Combinatorics in Stella Cunningham – Mentor: Dr. Jason Callahan

Tarot is a deck of cards used for games and fortune telling. To read for the reader to recognize it. constellation is present in an n-card spread. For each n we simply di- one’s fortune with Tarot cards, a reader presents a spread of n cards. Tarot Constellation Total Cards vide the number of ways a constellation can appear by the total number We use combinatorial methods to calculate how many spreads of n The Magician 1, 10, 19 10, 11 of spreads; the resulting probabilities are plotted below. Tarot cards exist. In the Tarot deck there are nine types of constellations 2, 11, 20 2 7 Probability of constellation in n-card spread comprising disjoint subsets of the deck. We write a Python program to 3, 12, 21 3 7 1 count how many n-card spreads contain at least three cards from any 4, 13, 22 4 7 0.9 one of these constellations and then find the probability that an n-card The 5, 14 5 6 0.8 spread contains such a subset of cards. 6, 15 6 6 0.7 7, 16 7 6 0.6 8, 17 8 6 Terminology 0.5 9, 18 9 6 The Tarot deck is comprised of 78 distinct cards divided 0.4 Probability into the major arcana and minor arcana. is Counting Constellations 0.3 an example of a Major Arcana card. Its name is at the Now that we know how many n-card spreads are possible, we wish to 0.2 bottom and number 0 through 21 at the top. The Minor know how many of those spreads contain a constellation. We start with 0.1 Arcana cards are broken into 4 suits: , , specific constellations and then generalize. For a constellation to not swords, and cups. Each suit contains 14 cards: the court cards (King, 0 appear means that 0, 1, or 2 cards from that constellation are present. 0 5 10 15 20 25 30 35 , Knight, and Page) and the number cards (numbered Ace and Thus the general formulas will find arrangements without the constella- Number of Cards Two through Ten). To find a card’s Identifying Digit decompose and tion and then subtract from total spreads to yield spreads with a specific sum the numbers until reaching a single digit. For example, the identi- Why is the probability 1 once we reach 35 cards? By the pigeonhole constellation. fying digit for 19 is 1 because → 1 + 9 → 10 → 1 + 0 → 1. Knowing a principle, we must have at least 19 of the 62 cards from the 9 constella- card’s identifying digit is crucial to placing it in a constellation. The Magician: tions to guarantee 3 belong to the same constellation. Together with the 78 67  67 11  67 11 − + + 2n · n! 16 cards not part of any constellation, we must have 16 + 19 = 35 cards Tarot Spreads and Constellations n n n − 1 1 n − 2 2 in the spread to guarantee the appearance of a constellation. The High Priestess, The Empress, or The Emperor: A spread is a collection of cards presented during a reading. The cards 78 71  71 7  71 7 − + + 2n · n! Future Research below represent a three-card spread configuration. The features that n n n − 1 1 n − 2 2 make a spread distinct are order and orientation, as seen below. Future research includes finding a general form to calculate the num- , The Lovers, The Chariot, Strength, or The ber of spreads with constellations for any n. We could also explore Hermit: 78 72  72 6  72 6 placing restrictions on the appearance of constellations. For example, − + + 2n · n! n n n − 1 1 n − 2 2 how many spreads contain The Magician constellation, where the cards The next question is how many spreads have at least one constella- are placed consecutively and have no outside cards breaking up the con- tion? We use Python to write a counting program that uses 9 nested stellation? for loops to iterate through all possible combinations of having 0, 1, or 2 cards present from each of the 9 constellations. The variables Acknowledgments When counting how many ways we can arrange n cards in a spread we x1, x2, x3, . . . , x9 represent how many cards are present from each of I would like to acknowledge that this research was directed by my re- proceed as follows. How many ways can we choose n cards from a deck the 9 constellations and x0 represents how many cards are present from search advisor, Dr. Callahan. Partial support for this project was pro- of 78? How many ways can we order those n cards? How many ways the remaining 16 cards not part of any constellation. The for loops in- vided by the Dr. M. Jean McKemie and Suzanne Mason Endowed Stu- can those n cards be oriented? We know the answer to these questions dex each variable through their possible values: x1, . . . , x9 ∈ {0, 1, 2}, dent/Faculty Fund for Innovative Mathematics Summer Scholarship. individually and just need to put it together: and x0 ∈ {0,..., 16}. Once the sum of the xi equals n, the variables 78 78! 78! · n! · 2n = · n! · 2n = · 2n are used in the formula below which is added to the existing count of n n!(78 − n)! (78 − n)! possible spreads so far. This formula counts all the ways a subset of the References Thus, this is our general form to find the total spreads possible with n cards can be chosen from each specific constellation and arranged, as [1] Fletcher, Peter, Patty C. Wayne (1987) Foundations of Higher cards. order and orientation matter. Mathematics. PWS-Kent, ISBN 978-0-87150-164-6. In the Tarot deck there are nine constellations formed around the major 11 7  7  7  6  6  6  6  6 16 [2] Greer, Mary K. (1988) Tarot Constellations: Patterns of Personal · n! · 2n arcana cards which have identifying digits from one to nine. The table x1 x2 x3 x4 x5 x6 x7 x8 x9 x0 Destiny. Borgo Press. below lists all the constellations and their cards. When a spread is pre- Combining these results from our Python program and the calcula- [3] Iiams, Joel E. (2002) Counting Trash in Poker, Mathematics Maga- sented there must be at least three cards from the constellation present tions of total spreads possible, we now compute the probability that a zine, 75:4, 263-274.