Ivars Peterson's MathTrek
December 9, 2002
Tired of playing the same old card games with the same old cards? One option is to expand the deck to include five suits instead of just four.
Normally, a standard deck's 52 cards are divided equally among four suits: spades, clubs, diamonds, and hearts. These suits come into play in a host of card games. In poker, for example, a player holds five cards and seeks a combination that has a higher value than those of his or her opponents. At the top of the heap is the straight flush, which consists of any sequence of five consecutive cards of the same suit. There are 40 ways of getting such a hand with a standard deck, so the probability of being dealt a straight flush is 40/2,598,960, or .000015. The next most valuable hand is four of a kind, then full house, flush, straight, three of a kind, two pair, and one pair. Whatever your hand, you can still bet and bluff your way into winning the pot, but the ranking (and value) of the hands reflects the probabilities of obtaining various combinations by random selection of five cards from a deck.
How does the introduction of a fifth suit affect poker odds? With a five-suited deck of 65 cards, the total number of possible five-card hands is 8,259,888. You can now draw five of a kind, a combination that's even rarer than a straight flush. Indeed, there are precisely 13 ways to obtain five of a kind, whereas there are 50 ways to get a straight flush.
Moreover, you can have other combinations that consist of one card of each suit. Such a combination is sometimes termed a rainbow. A rainbow straight, for instance, consists of five consecutive cards, each one a different suit. Its value is between that of a straight flush and four of a kind. A rainbow itself (any five cards, each one a different suit) would fall between three of a kind and two pair. It's also possible to introduce rainbow variants of four of a kind, three of a kind, two pair, and two of a kind.
People have worked out five-suit versions of other card games, including spades, bridge, hearts, and various types of solitaire.
If you're interested in trying out some of these variants, you can buy five-suit decks from a number of companies: Stardeck Games (http://www.stardeck.com/), Five Star Games (http://fivestargames.9f.com/), and USA Playing Card Co. (http://www.usaplayingcardco.com/). The Stardeck Web site provides the ranks and probabilities of five-suit poker hands. For general information about standard card games and their rules, go to http://pagat.com/.
If you want to go one step further, six-suit decks are available from Empire Cards (http://www.rightfast.com/empire/). The Empire Cards Web site includes rules for six-suit rummy and hearts (see http://www.rightfast.com/empire/CAcards.htm).
A five-suit deck of cards might be a suitable holiday gift for someone mathematically or probabilistically inclined. It can also serve as an unusual, conversation-generating token of appreciation.
Here are some other, mathematically oriented gift possibilities.
For those who enjoy solving puzzles, the Rush Hour family of games provides interesting challenges. I first encountered these puzzles earlier this year when I wrote an article for Science News about the computational complexity of solving sliding-block puzzles (see Logic in the Blocks at http://www.sciencenews.org/20020817/bob10.asp).
In Rush Hour, a player starts with rectangular blocks shaped like cars and trucks, each in a given location on a six-by-six square tray. Each vehicle is one square wide and either two or three squares long. It can travel only backward and forward along the row in which it's initially placed and can't change its orientation. The player's goal is to figure out a sequence of moves that clears a path for a designated car so it can reach the only exit on the grid.
Japanese designer Nob Yoshigahara came up with the idea for Rush Hour in the late 1970s. In Japan, the game was issued as a commercial product called Tokyo Parking Lot, and players were expected to solve a variety of tough traffic-jam challenges. For the subsequent U.S. edition of the game, Nob and his team developed four sets of puzzles specifically designed to offer challenges rated from "beginner to expert."
"The ordering of the problems was made on the basis of human perception of their difficulty," says puzzle designer Harry L. Nelson, who lives in Livermore, Calif., and represents Nob in the United States. No computers were involved in the analysis. "Our feeling was that we were making these puzzles for humans to solve, so human judgment as to their difficulty was best," Nelson notes. However, computer analyses were used to show that the published solutions are optimalrequiring the fewest possible moves.
"Problem 40 from the second deck [of challenge cards] has the longest known optimal solution of 53 moves, but it is not perceived to be exceptionally difficult," Nelson remarks. "The solution tree for this problem is quite narrow."
Additional information about Rush Hour can be found at http://www.puzzles.com/products/rushhour.htm. Safari Rush Hour and Railroad Rush Hour are each played on a seven-by-seven grid. Roadside Rescue brainteaser puzzles are presented on a three-by-ten grid.
On the aesthetic side, you might consider getting an elegant set of die-cast polyhedra. Crafted out of solid aluminum and tinted gold or blue, the set includes a tetrahedron, cube, octahedron, dodecahedron, icosahedron, cuboctahedron, icosidodecahedron, and rhombitruncated cuboctahedron, each one about 3.2 centimeters wide. Heftier and sturdier than analogous plastic or paper models, these glittering solids certainly garner attention when displayed and invite handling (see Pedagoguery Software at http://www.peda.com/).
Artist Bathsheba Grossman of Santa Cruz, Calif., has her own unique perspective on polyhedra, as seen in a variety of gracefully and intricately sculpted forms based on geometric shapes. See her Web site at http://www.bathsheba.com/ for examples of her fascinating creations in metal, wood, and plastic.For those fascinated by optical illusions, two books provide a dazzling array of conundrums for the mind: "The Art of Optical Illusions" and its sequel, "More Optical Illusions." Compiled by Al Seckel, who works in the computational and neuronal systems division at the California Institute of Technology, the books feature dozens of amazing illustrations. Seckel has Web pages at http://neuro.caltech.edu/~seckel/ and http://www.illusionworks.com.
For additional gift ideas, see "The Math Hatter and More" at http://www.maa.org/mathland/mathtrek_12_10_01.html.
Copyright 2002 by Ivars Peterson
Dworkin, P. 2002. A five suited deck of cards. Gathering for Gardner 5 (G4G5). April 5-7. Atlanta.
Peterson, I. 2002. Logic in the blocks. Science News 162(Aug. 17):106-108. Available at http://www.sciencenews.org/20020817/bob10.asp.
______. 2001. Polyhedron man. Science News 160(Dec. 22&29):396-398. Available at http://www.sciencenews.org/20011222/bob13.asp.
______. 2000. Möbius at Fermilab. MAA Online (Sept. 4).
______. 2001. The math hatter and more. MAA Online (Dec. 10).
______. 1997. A progression of primes. MAA Online (Nov. 17).
______. 1996. Trouble with wild-card poker. MAA Online (Sept. 9).
Seckel, A. 2002. More Optical Illusions. New York: Carlton Books.
______. 2000. The Art of Optical Illusions. New York: Carlton Books.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles.