Astonishing Math With Playing Cards

It is baffling how playing cards can lead us into a mystery land of wonder numbers. Perhaps the examples listed here are not even complete - it is up to you to enrich playing card math with new discoveries of your own.

We leave out the question whether we have to view the these mystery numbers as representing the intentions of sophisticated inventors or if these are just a random coincidences which some bright head found by looking at the playing cards from a mathematical angle.

A typical card deck has 52 cards - like the 52 weeks of the year. In the four symbol groups we may, with some right, recognize the four seasons (spring = Hearts, summer = Clubs, autumn = Spades, winter = Diamonds). Or order the classic four elements to the warm (red) and cold (black) seasons (fire = Hearts = spring, air = Diamonds = summer, water = Clubs = fall, earth = Spades = winter).

Fifty-two weeks times seven make 364 days - a year. Add the Joker and we get even closer to the solar year of about 365.25 days. The cipher sum of 364 is 3+6+4 = 13, the number of moon phases in a year, just like number of cards in each suit (4 * 13 = 52).

Now add the values of the cards, ace = 1, two = 2, three = 3, ..., nine = 9, ten = 10, Jack = 11, Queen = 12, King = 13. Surprise: the sum is 364 = 4 * ( (13) * (13+1) / 2 ) = 4 * 91.

It goes on: Divide the number of values of a single suit (91) by the holy number seven and we get 13 again! The cipher sum of the 52 cards is also 5 + 2 = 7.

To get even more astonished, now count the letters of the names of each card

English:Ace(3), Two(3), Three(5), Four(4), Five(4), Six(3), Seven(5), Eight(5), Nine(4), Ten(3), Jack(4), Queen(5), King(4).
There are 52 cards and the letter sum of one suit is
3 + 3 + 5 + 4 + 4 + 3 + 5 + 5 + 4 + 3 + 4 + 5 + 4 = 52.

This also holds for French: As(2), Deux(4), Trois(5), Quatre(6), Cinq(4), Six(3), Sept(4), Huit(4), Neuf(4), Dix(3), Valet(5), Reine(5), Roi(3).
2 + 4 + 5 + 6 + 4 + 3 + 4 + 4 + 4 + 3 + 5 + 5 + 3 = 52 (Reine = Dame)

And German:As(2), zwo(3), drei(4), vier(4), fünf(4), sechs(5), sieben(6), acht(4), neun(4), zehn(4), Bub'(3), Dame(4), König(5).
2 + 3 + 4 + 4 + 4 + 5 + 6 + 4 + 4 + 4 + 3 + 4 + 5 = 52 (zwo = zwei, Bub' = Bube) Don't force it - use a bigger hammer :-)

The sum of the "people" (Jack, Queen, King) is always 13, in these three languages. It follows that the sum of the rest of the cards must be 39, because 52 = 13 * 4 (the four seasons again!)

Intention or coincidence - now you are able to impress your friends with a knowledge of the astonishing mathematical properties hidden in the playing card deck.


Edited and translated from a German book on playing cards and card games:
Peter Trumpf; Spielkarten und Kartenspiele; Keysersche Verlagsbuchhandlung; Heidelberg, München (=Munich) 1958

If you found the above interesting, you may want to read about Karl Palmen's Playing Card Calendar.

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