Updated MAY30,01

Poly-Y, Star and *Star

This page is for those who want to know something of the evolution from Poly-Y to Star to *Star and of the close relationship between them. And for those who want to know what went into the design of the boards and some of the reasons I think *Star is the best of the three. For these enquiring souls I herewith present the

Official Apocryphal History of *Star

In the beginning was the "Y", and it was good ---especially on cracked mud. But while playing Y Craige thought: What if the board had more than 3 edges and corners? What if a player captured a corner by making a Y which touched the two edges meeting at that corner, and any third edge. As in Y, one player will always capture a corner when the board is filled up, but both players cannot capture the same corner. If the player who captures the most corners wins, and there are an odd number of corners, then there will always be a winner, never a tie. We could call it Poly-Y, and Y is the special case with 3 corners, all of which are captured by the player making a Y.

And so it was --- Poly-Y, the game Y had always wanted to be, was born. With the same tactics as Y, Poly-Y was deeper strategically, had another strategic layer.

Poly-Y

Craige tried boards with more and more corners, 5, 7, 9, 15 ... . At first it seemed that the more corners the better --- there were more points to contest and a beautiful global strategic picture emerged. But as the number of corners increased, of necessity the length of the edges decreased. When the edges became too short it was found that it was too easy to make a Y touching 3 consecutive edges, thus "capturing" the middle edge and the two corners bounding it. This "edge capture" tended to make the game more tactical and local, focused on quick gains along the edge, thus losing the elegant global strategic flavor . So the strategic depth increased at first as the number of corners increased, but then decreased. Finally a board with 9 corners and 7 cells along each edge was chosen as the ideal balance.

A board consisting entirely of hexagons has 6 corners. Changing one of the hexagons to a 7-sided region increases the number of corners to 7. For each 7-sided region one more corner is added. So the 9-sided board has three 7-sided regions. If the 7-sided regions were put on the outer edge of the board, then the board would have 262 cells. If they were moved halfway into the center then the board would have 208 cells. If they were moved all the way to the center of the board (where they become a single 9-sided region), the board would have 190 cells. In order to make the edges as long as possible for a given size board (to make "edge capture" more difficult) the 7-sided regions should be moved toward the center. But 7-sided regions are more powerful than 6-sided regions (Luck is a many sided region), and at the center of the board they have too dominating an influence. Craige chose the 208 cell board with the 7-sided regions halfway to the center as the standard Poly-Y board.

Star

A decade later R. Wayne Schmittberger contacted Craige about the possibility of publishing Poly-Y in Games magazine. This reactivated Craige's interest. He and Irene started playing Poly-Y. One day he thought "Suppose we had a situation like this.

Red has captured 5 corners and Blue has captured 4 corners.The game is over and Red has won. But both players have two groups which are not connected. They could continue to play, trying to connect their two groups. Essentially they would be playing Hex on a little 7X7 Hex board in the middle of the Poly-Y board. It seems as if the game should not be over until there are no more unconnected groups, like the Red and Blue groups here, that can still be connected."

"We could give a bonus for connecting groups together. The smallest bonus we could give is one point. But if Blue is the one who connects, then they would each have 5 points and it would be a tie. Furthermore the sum of their scores would no longer be 9 points. Aha! If whenever we add points to one player's score for connecting groups, we subtract the same number of points from the other player's score, then the sum of their scores would always be 9 points. The one with 5 or more points is the winner, there is never a tie."

"We could call any group which captures at least one corner a "star". At the end of the game if one player has been more effective at connecting their stars together and thus has fewer stars than the other player, we could give that player a bonus equal to the difference in the number of stars (subtracting the same number of points from the player with more stars of course). We could call the game Star."

And so it was --- Star, the game Poly-Y had always wanted to be ... But wait! That would be too simple. Boring. Our story needs a few twists and turns. What really happened was this: Craige discovered an elegant generalization of the "Y theorem" (when a board with 3 edges is entirely filled in with two colors, there will always be a connected group in one and only one of the two colors touching all three edges). What Craige discovered was that if you call any group which touches at least 3 edges a "star", and give that star two less points than the number of edges it touches, then the sum of the points for all of the stars is two less than the number of edges on the board.

And so it was --- Star, the game Poly-Y had always wanted to be, was born after all. A group which touches at least 3 edges even if it does not capture any corners is a star which is worth two points less than the number of edges it touches. The player with the most points wins. If the board has an odd number of edges, then there will never be a tie. The mathematical result underlying Star was so beautiful that Craige was not tempted to look into the less elegant version in which points are transferred from one player to the coplayer when the coplayer succeeds in connecting stars together --- an inelegant phoenix waiting to rise from the ashes. Much later it was found that the two versions are really the same game (see clarity ) --- and it was the the "inelegant" version that opened the door to *Star.

The Board for Star

Poly-Y and Star are two different ways of generalizing Y. Poly-Y is more of a brute force approach while Star is a more subtle elegant approach. They have much in common. In fact with boards having 3 edges or 5 edges they are the same game --- the same player wins whether you use Poly-Y scoring or Star scoring. But as the number of edges increases Star puts more and more emphasis on connecting groups together relative to Poly-Y. This gives Star a greater strategic depth.

Initially Star was played on the standard Poly-Y board. Even though Star was Poly-Y's daughter, the rules were so different that it was not clear just how closely related they were. Yet everyone agreed that Star was beautiful. Even more beautiful than her father.

The concern about "edge capture" fades away for Star. Even if you focus on the edges and capture some edges, you still have to connect them across the center of the board. So the ideal board for Poly-Y is not the ideal board for Star. For Star the edges can be much shorter. Once again Craige started considering boards with more and more edges. He got so obsessed that he came up with "crazy" boards that had a corner between two edges on every cell on the perimeter of the board (a few cells even had two corners). Craige vacillated. Sometimes it seemed as if "crazy" boards were the wave of the future. But sometimes he got nostalgic, thinking that something lush and rich was being lost. There was a great deal of back and forth between Craige and Wayne about just how short the edges should be. Finally Wayne, whom Craige admired greatly, convinced Craige to go with the new boards which were far more regular than the boards with fewer edges, and had more points to contest. When Wayne published a version of Star in Games magazine, the die was cast. But even so, Craige had

Lingering Doubts...

The Star board which most Star afficianodos have used is the one that appeared in the September 1983 issue of Games magazine. It is a rather small board with only 75 cells and 33 "edges". Drawn in the fashion of the Poly-Y boards it would have 33 cusps on the outer boundary representing the 33 corners with 33 arcs between them --- the 33 edges, as in the board on the left. As actually presented in Games the 33 edges were represented by partial hexagons in a darker color around the outer boundary. In Star with the "elegant" rules the sum of the scores of the two players is in general two less than the number of edges --- 31 in this case.

One of Craige's "lingering doubts" concerned the fact that a single play in any of the 6 corner cells made a star touching 3 edges and thus worth one point, accounting for 6 0f the 31 points. Craige had found that it seemed to add to the depth of the game when all of the cells have nearly equal value at the start of the game. It seems reasonable that the corner cells should have some compensation for having less access to the center than the other boundary cells, but had Craige and Wayne gone too far? Sometimes Craige thought that the third edge in these 6 corner cells should be eliminated. (It is interesting that Mark Waldow has written saying "My college friends and I used to play Star quite often before they moved away and we found we liked it better without the third dark edge sections touching the six corner cells ..."). At one point Craige even thought that the correct approach was to make a play in the corner cells worth one third of a point. Surprisingly this works very well, but it makes keeping track of the score too complicated (however this idea has been resurrected now in a new guise).

Another lingering doubt concerned the "trivial invasion". Here Red has surrounded a territory of 3 unoccupied cells on the boundary of the board. But Blue can "trivially" invade by occupying the middle cell and then following up by occupying either of the other two cells. It is so easy to invade territories that the focus is shifted strongly to play along the edge of the board. Star seemed better than Poly-Y on any board, and the "crazy" boards (with a corner at every cell on the boundary) seemed optimal for Star ... yet Craige was uncomfortable with what he saw as the lack of balance between edge and center.

Whoops!

Another decade passes and Craige notices that there is an alternative way to get the score for Star. You could give a Star one point for every "corner" (instead of "edge") it touches and then subtract one point and give it to the other player. It was quite amazing that this gave the players the same score, except that each player got one more point. With this way of scoring it is obvious that the sum of the two players' scores is equal to the number of "corners" on the board (remember that now there is a "corner" on every cell on the perimeter of the board, even two "corners" in the true corners). And it suggests a way to modify the game. Instead of taking away one point from a Star and giving it to the other player, suppose you take away two points (giving them to the other player)! This looked really promising! It increased the motivation for connecting stars together resulting in a desirable shift of focus toward playing in the center, redressing the imbalance between edge and center. It made the trivial invasion worthless. A single play in the corner no longer got a point. But (whoops!) Craige neglected to make the connection to Poly-Y where Star had started, and overlooked the fact that a Star not only contains a cell with a "corner" (thus touching the two "edges" meeting at the "corner") but also has to have a "tail" going to a third "edge" in order to avoid being enclosed by the other player. This oversight resulted in the possibility that occupying a region could result in lowering your score. This was a fatal flaw, completely foreign to the requirements of Star.

*Star

Another decade passes. It is the coming of a new millenium. Mark Thompson is planning to host an Internet *Star tournament. Kate Jones of Kadon, who already produces Y, decides to add Star to their catalog of exceptional games and puzzles. Craige, who has become Ea Ea, becomes involved. (Ea is a god from the Babylonian pantheon who doesn't always follow the conventional rules, but makes up his own. On reading Enki's description (Enki is what the Sumerians called Ea before the Babylonians took him over) in Descent to the Goddess Mariah thought "Oh. I know him.") Seething activity ensues. Rules and boards are tossed around like oranges (Babylonian snowballs). Finally Ea notices what Craige had missed (whoops!).

Ea mends the fatal flaw in "whoops!", requires that a star not only enclose a "corner" but not be itself enclosed --- and introduces a board that looks like a star. In order to avoid the scalloped lace doily look of a board with 50 "corners" he introduces the idea that each cell on the perimeter of the board, instead of having a "corner", contains a "peri" which a star "has" if it encloses the peri without being itself enclosed ( a "peri" is a supernatural being from Persian (Babylonian?) mythology which is formed of fire --- what a "star" might be composed of). Ea reduces the compensation given to the cells at the 5 points of the star board. Instead of containing two peries they contain one peri and a "quark", (a peri is composed of 3 quarks) --- the player who gets 3 or more of the 5 quarks has one more peri. A player scores one point for each peri they possess. But, so that the players will try to connect their stars, for each star a player has he must give the other player two of the points which that star has.

It became clear that Poly-Y, Star and *Star were members of a series, the only difference being the number of points transferred from one player to the other for each star --- zero in Poly-Y, one in Star and two in *Star. Looking at the situation carefully Ea came to a better understanding. He could see that the rules for *Star could be cast in a form very similar to the rules that were given for Star in the Games article.

And so *Star, the game that Star had always wanted to be, was born.

Now it is time for the next stage in the history of *Star to begin --- the one where you get involved, the one where you learn the rules for *Star and start to play online. And it is time for me to go to the back shore to bathe in the ocean. For Ea is the water god, and in Hawaii the name 'Ea'ea means sea spray.