One of the most common problems in computerized thinking is trying to solve chess, or create the perfect chess player. Such an aspiration for perfection is arguably within the bounds of possibility, and the different ways people approach the problem has always interested me, however, I realized I needed to start relatively small before building up to the grand challenge.
For that reason, I decided to create a chess set that one could use like a normal one. Cheating must still be monitored by the other player, castling is still done by hand, and en passant requires a helping hand to remove the captured pawn from the board. However, it functions past my expectations, and for that, I am willing to declare it a springboard for future improvements.
Here are videos of the code and the functioning board.
There are still some problems, like with promotion and win conditions, but overall it gets the job done.
In its full form, chess often appears as an unnecessarily high mountain. The work one must, compounded with the frustration of learning hiding behind repetition makes the game unwieldy and discouraging for most. For this reason, across the world of developing chess, specifically in some British schools, a variant on the game is being introduced, one with only 5 pieces and pawns for each side.
Upon hearing this, I was shocked such a thing even existed. But, this variant still seems to preserve the spirit of the game, with its complexity simply being reduced. And so, less appalled by this affront to chess, I considered what the simplest form of chess would be, without completely draining those aspects which make it still recognizable in chess. In other words, how much could I get rid of before chess lost its meaning.
To begin this, it is necessary to look to chess’s origins, with a lack of special moves like castling and en passant. Additionally, the bishop would only move one square in any direction, and pawns were limited to the sole one step forward (with the exception of captures) instead of having a jump start at the beginning of their lives.
Bishops and rooks were not needed, for the queen could easily mimic their actions. This meant that the only pieces I would need were a king, a queen and a knight. These three lonely pieces would each have one pawn for cover, and the board would be shrunk by a distance of 1 square from all sides to simplify the board but still allow ample room for strategy. Additionally, this bare-bones form of chess even allows for strategy in the placement of your pieces, and as such, maybe even heightens some of the strategic challenge of chess. Here are two possible starting positions:
The rules I developed are as follows:
The playing field is a 6 by 6 square of alternating colors
Placement of pieces rotates, with white placing first
Pawns may only move forward one step at a time
Pawns may only promote to a queen or a knight
All other applicable normal rules of chess are in effect
It is actually quite simple to play, but it still offers potential for a particularly motivated person to consistently beat one who lacks experience with the game. A sample game may be as follows:
To demonstrate the mind-altering nature of chess, and the pure hysteria it brings about on anyone unlucky enough to achieve some greatness in the game, I would like to point out a specific quote from arguably the most brilliant chess mind to have lived, Paul Morphy.
“The ability to play chess is the sign of a gentlemen. The ability to play chess well is the sign of a wasted life”
Superficially, there exists no argument against this. A life amounts to nothing if it carries itself to death focused on a sole goal, not noticing the world around it. But of course, I think differently then this. Chess is a form of expression. As such, I am convinced that there is some knowledge or creativity lying in the games of the best, just waiting to be unfolded into another field.
To test this, I decided to look at a widely-held game of great stature and brevity, and use it to compose a piece of music. If we find some coherence in the notes of the music, then undoubtedly, chess draws out universally-applicable creativity in subconcious ways.
The game I chose to base the composition on is Louis Paulsen vs. Paul Morphy (1857), due to its brevity, simplicity, and charming nature. How I decided the moves would dictate the notes of the piece follows a slightly circuitous logic, but indeed makes the most sense.
Firstly, I would only build the melody, so I decided to focus solely on Morphy’s moves. Secondly, the key of the piece would be decided by the letter of the file on which the first move was made. Thirdly, any time a piece moved to a square that was touching it, I would have to shift the tone 1 note up or down, and any time it moved further than a touching square, I would shift the tone the corresponding number of times. Additionally, pawn moves would mean shorter notes – a quarter note or less, while piece moves would correspond to longer notes – a quarter note or more. And finally, I allowed myself the luxury of adding one extra note at the end, symbolizing the tip of the king as the melody finishes.
And here is the piece I came up with that matches the game. Quite honestly, the results indicate a shocking connection between chess and music.
One thing I still wonder is if this effect holds true for all the great players, for while Morphy had a style that lent itself to exotic craziness, and a romantic era of chess, there are others who simply squeezed the life out of the game to outlast their opponent.
No matter the case, it is clear that there is some link to creativity, some indication that a mind hasn’t been wasted but has rather found meaning in something more pure than the pitiful concerns of the world.
The brilliancy of human ingenuity tackles on more challenging problems each day, lending a sense that perhaps we advance fast enough to evolve ourselves to some semblance of continued superiority over ourselves primarily, and also our environment.
But, although this course holds true for all of human history, we now reach a troubling time where the core of humanity must again prove its immortality against any outside influence that claims to match it.
In this case, the outside influence has quite a convincing claim.
Already the beauty of craftsmanship, the inspiring acrobatics of mental prowess and the general appreciation for life face a world where the common man does not need to feel so small in comparison to the greats. We mold everything around us to perfection, we navigate a complex world with no regard to the specifics, we indulge ourselves in sweetness on daily basis.
Separated no more from a rival to our genius, what can we cling to?
Unless we stand on the cliff of what we can relegate away, a stupid idea to hold, then the only hope we may keep as comfort is that we constitute the sole judge of the world around us. Everything takes place because of our mind. So maybe then, we can beat the new age of our challengers in our perception, in our derivation of meaning. A slim slice of life, but with just that, we can relax again comfortably, knowing we conquered the world again.
With this in mind, I thought about a classic problem in the world of chess, which involves using computers to create compositions of positions with a solution. For me, however, this felt too closed, too easy for the machine to dominate in some eventuality, so I decided that simply using a chessboard as a canvas, I would try to compete with a simple program to see who could create the more beautiful chess position.
While I would be free to create whatever I liked, the machine would be forced into some intelligence approximation to churn out positions in the hopes at least one could be confused for something made by a person.
So, after running a fair amount of trials, I compiled what I feel is a fairly representative sample of what a simple, more stupid computer could do. To make things interesting, I have also placed three positions I created myself: one was made to outshine the rest in aesthetic, another to be starkly human, and the third to be grouped as one of the machines, disguised to even the trained eye.
To anyone invested slightly too much in chess knowledge, the fact that some of the most brilliant players have also toed the line of insanity outside the chess world seems to be a common motif. And although this parallelism could be developed further, and perhaps argued for and against, there is a more special type of player, who has embraced their brilliance in such a characteristic manner that there is no doubt of their insanity, and yet their brilliance can’t be argued by minds restrained by contemporary logic.
The most prominent example of this special case of intellect is one of my favorite chess players Aron Nimzowitsch. Author of a multitude of books that codified modern chess thinking, he is also the author of an equally admirable collection of ideas that are so far past our understanding, we view them as utterly ridiculous.
As an example of this, the following post goes into more depth what the story of Nimzowitsch is, more inclined to ridiculing his thought rather than celebrating it.
Taking inspiration from this, I decided to create my own revolutionary chess strategy, and both create my ideal game of how it would work out, as well as play a real game to see how it really interacted with the real world. Then, I wanted to compare this to how my attempt to replicate Nimzowitsch’s idea of over-protection in a real game went.
The idea I came up with was partitioning – you delegate some of your pieces to only defend the king, while the others must recklessly attack. Although not as concise as Nimzowitsch’s idea, I feel it will have the same impact on my game play.
And, after experimenting with this method in three games, although there were some close calls, I ended up with three horrible losses, where in each game I was aching to play a certain move I knew to be good, but forcefully constrained myself to follow the newest improvement to chess. To give myself more of a fighting chance, I loosened the restrictions of my game play a little bit, but even then the plan was simply doomed from the start. I have attached the games below for instructive purposes.
And as the grand finale, I will match up with the idea of over-protection to see if it perhaps has more hope than the idea I put forth.
Weirdly enough, on the first game I tried this new concept, it worked. I will admit, I had to win the game on time, but at some parts of the game, I did have a better position.
One characteristic that has fascinated me is the ability to conjure up an idea from the rubble of knowledge. In essence, it is to see what isn’t there. Taking the leap past what is truly present is the most impressive part, with being able to act upon what you see in a manner that elevates the world, or at the very least your position in it, radiating a more mundane and trivial aura. And yet, this may be the unrecognized struggle of many, who see a unique way to approach the world only to have its limitations realize themselves too quickly and strongly for any action to be taken. In fact, having visions that the normal eye can’t see, and then having the torture of sitting idly while you can never enact them into the view of all might be the torment of great minds.
To pay homage to this art, and perhaps to test if I had such a gift, I decided to play blindfold chess. I decided to force myself to see a board that existed only in my imagination, and to interact with the objects only I could see.
As a quick explanation, blindfold chess is a form of chess where the player is not able to look at a chess board, they must hold the position in their head and call out moves to another person who makes their moves for them such that the person they are playing does have access to the board and pieces. The idea is to test the limits of envisioning a board and calculating.
Going into my first game, my confidence was low, mostly due to the fact that of all the times I have played blindfold chess, I have played against someone better than me, and therefore always suffered a loss, or not been able to keep track of the board for more than 10 moves.
With this in mind, I realized that I could not approach the game in a safe, classical manner against people who didn’t have a lot of chess experience, as I needed to take advantage of my chess abilities while I could still properly envision the board. For this reason, I attempted a theme similar to the scholar’s mate for nearly all my games. Additionally, I played as white so as to have the first move and actually be able to have a hope of successfully carrying out the mate.
Each game consists of who was playing, the result, a link to the game, a picture, and a computer summary of how both players played.
After finishing this game, I felt fairly confident, for I was able to envision the board fairly well. However, my initial plan of attack of the scholar’s mate didn’t work out to well. Looking at it retrospectively, it was one of the few games I have played without any inaccuracies.
That being said, my play went downhill from there.
In this game as well, I had to modify the scholar’s mate plan, and my play was markedly worse. I still won though, and starting now was when I felt the most confident throughout this process. I really felt that life wasn’t so much worse when you are forced to see things that aren’t there.
Yet another game where the scholar’s mate attack plan was denied. This time however, I felt that although my play wasn’t bad, my mind was slipping, and I couldn’t as clearly see the board towards the end of the game. We declared it a draw at the end due to the fact that I thought there was no pawn on g6, but that I still maintained a slightly better position.
This could be seen as the true turning point from a successful challenge to one full of learning opportunities.
This game hurts. I was so confident throughout the whole ordeal, and even declared mate when I got my queen to f7. Alas, although I remember thinking not to take the Bishop on f8 as it was protected by the queen on d8, I completely forgot about its existence the very next move and captured the pawn on g7, to which my opponent took my queen, and I promptly resigned.
This game was perhaps the ideal of what I hoped would happen. Here we get to observe the scholar’s mate happening in action, and it was a much needed game to give me the strength to finish my journey of hallucinations.
As a bit of a disclaimer, in this game, as well as the one that follows, both players played blindfolded instead of just one, as I realized that trying to win a game like so against higher caliber players would not turn out so well given my record against players of less skill.
This game is a bit sad for me, but nonetheless, demonstrative of the power of humans to put themselves through suffering for no good reason.
In this game, in addition to both player’s playing blindfolded, I chose to play as black, just to see how things would turn out. The results were about as I expected at this point, another loss to learn from.
All in all, my record was 3.5 / 7. I was more interested, however, in how well I played rather than in the game to game results. So, using a feature called CAPS by chess.com, I decided to approximate my rating from these seven games. My average CAPS score was 74.327, which corresponds to a rating of around 1100. For comparison, my true rating is around 1800.
In conclusion, you do better when you can see what’s there, instead of glimpsing at visions of what’s not.
There seems to be a consensus that chess is constrained in such a manner that it is easily recognizable to be a game of skill. If you invest enough time, you may supplement this disappointment with the knowledge that even the most creative versions of chess still apply a larger amount of skill than luck, in fact in such a manner that these games all comprise of the same plan-based thinking that characterizes our intuitive feeling of skill.
Fear not.
Past all the calculations and intelligence that clouds the mysteries of the game, I have seen a glimpse of beauty that lies more in a field of luck than skill.
What is of more surprise is that it is such a simple alteration. To rewrite the rules of chess such that a person would find this new version and the true version to be irreconcilable would be one thing. I feel as though I have accomplished quite another task, where a casual onlooker may believe the two games to be one and the same, where a frame of the game could be indistinguishable from its counterpart, but where the great minds of the original may flounder on the premise of the new.
Even more interesting is that the same themes of chess can be superimposed on this new version. On the surface level, the ideas seem to be directly transferable. This is the component I appreciate most, because of its implications. A world where all of our ideas of the way things work are consistent, but where the outcome of a scenario is unpredictable is a fascinating thought, especially if you consider that such a world may be the one we live in.
An ill-analyzed portion of chess is its reflection of how we can prove knowledge. Chess mirrors the same bounds that govern how we can think, and how we know something is true. Briefly discussed, this is apparent when considering that a true chess position may be one where both sides have made the best moves, or where the question and answer forces that drive reasoning have reached harmony in a perfect manner. For this reason it is clear to me that if it is the framework of chess rather than all that can be logically derived that controls whether or not you control the destiny of your game, the same is true for more abstracted systems of applied thinking.
And so, it pleases me that it is possible to alter the outcomes of our thought by changing not the way of thinking itself, but rather the way we interpret thought.
It is this motion of thinking, the question and answer, that inspired me to develop this new version. The procedural flow of this action is what dilutes our ability to understand the world in a way that unlocks infinitely more possibilities. And so, the procedural motion of chess blocks its own power as a system.
While the general concept of this new game has no doubt entered the minds of many, I will be the first to introduce it in a way where its applications are useful, though not easily comprehensible, and its result shown to be that we live with only the ability to maximize our chances for, not to guarantee, anything.
Both sides may move simultaneously.
Logistically, this means each simply writing down your move while your opponent writes theirs down, and then you will both view what has been written down and move the corresponding pieces. In such a manner, the chain of thought is reconstructed to be pairs of complementary thoughts. There are in fact a bit more logistics to work through, and I will do so in the {} such that the minor details can bore those who chose to partake in them.
{
If two pieces attempt to capture each other simultaneously, then the move is nullified, the pieces return back to where they were at the start of the move, and the players proceed. In yet another similarity with life, the cyclical nature of attempting to capture a piece over and over again as both players refuse to try anything else is reminiscent of the criminal repetitiveness that interferes with our enjoyment of life. But perhaps to encourage development as life punishes those who commit repeated errors, an enforcement of disallowing trying the same move more than once per turn could be applied.
If a piece attempts to move in such a manner that an illegal position arises at the end of the turn, that piece will be placed back to where it was originally, as well as any other pieces involved in this part of the move. If it happens that this replacement still leaves an illegal position, then the piece will be lost, and in the case that this piece was a king or the illegal nature of the position involves a check, the player who made the illegal move will lose. Life is a large portion risk, and while risky maneuvers are easier to make in chess, where the consequence is decidedly less, there is a deadly reward awaiting both success and failure. Either you are rewarded and your opponent is cheated of something, or your mistake is retracted in full view of the world, or the aspect you sacrificed has been claimed by someone else — leaving you with nothing of what you started, or a debt even steeper is incurred.
You may pass your turn. To sit idly as the world flashes before may not be advisable, yet its abundance in practice is past the point of denial, and so I feel it is necessary that it be included to truly encompass the human condition
}
Following are a few short games I imagined could take place.
Game 1
And Black has lost. Ng4 leads to an illegal position, as the king is left in check. To reconcile this by placing the knight back on f6 would still leave the king in check, and so the game is over.
Game 2
The two pawns attempt to capture each other. As described, they are therefore reset.The reset position.
The queen captures the f2 pawn. The king also captures this pawn. Since the king was capturing a piece of its own color, it gets preference as to which piece stays on the board.
An illegal position arises as both players check each other.When reset, the board is legal, so play continues.And Black has won.
What is important to note is that in each of these games, while both sides maintained a plan, the outcome of that plan was completely up to chance. As seen in the first game, by playing Qe6, White gambled that Black would not play dxe6 and win a queen. Or as in the second game, instead of playing Ke1 at the end to avoid the check, white gambled Qg4 in hopes to capture on d7 some day.
No doubt, the game is convoluted. But it offers such a lesson, and is such a device to view life through, that I believe this value gives the game its own claim to a share in humanity.
Astute readers might claim that this game bears no resemblance to the concepts of chess, in pawn structure, endgames, tactics, or even the core of what the game is about. You simply have not looked deeply enough at the game. On the surface it looks like chess. If you view it slightly deeper, the game reveals that its nature is much faster paced and based around different concepts than chess. But look at it even further, and you understand that it is a game of ideas, a game of the application of knowledge and skill. The only difference is that no matter who you are, the next move can either be brilliantly lucky, or foolishly fateful.
There are more games of chess than there are atoms in the universe. That statement made me consider whether there was more than one way to apply a chess set than as normally prescribed, specifically in regards to determining whether a chess set could replace another board game for a mathematically equivalent experience.
The first board game my thoughts chose to challenge me with was Monopoly, and so I began embarking on a quest to first determine if it was even possible for every possible combination of a monopoly game board to be represented on a chess set. To maintain the integrity of my rationale, I decided to test out the conditions for a two player game of monopoly, as it could be argued that if this was possible, then a monopoly game of more people could than be represented by the corresponding version of chess that encompasses that number of people. For example, a four player monopoly game could theoretically be represented by a 4 player version of chess given that a two player game of monopoly can be represented using a standard chess set.
To calculate how many monopoly positions there are, I decided to do some math, and my thought process was applied as follows. If you would like to skip this process shown in {} the gist was multiplying all the different number of possibilities for each sub component of monopoly (position, money, chance and community chest card, and property ownership).
p1Pos = 40 possible states corresponding to 40 possible spaces
p2Pos = 40 as well
p1$ = I said 10,000 possible states since in no monopoly game I have played has anyone reached near this amount
p2$ = 10,000 as well
chanceCards = 16 cards
communityCards = 16 cards
propertyOwn = …
This was definitely the hardest variable for me to calculate. what makes it so weird is that there are 7 possible property levels (ranging from unowned to zero houses to three houses to a hotel) for 22 of the properties, and there are 6 other properties that have 2 levels of either owned or unowned (the railroads and utilities). In addition to this, I had to account for whether a property was owned by the first player or the second player.
Doing some slightly dubious mathematics I came up with a number of permutations of property ownership states of (13)22 * (3)6.
A quick reiteration of the above expression: 22 properties have 13 possible states and 6 properties have 3 possible states. The list below shows each of those states.
Normal Property
Unowned ————- ————
Player 1 0 houses 0 hotels
Player 1 1 houses 0 hotels
Player 1 2 houses 0 hotels
Player 1 3 houses 0 hotels
Player 1 4 houses 0 hotels
Player 1 0 houses 1 hotels
Player 1 0 houses 0 hotels
Player 1 1 houses 0 hotels
Player 1 2 houses 0 hotels
Player 1 3 houses 0 hotels
Player 1 4 houses 0 hotels
Player 1 0 houses 1 hotels
Railroad or Utility
Unowned
Owned by Player 1
Owned by Player 2
So, plugging in all of these values we come back to the original equation with:
With a very conservative estimate of the total number of chess positions being greater than 1050, I determined that it was theoretically possible to represent any game of monopoly through a chess board.
That being said, I had just cracked the surface of how I could use a chess set to represent something else. Up until now I had been only considering classically legal positions, and so I ran into a problem trying to construct a way to have the players roll dice in some analogous way using a chess set.
This issue plagued me, for as I began to see its impossibility, I realized the great number of games that used dice, and I began to believe perhaps my efforts would be unwritten.
By a great amount of luck however, I was inspired by the way computers generate random numbers, and I realized if I broke out of the mold of using chess pieces as they were intended, I could unlock the roll of a die.
So, I thought of tossing two pawns of differing colors into the air, and having whichever pawn landed the most right represent a digit in the binary system. If it was a black pawn, it would be a zero, and a white pawn would be a one. Now, using a representation scheme where a person would toss up the pawns say ten times, we could treat that as a decimal between zero and one with 10 binary “decimal” places. Once a sufficiently precise decimal is created, then I could utilize a simple random number generation algorithm of multiplying this decimal by the range of numbers I needed (from 2 to 12 is a range of 10) and adding two so that my random number started from two instead of zero.
Phrasing that last paragraph in the context of an example is critical, I feel, to properly demonstrating the possibility of this method. So, lets just say that after tossing up the pawns 10 times, a player got the following results
Pawn on the right : Black, White, Black, Black, Black, White, Black, White, White, Black
This would translate to .10111010012
Which in decimal would be 1/2 + 1/8 + 1/16 + 1/32 + 1/128 + 1/1024 = .7275390625
Multiplying this by 10, the corresponding number would be 7.275390625
Adding two to this would give 9.275390625
Rounding this to the nearest whole number would result in the player having artificially rolled a 9 on a dice.
In my calculations, I found that it was technically only necessary to have 5 tosses to be able to reach 12 (11.6875 rounded up), but to maintain a distribution more similar to what an actual dice roll provides, I feel that 10 tosses is a good number.
The interesting thing I found about this method however is that it does not properly achieve the distribution of dice rolls I had hoped would be possible. For that reason, I took a little bit more inspiration, this time from a technique called rejection sampling, and now the calculation of a dice roll is much simpler, and provides for a more dice consistent rolling as follows:
Use the same tossing method prescribed above, and come up with a binary number. If that number is within the desired range, then that is the equivalent roll, if it is outside the range, then restart the process. This would be done three times to simulate each die, meaning that this method requires six tosses at the least, and takes out much of the need for having a calculator. Some context:
Pawn on the right for toss set 1: Black, White, White
Translates to 1002
Which in decimal is four, so the player would have artificially rolled a four for the first dice.
The player would then proceed to repeat this process for the second dice.
The only case where some “complexity” arises is in a case out of the range, so for example:
Pawn on the right: Black, Black, Black
Translates to 1112
Which in decimal is 7, so the player would need to do a re-roll and restart the pawn tosses.
This method provides a more authentic game. Those of you who are a bit too interested in the methods I described above might have noticed that if I implemented some parts of the second method into the first, I could have created another artificial dice rolling method that would have the same percent chance of landing on a specific number as a real dice, but I decided not to do this due to the fact that the second method is simply much easier to implement during game play. If interested, code I used to simulate these two dice rolling methods is accessible by clicking here.
Now that I was much more open to configuring pieces in ways other than intended, I quickly moved through representing each component of the game I discussed towards the beginning.
I could create a distinguishable base 10 system using a piece (where pieces could lie down pointing to a certain direction) as follows:
No piece = 0
Piece pointing NE = 1
Piece pointing E = 2
Piece pointing SE = 3
Piece pointing S = 4
Piece pointing SW = 5
Piece pointing W = 6
Piece pointing NW = 7
Piece pointing N = 8
Piece standing up = 9
The players positions could be represented by two pieces each, with the number corresponding to a square on the board, and 60 of the possible permutations not being used.
Using this system, money could be represented using four pieces for each player different colors to symbolize the different players.
For chance and community cards I had to stray from the game’s methodology of cycling cards in the same order over and over, and chose to have only two pawns represent both the chance and community cards. Each number (0-95) would correspond to a specific card (the last 4 numbers wouldn’t be used), and every time a player landed on a chance or community card space, the two dice would be shuffled to a random number between one and a hundred using the second method of dice rolling described above, adapted to fit 100 cases (essentially rolling a 100 sided die by using 7 tosses). If a player obtained a get out of jail free card, they could take possession of the queen of their corresponding color, and a similar permutation method as I have already mentioned could be used to represent multiple get out of jail free cards being owned.
And so I believed I was almost done, that too only having used 18 pieces.
But I struggled to come up with a way to represent 28 properties ownership using 14 pieces. Although this is theoretically easily achievable, for the board to maintain some sort of resemblance to monopoly, I didn’t want to simply say that every single possible permutation of board states could be modeled using the base ten piece system I described above by a direct mapping of a number to a game state.
So, I decided to group properties together, with every two consecutive properties now being an amalgamation of one distinct object. This object would then, at most, have the 13 possible states described above for a single property, times the 13 possible states its partner property would also have. So, I needed to find a way to have a single piece represent 169 distinct permutations. The 14 remaining pieces would be these super-descriptive pieces.
From there, only one more creative aspect was required, I used the chess board as a sort of counter, and used pieces whose direction could be noted standing up (these pieces are the knight (X2) , bishop (X2) , and king (X1) , totaling 5 of these pieces for each side). These pieces have 17 easily distinguishable orientations (8 while standing up, 8 while laying down, and 1 more while lying down oriented to face a different direction), and using the 8 rows of a chess board as well as counting one row off for each side, it is easy to code 170 possible orientations for 10 of these double-property objects. Three of the last four can be represented using three rooks either being upright or upside down on the eight chess board rows plus the two outside board rows to make 20 permutations which is more than enough to represent the nine possible permutations a railroad-utility object could have. The last pair object could be represented using a rook that takes up two columns of the board. This rook would have 19 possible rows to be on (by splitting each chess board row into two distinct parts and adding two rows for the pawn being off of either side of the board), 3 orientations (one standing up, one laying down, and one upside down), and 3 places laterally (on the left square, in the middle, and on the right square), and would represent the last property pair by having 171 representations.
This new take on monopoly has thus been created, and all that each player needs is knowledge of how to play, which, along with property prices, a run-down for how to role the “dice”, and any other needed information, can be found in the rule books. As I mentioned earlier, it would have been easier to simply state that each position of chess would correspond directly to a possible orientation of monopoly, but such a claim, although well-founded, wouldn’t have the same meaning as a version which two people could feasibly play without need to store positions in a dictionary with over 1050 entries.
Once I found out that monopoly could be implemented in a way that a person could interpret, it seemed irresponsible to go out and codify every game I could think of in such a manner, as the “proof” it is possible was enough to satisfy my curiosity. A simple list of all the games I think could be represented in a similar fashion include: Sorry, Checkers, Clue (piece color wouldn’t have meaning), Battleship (if players hid their pieces behind themselves), Connect 4 (using a pairing method), Yahtzee, Chutes and Ladders (a “rule book” would define where the chutes and ladders are), and Life.
Some example positions in case you would like to try to play (not recommended).
Starting position:
From left to right, the pieces represent these property pairs: Mediterranean Avenue and Baltic Avenue, Reading Railroad and Electric Company, Oriental Avenue and Vermont Avenue, Connecticut Avenue and St. Charles Place, States Avenue and Virginia Avenue, Pennsylvania Railroad and B. & O. Railroad, St. James Place and Tennessee Avenue, New York Avenue and Kentucky Avenue, Indiana Avenue and Illinois Avenue, Atlantic Avenue and Ventnor Avenue, Water Works and Short Line, Marvin Gardens and Pacific Avenue, North Carolina Avenue and Pennsylvania Avenue, and Park Place and Boardwalk. The pawns are as follows: first two = P1 position, next 4 = P1 money, next 2 = chance and community chest cards (the queens are the get out of jail indicators), next 4 = P2 money, last 2 = P2 position. Take notice that to represent a zero, the pawns are kept out of line of the chance cards. It is not necessary to have the chance cards be 0 at the start of the game because they will be randomized every time they are used anyways. The two pawns that are interlocked in the center of the two rows will serve as the dice. Additionally, in the starting position both player 1 and 2 have $1500.
Some close up shots of the set up.
To get a better idea of the orientation of the super-descriptive pieces, I have attached some examples below.
Now I will run through 1 turn of a game to give a little bit more context.
Here P1’s position is shown to be 7, which means that the dice roll could have been Black, White, White for the first toss and White, Black, Black for the second toss. Because position 7 is a chance space, P1 would take a chance card.The chance card was randomized to be above. For this game, the number depicted (13) was chosen to represent a get out of jail free card if the player was on a chance space.Because P1 obtained a get out of jail free card, the corresponding color queen was relocated from next to chance deck to being to the left of the position marker, and the queen was oriented such that it represented a 1 (the number of get out of jail free cards P1 now has).Here, P2 is shown to have a position of 5, which means the dice roll could have been White, White, Black for the first toss and Black, White, White for the second.Since P2 landed on Reading Railroad and chose to purchase it, the money of P2 is decreased to 1300 as shown.To represent the new property ownership level, the Reading Railroad and Electric Company conglomerate piece is moved to its 7th possible position, which if you create a probability tree for this property type using the structuring described previously, corresponds to Reading Railroad being owned by P2 and Electric Company being unowned. One important thing to note is that as long as the relative order of a piece to its duplicate is kept, then the piece can be translated anywhere on the field laterally, which is why I can move the rook right next to the board.
Some code I used to verify the dice rolling methods is below. The first set corresponds to the first method I described.
import java.math.RoundingMode;
import java.text.DecimalFormat;
public class BinaryToDice {
public static void main(String[] args) {
DecimalFormat df = new DecimalFormat("##.##");
df.setRoundingMode(RoundingMode.DOWN);
int numOfRolls = 100000;
int numOfTosses = 10;
int[] rolls = new int[numOfRolls]; //array of all the rolls that are generated from the binary dice
for (int counter = 0; counter < numOfRolls; counter++){
double biDi = 0;
for(int i = 0; i < numOfTosses; i++) {
biDi += (Math.round(Math.random()))/Math.pow(2, i + 1);
}
rolls[counter] = (int) Math.round(biDi * 10 + 2); // assigns the simulated roll to its corresponding slot in array
//System.out.println(rolls[counter]);
}
// outputs how many of each type of roll was achieved by the analogous toss dice
for(int compare = 2; compare <= 12; compare++) {
int total = 0;
for(int j = 0; j < numOfRolls; j++) {
if(compare == rolls[j]) {
total++;
}
}
double percentage = ((total+0.0)/numOfRolls) * 100;
String percent = df.format(percentage);
System.out.println(total + "\t(" + percent + "%)\t\tis the number of the rolls that were a " + compare);
}
}
}
And the second method.
import java.math.RoundingMode;
import java.text.DecimalFormat;
public class BinaryToDiceImproved {
static int numOfTosses = 3;
public static void main(String[] args) {
DecimalFormat df = new DecimalFormat("##.##");
df.setRoundingMode(RoundingMode.DOWN);
int trialSize = 1000000;
int[] rolls = new int[11];
for(int i = 0; i < trialSize; i++) {
int rollSum = 0;
for(int j = 0; j < 2; j++) {
int roll = tossUp();
while(roll < 1 || roll > 6) {
roll = tossUp();
}
rollSum+=roll;
}
rolls[rollSum - 2]++;
}
for(int i = 0; i < rolls.length; i++) {
double percentage = (rolls[i] + 0.0)/trialSize * 100;
String percent = df.format(percentage);
System.out.println(rolls[i] + "\t(" + percent + "%)\t\tis the number of the rolls that were a " + i + 2);
}
}
public static int tossUp() {
int sum = 0;
for(int i = 0; i < numOfTosses; i++) {
int random = (int) Math.round(Math.random());
sum+=(int) (random * Math.pow(2, i));
}
return sum;
}
}
1. Teach someone how to play chess incorrectly and having them play in a tournament to judge how well they do with limitations compared to a more free method of thinking.
2. Judge which form of piece movement is best, if done so randomly, to get from one random square to another random square. Contrast this with a thinking piece who takes the quickest path. Though the results for this latter trial seem predictable, there might be some unexpected result.
3. Mirror the concept of a work of art through mapping pieces to part of the work. The goal of this would be to create a coherent game that could be use to distinguish a very different piece of work from the one that is represented.
4. Create a variation on chess that seems to be skill based, but is comprised of a verifiable majority of luck.
5. Attempt to make aesthetically pleasing positions and then try to have a computer also create its own idea of what such positions may look like. In the end, people would judge which positions looked better, and it would be revealed whether the computer or human could create the best positions.
6. See how many other board games could be represented, and played, using a chess board and pieces.
7. Analyze the squares that once visited, most often led to victory. As a further point of study, observe what squares corresponded to certain pieces in regards to victory.
8. Define the game in such a way that it becomes a formal axiomatic system under the current rules, and then come up with a corresponding position that is an unprovable truth.
Chess 