Pythagorean Tic Tac Toe

Copyright (c) 2018 (All Rights Reserved) by Tony Berard

Pythagorean Tic Tac Toe--The of a Thousand Universes in Each and Every Go Position.

Introduction

Pythagorean Tic Tac Toe is a new game built upon the old game of Tic Tac Toe. Each player gets 85 or 90 dice. One player is White, who always goes first, gets white dice. The other player is Black, who always goes second, gets black dice. We also have 70 Grey dice, which will be explained later. Pythagorean Tic Tac Toe is played on three boards--a 6x6 board, an 8x8 board, and a 10x10 board. It is a mathematical fact that 6x6 + 8x8 = 10x10. A 6-8-10 triple of numbers is known as a Pythagorean Triple. Thus, because of the similarity of the boards to this triple of numbers, the name Pythagorean Tic Tac Toe was chosen. The logo will be a 6-8-10 triangle with the sides of the triangle made into squares. Each of the squares will host a board of Tic Tac Toe of that size. There are infinitely many Pythagorean Triples. The 6-8-10 triple was chosen for several reasons and is the best choice as a result. It provides sufficient complexity for the game to make it the most complex game on the planet, and it is still small enough that the three boards are easy enough for people to understand.

The Boneyards and Throwing Etiquette

The Boneyard is a term borrowed from Dominoes, and it is the place to store White's dice, Black's dice, and the two areas to store the Grey dice. If some kind of dishes or bowls are used, we can still call them Boneyards. White's Boneyard is just out of play on White's side. Black's Boneyard is just out of play on Black's side. And the two Grey Boneyards holding 35 grey dice each at the outset of the game are on the left and right sides of the three boards just out of play. Each player has a rolling cup to remove hand throwing as a skill over the chance element intended by the use of dice. Place the dice in the throwing cup and cap it. Then, shake it for at least three seconds. Your opponent can demand that you shake it for at least three seconds if it seems less than this. Disputes on this can be determined with a stop watch. Finally, pour out the dice onto an empty area of your Boneyard (or on the bare table if dishes are used to hold the dice). Record on the scoresheet what your throw was in terms of how many 1's, 2's, 3's, etc. you got. The time clocks will have a delay of one minute to give you time to make your throw and record it.

The Power of the Dice and Forming Chains

The numbers 1, 2, and 3 are "weak" in this game. The numbers 4, 5, and 6 are "strong." The goal of the game is to form chains the full length of the board. Thus, the 6x6 board requires a chain of length 6. As the dice are loaded onto the boards, the goal is to try to form a chain on that board. On the 6x6 board, there are 14 ways to form a chain--6 vertical, 6 horizontal, and 2 diagonal. On the 8x8 board, there are 18 ways to form a chain--8 vertical, 8 horizontal, and 2 diagonal. On the 10x10, there are 22 ways to form a chain--10 vertical, 10 horizontal, and 2 diagonal. A chain is formed when a player has filled up a row, column or main diagonal of a board with either his own or Grey dice. A chain with even one weak die in it is a weak chain because a chain is only as strong as its weakest link. Thus, a chain with only strong dice in it (i.e. strong links) is a strong chain. To win on a board requires one strong chain or two weak chains.

Placing the Dice

On White's first turn, he or she tosses six dice. One must be placed on the 6x6 board, two must be placed on the 8x8 board, and 3 must be placed on the 10x10 board. After that, each player will throw 12 dice per turn, and three go on the small 6x6 board, four go on the medium 8x8 board, and five go on the big 10x10 board. The dice may be placed upon the boards following a few simple rules. Of course, a die may be placed on any empty square. But, you can also displace dice already there--either the opponent's dice or your own. There are two methods of displacing dice. The first method is the normal method. With this method you can replace a weak opposing die with anything of higher value. Thus, an opposing 1 can be replaced by a 2, 3, 4, 5, or 6. An opposing 2 can be replaced by a 3, 4, 5, or 6. And, an opposing 3 can be replaced by a 4, 5, or 6. If you replace an opponent's weak die with this method, it is called a "Claim." If you replace your own die with this method, it is called an "Upgrade." Upgrades, though, have no restrictions at all. You may Upgrade any die belonging to you with anything of higher value on your roll--even strong dice.

The other way to replace a die is through the tossing of a four of a kind. If you get four of a kind, you can optionally replace one of them with its equivalent with a Grey Die (or, as we like to say more simply, a Grey). You don't have to replace like this with a Grey Die because it is optional. But, if you do, this Grey Die can displace an opposing die of equal value or less. Thus, a Grey 1 can displace an opposing 1. A Grey 2 can displace an opposing 1 or 2. A Grey 3 can displace an opposing 1, 2, or 3. A Grey 4 can displace an opposing 1, 2, 3, or 4. A Grey 5 can displace an opposing 1, 2, 3, 4, or 5. And, a Grey 6 can displace any opposing die--even a six. Note that it is not allowed that a Grey Die displace a player's own die. A Grey Die may only displace an opposing die. The strength of a Grey die in a chain must also be accounted for as well as the strength of the player's own dice. Note, for example, that a chain with all strong dice and a Grey 3 is a weak chain.

The Greys are Weak and Strong, Too (Overrides) [Greys in this game set are the Brown Dice]

A Grey 1, 2, or 3 is weak and may be Overridden by either player with a 4 or higher (a strong die). More specifically, a Grey 1 can be Overridden by a 2 or higher. A Grey 2 can be Overridden by a 3 or higher. Any strong die can Override a Grey 3. However, a Grey 4, 5, or 6 is strong. No reason exists to strengthen a strong Grey Die to a higher strong Grey, so a Strong Grey stands as immutable and inviolate. Thus, a Grey 4, 5, or 6 is the highest and therefore strongest piece on the board since nothing can change it. By comparison, a Black or White 6 can be changed into a Grey 6 by a player rolling four 6's. Thus, a strong Grey (4, 5, or 6) is our queen even though it doesn't move.

So, we have Claims, Upgrades, and Overrides as our mechanisms to change the colors or values of the dice on the boards as game play progresses. Claims are when you displace an opposing weak die with your own higher valued die on a roll. Upgrades are when you increase the value of your own dice with a higher valued die on your roll, and Upgrading can be done on weak or strong dice of your own color. And, Overrides are when you displace a weak Grey Die with a higher valued die of your own on your roll. A strong Grey Die cannot be Overridden. And, a Claim may not be made on a strong opposing die.

Finishing a Board

When a board is full with no strong chains and neither player having two weak chains as well, that board is a Cat or a draw. No more dice may be placed on that board, but each player still throws 12 dice each time, but will only use the correct allotment of dice to use on each of the remaining boards in play. For example, suppose the small and large boards are still in play. Then, a player will only use 8 of the 12 dice thrown to put on the boards--three dice to be placed on the small board and five dice to be placed on the large board. When a player gets a strong chain or two weak chains, that board is not available for further play. So, a Win finishes a board. Play stops when all three boards are Finished and become unavailable for further play.

Scoring

We split up a point using decimals for scoring in this game. The small board offers 0.2 points. A winner on this board takes all of this allotment. If this board is a Cat, then each player gets 0.1 points. The medium board offers 0.4 points. A winner on this board takes all of this allotment. If this board is a Cat, then each player gets 0.2 points. The large board scores the same as the medium board. The total is 1.0 or the full point. In order to take a full point, a player needs to win on all three boards. A draw will be defined to be 0.5 points each (which can happen with a Cat on the small board and splitting the Wins on the other two boards, or it can happen with a Cat on all three boards). Thus, if a player gets more than 0.5, that will be considered a victory. A player getting less than 0.5 points will be considered to have lost. Here is an interesting scenario that is possible due to this finer resolution of splitting the point. Players can play a match of 6 games or 10 games etc. Let us suppose that they play a 10 game match, and the score is 4.3 points to 4.7 points going into the tenth game. The player with 4.7 points needs fewer points to break 5 points total. Thus, suppose the player with 4.7 points loses the 10th game by getting 0.4 points. This player did lose the final game but still won the match 5.1 to 4.9.

Immediate Equality

During my battles with discovering how to invent a game properly, it has come to my attention that a game needs to have equality at the outset. This is important because Chess and Go have statistics at the highest levels that reflect a disparity in the colors, and this is directly the fault of having an imbalance at the start of the game. escapes this fate with 50 percent White wins, 48 percent Black wins, and 2 percent draws at the highest level. Shogi has a drop rule which allows a captured piece to be returned as one's own piece instead of a turn involving moving a piece. This keeps all the pieces in play the whole game long. This apparently leads to the balanced outcome statistics seen in Shogi because the complexity is too high for a player to see how to maintain the initial advantage bestowed upon the player moving first. Chess and Go fall short of this as they have clear disparities in their outcome statistics. So, how do I achieve immediate equality in my new game of PTTT? I restrict White to half the dice on the first turn. Since 12 are used in a normal turn, White's first turn is restricted to just 6 dice. By doing this, we get the immediate equality, and the outcome statistics should be balanced. But note that there are eleven possible outcomes in PTTT: 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. Note also that White's score plus Black's score must equal 1.0. So, we should see over millions of master level games that there is approximate equality between the White and Black outcomes--even at this higher resolution of outcomes. Note that White could get 6 weak dice on the first throw, which Black could Claim on the very first turn. This does not negate the equality argument for two reasons. First, because this is just one game, it will have little to no effect on the balance seen in the outcomes of the other millions of games. And second, White could have several throws down the line in this game in his or her favor as some sort of compensation offered by the Law of Large Numbers. Such an offering, of course, is not guaranteed; but, I am just saying that White's loss of equality in this example may not really be lost after all. Anyway, with the Law of Large Numbers and a balanced game we should see a fine balance in the outcomes of master level games in PTTT.

An Advancement in the Time Clocks

Regarding the time clocks, we require an advancement over the standard chess time clock. I have already mentioned the one minute delay before the player's actual time starts. There is also the time for the game itself. We will have three speeds: Fast, Medium, and Slow. The fast game will give ten minutes of thinking time per side. The medium game will give 30 minutes of thinking time per side. And, the slow game will give 2.5 hours of thinking time per side. We will have three ratings for a player--one for each speed. And, we will have three world champions--one for each speed.

Now, let us discuss the advancement in the time clocks--The Reservoir. Upon expiration of time, the clock will flag as usual to let both players know that the official time for that player has expired. The Reservoir is an additional small block of time we will allow for the player whose time has expired. It shall be ten percent of the actual playing time. Hence, for a Fast Game, the Reservoir will be one minute. For a Medium Game, the Reservoir will be 3 minutes. And for the Slow Game, the Reservoir will be 15 minutes. Once a player uses up the Reservoir and flags on it, that player will be said to have "lost on time." However, since there are three boards, if any of those boards were resolved as wins or cats, those stand. It is only on the unresolved boards that the player who loses on time loses.

You also must record your placements. If you get a 4 of a kind, circle that die on the scoresheet to indicate you replaced one of those die in the 4 of a kind with a Grey of that same value. But, do not circle it if you choose not to use a Grey Die since it is optional. The notation of PTTT is described in the next paragraph for how to write your plays on the scoresheet. The notation describes all that can happen during a player's turn. Since there are twelve placements each turn (except for White's first turn as noted previously), it obviously makes sense to record the placement immediately following the actual placement. You might do a couple of placements and then record those. But, if you do nine placements, say, and then try to record them all, you risk forgetting some of them. Your opponent must be recording them as well to be able to have a complete game score. You may ask your opponent to see his or her scoresheet, but it must be on your time, not the opponent's time!! After a player makes the twelfth placement and records it (and White's first turn of making the sixth placement and recording it), then the player punches the time clock to start the opponent's one minute delay for shaking the dice, pouring them out, and recording the throw. Note: It is ok for a player to put 12 dice in the shaker and cap it on the opponent's time. If you get good at throwing the dice and recording them and recording your placements, you can pocket that as extra time to think during the game.

The Notation of PTTT

A placement will involve 3 characters for normal placements and 4 characters for placements involving a Grey Die. The fourth character for a Grey will be a simple asterisk in the fourth position to show that the die placed was a Grey Die. The first character denotes the value of the die, and the next two characters show the placement (the square location). There are three boards, and they will use different letters for the files. The 6x6 board will use the letters a through f for the files (the vertical stacks of squares), and the numbers 1 through 6 for the ranks (the horizontal row of squares). The 8x8 board will use the letters g through n for the files, and the numbers 1 to 8 for the ranks. And, the 10x10 board will use the letters o through x for the files, and the numbers 1 to 9 and 0 for the ranks. The tenth rank will be called the zero rank so as to be able to record any placement with two characters in this game. The zero rank will be closest to Black, and all the first ranks will be closest to White. All the files are labeled in consecutive alphabetical order from left to right from White's perspective. So, from White's perspective, the smallest board is on the left, and increases in size to the right. From Black's perspective, the largest board is on the left and decreases in size to the right. A 12 placement turn might be the following: 5a5, 2c3, 6f6, 1g7, 1i2, 6g5, 5n3, 1o0*, 4o2, 3x9, 4x0, 1p6. This reflects three placements on the 6x6 board, 4 placements on the 8x8 board, and 5 placements on the 10x10 board. Note that a Grey 1 was placed on the o0 square of the 10x10 board (the zero rank is the tenth rank). White's first turn will show only 6 placements because of the restriction to six dice for that turn.

The Practice Game of PTTT and the Real Game of PTTT

The Practice PTTT is played on a 3x3, 4x4, and 5x5 set of boards. White gets 3 dice to throw on the first turn with a single die placement on each board. After that, both players get 6 dice to throw on each turn with a placement of 1, 2, and 3 dice, respectively for the three boards in ascending size. To make a Grey requires a 3 of a kind among the 6 dice thrown. It is possible to set 2 Greys in a single turn because 6 / 3 = 2. The practice game probably won't last more than 20 turns. The scoring is 0.2 points for the smallest board and 0.4 points for each of the two larger boards for a total of 1.0 points up for grabs each game. We will have an option for you to play the Practice PTTT on our site www.tinesandbarbs.com, but it will be a casual game that is not rated.

The Real Game of PTTT is played on a 6x6, 8x8, and 10x10 set of boards. White gets 6 dice to throw on the first turn with a placement of 1, 2, and 3 dice for the three boards in ascending order of size. After that, both players get 12 dice to throw on each turn with a placement of 3, 4, and 5 dice for the three boards in ascending size. To make a Grey requires a 4 of a kind among the 12 dice thrown. It is possible to set 3 Greys in a single turn because 12 / 4 = 3. The real game probably won't last more than 45 turns. The scoring is 0.2 points for the smallest board and 0.4 points for each of the two larger boards for a total of 1.0 points up for grabs each game. When you find an opponent on the site, there will be a quick set of questions for you both to jump through to decide if the game will be a rated one or a training game, and it will involve what I call the RC coin. But, this RC coin idea is explained on the site www.tinesandbarbs.com in an article I wrote on that topic.

A Comment on the Speed of PTTT

In chess, you get to move just one piece, except for castling where you move the king and a rook in a specific limited fashion. In Go, you get to place just one "stone." But, there are times when sections of the board get removed due to rules of the game. One practitioner explained to me, "Now, you get to fight for it all over again!!" But, barring a removal turn, the speed of Go is quite slow. Without removal, it takes 180 turns by both players to fill up the board and one more empty spot to fill with White's 181st turn. In Shogi, you get to move just one piece at a time, and these pieces are quite limited in mobility anyway. The drop rule does bring in some life to the game, but it is mostly a plodding tedium. In Aarima, I think you also just get to move one piece at a time, but I need to know more about that one. What I can say is that it was a more confusing game to me. Checkers has that you can move just one piece at a time, but sometimes, you can jump several of the opponent's checkers in a single turn, which is kind of exciting (and frightening for the recipient). But, all in all, checkers is pretty plodding, too. Tines and Barbs (a game I invented) has the ability to do multiple things on a turn. By using a Turn Scale, the pace of the game is sped up over chess. Some of the turn combinations in Tines and Barbs are quite exciting as well, an MRW turn for example is when a piece is moved, a piece is rotated, and a shot is fired. This can be the same piece (a Lance with a gun) that moves, rotates, and fires a shot at a piece (say, the Grand One--that game's king) can be a demoralizing maneuver. I would say Tines and Barbs is pretty exciting in terms of the pace of progression of events, but it still seems like it could go higher and still be ok to process over the board for players. Well, in PTTT I think I can say I have struck gold. With 12 dice to place on each turn where a player must place 3 on the small board, 4 on the medium sized board, and 5 on the large board, the pace of the game is really spectacular. I should say that it is important to allocate required amounts on the three boards because a player can win on a board right off the bat. So, we must restrict to 3, 4, and 5, respectively for the three boards in ascending size. Think of Go with the placement of a single stone, and now think of 12 dice with a variety of 1's, 2's, 3's, 4's, 5's, and 6's to distribute among 3 boards. The decision tree for one turn is quite impressive, let alone trying to forecast for very many turns in the future. However, the simplicity of the game--making a strong chain or two weak chains--gives us an anchor in which to focus our thoughts. Thus, the speed of the game is very fast, yet it isn't out of control because of the game's simplicity. The main attraction of PTTT will be its great speed of development and game play.

Pythagorean Tic Tac Toe is Almost as Simple as Tic Tac Toe but More Complex than Go

(Warning!! This Paragraph Contains Math--Skip it if You Like)

Using the formula S = N^(1/e), where S is the speed of permutation, N is the Boundary, and e is the exhaustion tabulation, we can find the speed of permutation of PTTT. However, because of the chance element in PTTT, we must make a few simplifying assumptions. First, N = 19^200 = 5.63 x 10^255 (this far exceeds Go's N of 3^361 = 1.74 x 10^172). Next, we find e to be technically 40, but due to Claims and Upgrades and Overrides, we shall say 80--a simple doubling. Note that this is far less than Go's e of 361 (361 half turns to fill up the board). But, I am told that e for Go is actually about 10^170 (filling and emptying the board many times over during a single game), which is a horrible reality for Go because I declare Go to be a game construction disaster because of this fact. I propose the rule if by the 200th turn of Black no winner is declared, then it shall be an automatic draw. But, who am I in the world of Go? Anyway, using e of 80 for PTTT, we get that S = (5.63 x 10^255) ^ (1 / 80) = 1573.6. S for Go (to compare) is 3. S = N^(1/e) = 3^361 ^ (1/361) = 3 (The 361's cancel multiplicatively). Thus, we can declare with great confidence that PTTT is vastly more complex than Go because its S is 524.5 times the S for Go. This means SxSxSxS... (a total of e multiplications) grows much much faster for PTTT than for Go. This should mean that a computer, such as that developed by Deep Mind (a division of Google) which produced Alpha Zero and Alpha Go that became World Champions in Go, Shogi, and Chess (the triple threat lol), would not be able to do the same in PTTT because of the great increase in complexity, unless it is able to find a shortcut to victory in the great tree of possiblities. The great game of Tines and Barbs (also invented by Mr. Berard) is also more complex than Go by measurement of its S, but even that game does not compare to PTTT by looking at S. This formula of mine: S = N ^ (1 / e) is an advancement in game complexity assessment.

Strategy and the Defensive Stance

My advancement to Tic Tac Toe does require the use of to play it well. We will first note that it is possible to fill a board with O's, and then place a single X in each row and column such that each X blocks a chain from forming in that row and column. A nxn board will thus require n such placements of these X's to block the opponent. One can replace n by 6, 8, and 10 to create three statements for the three board sizes in PTTT. We will call this minimalist blocking The Defensive Stance. This is why the grey dice rules were created to enable a player to overcome the defensive stance posture. So, if a player is getting too many weak dice, it is advised to place the strong dice on that board in a defensive stance. A player can only break a defensive stance with a grey die, which requires a four of a kind. Below is pictured a defensive stance on the 6x6 board. The O's outnumber the X's greatly, but no chain is formed. A similar defensive stance can be made on the 8x8 board (with 8 X's) and the 10x10 board (with 10 X's) with very little difficulty.

XOOOOO

OOXOOO

OOOOOX

OOOOXO

OXOOOO

OOOXOO

A Few Other Ideas on Strategy

This is an exceedingly complex game, and as such, it will have people with superlative ability develop some amazing stratgies. While we wait for that, I will say a few obvious things. The first is in regards to the diagonals. Each square on a diagonal offers three ways to win, while a square anywhere else only offers two ways to win. So, the diagonals are more important than the other squares for this reason. Of course, if you only place your dice on the diagonals, your are sure to lose. So, when we say more important, we do not mean that you can neglect the other squares with impunity.

Next, it can be important to upgrade your own dice to reduce the liklihood that your opponent will claim them or turn them into Greys. However, this must be viewed with an eye to what your opponent is doing as well. If your opponent threatens to make a strong chain, you must defend it and not upgrade your own dice elsewhere. You will lose on that board if you don't spend your resources in defense when you need to. Similarly, you can mop up your opponent's weak dice with a bunch of claims, but watch what is going on. You may have missed a threat that you should have spent your resources on, and it would have been better to leave your opponent's weak dice alone.

Finally, a single 4 as your lone defense against your opponent forming a strong chain on a board is not much of a defense. If he or she throws a four of a kind of 4's, 5's, or 6's, you lose on that board. Here, it is imperative that you upgrade that 4 to a 6. Then, only a four of kind of 6's can cause you to lose on that board. Yes, it is still possible to lose, but it is LESS LIKELY. You have to think in terms of probabilities in this game to do well. If you have a pair of 6's as your defense against the opponent's strong chain, that is even better. So, try not to let your opponent develop near full length strong chains because it is easier to turn your few blockers into Greys, winning, than to develop new chains elsewhere. Of course, as the pace of the game is so fast, developing two weak chains must be watched for as well. Those opposing weak dice that you leave laying around can come back to bite you!!

The Rating System for PTTT

I have already developed a superior rating system to the Elo rating system used in Chess and Shogi. It is called the PRA (Performance Rating Algorithm), and it is truly a powerful rating system. In my game of Tines and Barbs, we will use a previous rating from Chess or Shogi since both of those populations use the Elo rating system and also because there is considerable skill transfer from Chess or Shogi to Tines and Barbs. But, with PTTT, we do not have a skill transfer to bank upon, and we will thus be in the dark for new players of this game. But, fear not!! The PRA is a powerful rating system as I have already stated (and just in case, I will say it again--the PRA is a powerful rating system!!). This system has what I call The Boosting System. It will provide crashes or boosts to a player's rating based on the trends it sees. We will start each player off at 1500 points. Then, as the players play games, the rating system will adjust their ratings normally. But, at certain intervals, the Boosting System acts, and sometimes it dramatically acts to give a big boost or big crash to a rating. This is necessary to get correct the players who are far away from the mean of 1500. A player who is really a 700 player has to lose 800 points, and a player who is really a member of the 2700 club, has to gain about 1200 points. Players needing these gains and losses to get their ratings correct can get there in about 160 to 200 games with the Boosting System. Without the Boosting System, it could take as long as 900 games. So, the Boosting System will be used in the initial population of players only. After some point (maybe 6 or 8 months or so), we will look to see if we have developed a stable bell curve of ratings. If so, we will turn off the Boosting System--it will have done its job. Then, we will use The Tournament App (the powerful part of my rating system used when the population engages in tournaments) to its full effect to conduct tournaments of PTTT.

The Block Transition Tournament

I have already designed the best type of tournaments to play as well--The Block Transition Tournament. We will conduct tournaments with this design with PTTT. Our site will also host Tines and Barbs, and these tournaments will be conducted with a Block Transition Design as well. I will briefly describe what The Block Transition Tournament design can do.

To this end, I will compare our modern tournaments to it. In a round robin, you might have the two highest rated players face off in round 3 of a 13 round event. Then, you watch those two as they battle the rest of the field. One might be a half point behind the other going into the last round game. We have these "what if" scenarios to spin. What if the leader draws? Then, if the second place player wins the final round, we will have a draw. What if the leader loses? Well, that opens the door for the second place player to win the whole thing!! The exclamation marks are there for sarcastic reasons!! (And, so were those.) The Block Transition Tournament provides a better way.

In a Block Transition Tournament, we break the players up in groups of size 4 for the most part. Different designs can use different sizes, but let us be simple here. Within each group, a mini round robin is played that lasts 3 rounds. Then, we rerank the whole roster of players by Points, PZero scores, G-scores, and Seniority. These are technical parts of the Block Transition Tournament that develop mathematically as the tournament progresses. We measure all of them and use them at the appropriate times to move the tournament along correctly. In the final grouping, we put all the stars together, and the final game of that grouping pits #1 against #2, which is where the climax should be. All modern tournaments fail in this most basic regard.

Another type of Block Transition design has what I call a migrating 2-Block. In this type of design, we can have #1 play #2 up to three times in the final three rounds of the tournament. Now, that is some exciting stuff right there!! We will use the Block Transition Tournament to host our PTTT events on our website.

The Naming of the Groups

I have just called these sides White, Black, and Grey. It is not lost upon me to call them Democrats, Republicans, and Independents. Or, they could have been called Managers, Workers, and Executives. How about calling them Plaintiffs, Defendents, and Witnesses? There are many such possibilities if one thinks about it, but I have decided to stick with tradition and use White and Black, and the neutral party in light of the names White and Black is logically, of course, Grey. Nevertheless, I am open to change the Greys to Red or some other color like Brown to stand out more. However, I don't want it dominate the attention like a neon color or some such thing. A lighter Grey with black dots is probably the best choice, and it is therefore, my preference.

So, What Are All the New Things?

First and foremost, we have a new game!! As part of this new game, we have an entirely new concept- -a neutral piece that belongs to both players. Chess and Go have partisan pieces only. Shogi has pieces that have transferred partisanship to the other player through the drop rule (but a Shogi piece has single partisanship only at any given time in the game). Thus, only PTTT has truly neutral pieces--The Greys. Tines and Barbs has a conflicted square with shields belonging to both players on it, but a conflicted square is "hostile" to both players. A Grey in PTTT is "friendly" to both players. Next, we have a new rating system. And, we have a new tournament design. We also have a new outcome system in PTTT (in tenths instead of halves). So, enjoy all this newness because all of it is better than what has gone on before!!

Final Thoughts

We will host this game on the Tines and Barbs website at www.tinesandbarbs.com for any subscribers to the service. It will be free to play this game as a subscriber to Tines and Barbs, but it will be unavailable to those with a free account. Thus, you get a rating if you subscribe and you get access to Pythagorean Tic Tac Toe!! You can play this game with others and get a separate PTTT rating from your Tines and Barbs rating. We will see what kind of demand there is for the game along with how the ratings develop for the population of Tines and Barbs players that wind up also playing Pythagorean Tic Tac Toe.

Now, PTTT was conceived of as a game of chance. However, with 12 dice per throw, the chance element paradoxically is reduced greatly. On average, a player would see approximately 2 of each number per throw. Of course, many turns will have a 4 of a kind. I threw 12 dice for 80 throws and found about 2/3 of the throws had a four of a kind. Thus, if a game lasts 40 turns, then there would be 80 throws. If we say 81 throws, the math works out really nicely when multiplying by 2/3. Thus, 2/3 of 81 is 54 Grey dice in the game. Half of these is 27 Weak Greys, which may or may not get Overridden during the game, but the other half is 27 Strong Greys, which is plenty to allow victories on the three boards while still allowing for Cats.

Having said the above, we COULD make this a pure skill game by allowing each player the predetermined average throw of two of a kind for each number. And, we could say each player gets an optional Grey of the value of the turn's remainder when divided by 6. Thus, on turns 1 through 6, the optional Greys are 1 through 6 in that order. But, on turn 7, we divide by 6 and get a remainder of 1. On turn 8, we divide by 6 and get a remainder of 2, and so on. Thus, the optional Greys go in the order of 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, and so on. If a player wants a Grey on that turn, replace one of the two dice in the pair for that value with the Grey. Then, place the 12 dice. I do not prefer this mechanistic way making PTTT a pure skill game. The chance element will bring in a much enjoyed element of fun. The skill will still be required to win as many decisions will have to be made especially in how to allocate the dice to the three boards. I put this in so that people will know that PTTT COULD be a pure skill game, but it is a game of chance by the choice of the inventor.