The game of Barcelona databases, evolution, and reverse tessellation through the eyes of a gambler Keith Clay GRCC

First a word from… Excessive Gambling is not cool!

And now a word from… Excessive study of gambling is a time honored tradition. Laplace Poisson Cauchy Excessive study of gambling is a time honored tradition.

The game of Barcelona

According to legend… It was invented on a Mediterranean island when two brilliant mathematicians were shipwrecked. ( island )

The game of Barcelona Their names are lost to history. We know them only as Mr. Red and Mr. Blue.

The game of Barcelona They played a game using only blue and red seashells… … drawn blindly, one at a time.

The game of Barcelona Shells are drawn one at a time. Whichever color has more shells showing after a given draw wins that round. What was the score for this game? Let’s do it again.

The game of Barcelona Shells are drawn one at a time… … so we draw one. The score is: RED BLUE The first shell is red. There are more red shells than blue shells. Red wins round one. 10 The second shell is blue. There is one red shell and one blue shell. Round two is a tie. The score is unchanged. The third shell is blue. There is one red shell and two blue shells. N blue > N red. Blue scores one point. 1 The fourth shell is red. There are 2 red shells and 2 blue shells. N blue = N red. The score is unchanged. The fifth shell is blue. There are 2 red shells and 3 blue shells. N blue > N red. Blue scores one point. 2 The sixth shell is red. There are 3 red shells and 3 blue shells. N blue = N red. The score is unchanged. This could also be called a score of -1 for red or a score of +1 for blue This game can be played solitaire. Or against a casino.

The game of Barcelona In a casino: The game can be played with cards. Only the color matters. If you are betting in a casino… Do you want to be red or black? Does it matter? (If not why not?)

The game of Barcelona This game would not be very interesting... …if it were not for a seagull. Now would you want to be blue or red?

The game of Barcelona With n blue shells and n + 1 red shells, Mr. Red had an advantage. To even the odds, it was agreed that blue would score on turns when drawing a shell produced a tie.

The game of Barcelona Under the new rules, who has the advantage? The score is: RED BLUE This new game is called Barcelona. Would you rather be blue or red?

The game of Barcelona Casinos can play Barcelona with decks of 51 cards (26 red, 25 black) You get $1 for each round you win. You lose $1 for each round you lose. Do you want red? Black? What are your odds?

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game?

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game?

Red or Black? POSSIBLE SCORES: With 26 red cards and 25 black… You can win 51 rounds ( +$51 ahead) You can win 50 and lose 1 ( + $49 ) You can win 49 and lose 2 ( + $47 ) … You can lose 51 rounds ( - $51 ) Can BOTH COLORS win $51? Lose $51?

Red or Black? Observation: Given (n+1) red cards and (n) black cards, Red can win 2n+1 rounds. Black cannot. Red wins $5, loses $0. Score = +$5 for red. Now reverse the order. Red wins $1, loses $4. Score = - $3 for red. Red can win (2n+1) dollars but can only lose (2n-1) dollars. Both appear equally likely.

Red or Black? Remember these shells? Try reversing the order again. The score was: RED BLUE 32 The score is: RED BLUE 14

Red or Black? Conjecture (not a proof): Reversing the order of a set of cards turns a “good” hand into a “bad” hand. The midpoint of “good” and “bad” is: [ (2n+1) + (-(2n-1)) ] / 2 = 1 net win for red. “Forward” and “Reverse” orders of a set of cards are equally likely. So on average, red gains 1 win per hand. Does Barcelona favor Red?

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game?

Running the Table To win $51, red has to win every round. That means there will always be more red cards showing than black. How likely is that? One way to find an answer: Count the number of arrangements where there are always more red the black ( reading right to left) Divide by the number of possible arrangements.

Running the Table How many ways are there to arrange 51 cards? 51 choices for the first card, 50 choices for the second card, 49 choices for the third… 51  50  49  …  3  2  1 = 51!  But wait! Not all of those arrangements are different! We might not need to check arrangements.

Running the Table Rearranging the cards (or seashells) of one color does not change the score. All 3! arrangements of these shells produce the same score. All 2! arrangements of these shells produce the same score. So the number of distinct arrangements of these five shells is:

Running the Table The number of distinct arrangements of 26 red cards and 25 black cards is: Calculating the Barcelona score of each arrangement would take a very long time.

Running the Table CRUCIAL QUESTION: How many ways are there to run the table in a Barcelona game with (2n+1) cards? This is a question with many applications.

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game?

Barcelona-like problems: The parentheses problem: Given n left and n right parentheses, how many arrangements are possible that leave all parentheses closed? ( ( ) ) ( ) Good ( ) ) ( ) ( Bad

Barcelona-like problems: The parentheses problem: In information science: How many possible relationships are there for n categories & subcategories? ( a ( b ) ) ( c ) One relationship ( a ( b ( c ) ) ) Another The problem is crucial to allocation of memory in databases and networks.

Barcelona-like problems: The parentheses problem: Color coding shows this is the same as “running the table” in Barcelona

Barcelona-like problems: Evolution and Genetics: How are creatures (or people) related? What is the family tree of this group? Who branched off when? Dinosaur Reptile Amphibian Bird

Barcelona-like problems: Dinosaur ReptileAmphibian Bird Dinosaur Reptile Amphibian Bird A possible family treeAnother possibility How many possible trees are there?

Barcelona-like problems: “Planted binomial trees” Start at the bottom When you come to a node, turn left At the end of a branch, turn around Left branches are red, right branches are blue The number of family trees for (n+1) critters is the same as the number of ways to run the table with (2n+1) cards (or seashells).

Barcelona-like problems: Reverse tessellation: How many ways are there to dissect a polygon into triangles using only non- intersecting diagonals? Leonhard Euler Euler solved the problem by induction in a process he called “ quite laborious.” Eugene Catalan returned to the problem a century later. The answers to this day are called the “Catalan numbers.” Eugene Catalan

Barcelona-like problems: Reverse tessellation: The connection to Barcelona, databases, and trees?

Solving Barcelona Given (n+1) red shells, and (n) blue, “cut the deck” One of the new groups must have more red than blue. Call it group A. A B The other group cannot have more red than blue. Call it group B.

Solving Barcelona If A precedes B, the score (for red) must be higher… A B … than it would be if B preceded A. A B

Solving Barcelona For any arrangement of cards or shells, a “cut of the deck” will change the score. Math lingo: “cutting the deck” = “cyclic permutation” No arrangements connected by a cyclic permutation will have the same score.

Solving Barcelona The “Pigeonhole Principle” Snow White lived with little people. Name them: Happy, Dopey, Sleepy, Gumpy, Sneezy, Bashful, and Doc. Is that all of them? The “Pigeonhole Principle” Snow White lived with little people. Name them: Happy, Dopey, Sleepy, Gumpy, Sneezy, Bashful, and Doc.

Solving Barcelona

The “Pigeonhole Principle” If you need N answers, all different… And you find N answers, all different… You’re done.

Solving Barcelona Given (n+1) red cards and (n) black cards: There are (2n+1) possible scores for red: +(2n+1), +(2n-1), +(2n-3), … -(2n-3), -(2n-1) Every cyclic permutation (cut of the cards) produces a different score. There are (2n+1) cyclic permutations. Every possible score appears once. Each cyclic permutation is equally likely.

Solving Barcelona Every possible score is equally likely! Every score has a probability of 1/(2n+1).

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game? There are 51 possible scores. The probability is 1/51.

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game? There are 51 possible scores. The probability is 1/51.

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game? None. All scores are equally likely.

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game? Each score is equally likely. Add the scores and divide by (2n+1). The scores add up to (2n+1). On average, red wins a dollar.

The game of Barcelona With 26 red cards and 25 black… Do you want to be red or black? What is the probability of winning $51? What is the probability of winning $11? Which scores are most likely? What will be the average score? Who cares? Is this any more than a game? How many databases, family trees, reverse tessellations, and all that?

Acknowledgements: Probability Combinatorics Graph theory Group theory Information theory Formal logic Set theory Fun I’d like to thank the following branches of mathematics for appearing in this talk:

Acknowledgements: Martin Gardner Possibly the world’s best mathematical author. Read his books. Most of all I’d like to thank …

Questions?