deal to carefully deliberate upon. Two 5-card row 1 stacks could be cleared against a 10-column lower walk before having to file upon the upper. 5 upper walk stacks could be led from simultaneously against 10 lower walk columns, giving rise to an exponential development in multiple-lead-stack possibilities (off of a 9-column upper walk). The skill floor for 100-card Boardwalk would (I would imagine) have to be higher than for its 52-card counterpart, but I think its ceiling would rise considerably more. The skill floor goes up because of the number of ranks simultaneously involved.

With the second idea, the player would always be able to see all of the instances of each suit in the deck, whether face up or face down. The element of (deterministically) available information rises dramatically. In 52-card variations, each face down card is transformed from 1 in 13 possible values to 1 in 52. With 100-card variations, each face down card goes from 1 in 20 possible values to 1 in 100. Conversely, possible values for face down cards can be narrowed based upon how many of the same suit is already showing, for those who wish to count them. The really fantastic thing about this “visible suits” idea is that it would have a negligible effect upon the skill floor – the additional information would not have to be used, whereas its effect on the skill ceiling would be considerable. In the context of this writing, it could be a design freebie.

The architectural vision of Boardwalk can be difficult to implement against, such that the game will be both simple and skillful in its range. While 52-card Boardwalk has been shown to be capable of pitting significant skill against 52 cards worth of random possibilities, it may also be too limiting when wanting to push the potential skill to man’s height. 4 each of 13 ranks just isn’t enough. These new ideas, such as a larger deck of cards and visible suits make great strides in doing so. Of particular interest to me is the idea of visible suits, since it would not affect the skill floor while greatly increasing its ceiling. I am also very interested in the idea of 100-card variations, as this would facilitate an exponential expansion in complex, upper walk multi-stack lead possibilities, the likes of which novices would not have to apply. Nevertheless, it would appear to increase the skill floor to some degree, by way of a longer lower walk and a larger number of face values to cope with simultaneously.

The goal in these genre of games is quite grand. It is to provide for the entire ship of man’s skill to sail within an infinite sea of possibilities. I know of no other game which attempts to do this, except perhaps for any other game that may dream of doing so. In solitaire, this is the finite man against infinite possibilities. In competition, where the same arrangement is dealt to multiple players, man may play against man, where each swims in the same sea.


More Progressive Boardwalk Variations

Since first writing this software and using the options it provides, I have experienced new ideas for creating even more engineered and balanced Boardwalk variations. 

All possible rule combinations: Slides to the upperwalk can be provided for.  Coaxing to the lowerwalk can also be provided for.  This much has been added to the software to date.

Wraps: With this idea, ascending order upon a walk may "wrap" once along its length.  That is to say one can declare Kings at any given location, and then play Aces, Deuces or Threes directly to the right of that location, thus wrapping from high rank to low (once) over the course of walk.  A greater number of possibilities are made available to the player with this rule.  What one may find exciting with this ability is its potential to scramble a player's orientation during play.  The very first play in a game - the first live play to the lowerwalk - could, to a great extent, be arbitrary in its strategic value. 

Redistributed Upperwalk: (Re-du)  Each lower walk source stack needs to begin with an exact number of cards for the game to play out properly.  This is not so on the upperwalk.  The only upperwalk requirement is that an exact number of foundations must be provided for.  Enter "redu", or a "redistributed upperwalk". 

I have tested this rule-making ability with a deck of cards and created my first original or root variation since Cheshire, called "Hopshire".   Let's say the game is 52-card Hopscotch, with 5 upper and 8 lower columns.  Now, deal the 20 cards for row 1 into 6 stacks instead of 5.  Deal 4 cards each to the first and last stacks, and 3 cards each to the 4 stacks in between.  At the end of a perfect game, the player only needs to have declared 5 of those 6 columns, leaving the one remaining upper column empty after having scored every card on the board.   Add in with this the stipulation that the upperwalk must be ascending, and the ability to slide cards to the upperwalk (to a clear row 1 position, from row 1 or row 3) and you have played 52-card Hopshire.  Compared to Hopscotch, this game is forgiving. 

In Hopscotch, one bad card can produce the most stubborn blockage in the game, where only a lowerwalk slide can relieve the condition.  Now, with upperwalk slides, an open upperwalk column can be used to provide relief for one bad card.  Also, the stipulation that the upperwalk be in ascending order is compensated by the "redu" upperwalk.  An interesting avenue in strategy is first opened here.  This new avenue is an expanded playing area within which the player chooses which way to go and which way or ways not to.   I imagine this avenue of expansion to hold potential for future variations beyond present day belief!

As it is, redu gives game engineering a greatly expanded ability to more acutely craft variations with specific balances between the player and the deal.

Stellar Walk: (starwalk)  This is an old idea, being given some note here.  Perhaps more thought can be applied to this in the future.  With 4 suits, one column on the starwalk has its open foundation at center, with one source card dealt to each point around it - above, below, left and right; these would be the North, South, East and West points of the star, respectively.  The North point would correspond to the upperwalk, the South to the lowerwalk, the West to the Hand, and the East to the starwalk.  I do not, at this time, recall the specific mechanics for making plays with these board elements at this time.  At this time, it seems there could be many mechanical possibilities.  The original purpose of the starwalk was to expand to 4 primary board entities instead of only 3 (upper, lower, hand and stars). 


Finite Vs Infinite Revisited:
Strategy in the Player Versus Strategy in the Cards

Players generally begin by developing some form of premeditated strategy as a result of the games they have played.  These strategies exist in a finite capacity, in the mind.  They may become more sophisticated over time, but still within the finite, mental capacity. 

The cards do not appear in a finite capacity.   They are random in their arrangement - they are real.  When the player sees the cards, they may find that they are seeing their own mind.  As such, the possibilities being seen in the cards is actually the possibilities that arise in the mind and in a finite capacity. 

How does the player expand in their ability?  The mind is a reference manual for scenarios that have been weighed, measured and contained.  The cards - they never occur this way.  What part of a player can approach the infinite playing field presented by the cards themselves?  The player is real.  This is not the player's mind - it is the player directly. 

Beyond any existing players strategy there is creative play, such that avenues of play without precedent can be taken.  In these cases, the player needs balance, which is also why any player, no matter how good in the past, can strategically fall flat on their face at any time in the future.  Balance will come with a mathematical sense of the total playing area - of its limits and of its tipping points, or points at which there is only a very fine line between left and right, or up and down. 

To all of that I say good luck and have fun!