Pythagorean Tic Tac Toe Copyright (c) 2018 (All Rights Reserved) by Tony Berard Pythagorean Tic Tac Toe--The Complexity 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 chess 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.