The KRYPTO "Arithmetic" Card Game

From a deck of numbered cards, select 5 (the "hand") and a 6th (the "objective"). The task is to find a way to combine the five (each card used once and only once) using the four basic arithmetic operations to yield the objective value.

As of September 2004 the game is available from The Making People Happy Company. (The game was long available from Dale Seymour Publications, who were then bought by Pearson Learning, who carried the product until mid-2004 I think, but then let it go "out of print". Copies continue to appear on Ebay for about five bucks.)

It is possible to play Krypto online through the company's website,

A computer simulation is available (it's pretty simple-minded); this is source code for the UBASIC language, an excellent variation of BASIC.

A printout of the precisely 500 formulas which can arise when combining five numbers using arithmetic operations. (Sample "formulas" are "x1+x2+x3+x4+x5" and "(x5-x4)-(x2/(x3-x1))"; given five card values x1 through x5, the first formula gives one way to combine them; the second gives as many as 120 ways to combine them, depending on which card is called "x1", or "x2", or...) Also available: working notes and odds and ends from the Maple computations.

You can also pick up a complete scan of the rules, which are quite short. A few comments of analysis are also available. I've got some scribbled notes to which you are also welcome. One of these days I'll polish this analysis. Suggestions welcome.

Got no cards? (Admiral Jota) writes, "I've played the game with a normal 52 deck. A=1, numbers=face value, J=11, Q=12, and K=0. Of course, when i played it on a normal deck, we spelled it Crypto."

Simpler version: scramble just _four_ (or fewer!) cards to match an objective value. (It turns out there are only 1170 possible combinations to check; see this Maple program and analysis to generate all 1170 of them and solve hands.) Here's a related post.

This program can also be modified to play the "four fours" game or the game "24" but you must adjust it to allow intermediate results which are not integral, if that is your intent. For example, in the game "24", the hand 3, 3, 7, 7 is insoluble with the restricted rule but without it there is the solution 24=7*(3+3/7). Thanks to Wilbert Dijkhof ( for this observation. Here are some USENET conversations regarding the 4 fours game -- combine four 4's to make all the integers up to 100 ? -- and some more comments and a bit of history and conjecture

Similarly one can try to make values from the four digits of the current year (which has gotten harder since the new millennium!); here is a "Year Puzzle" website.

Messages on related issues:

I have more mail lying around here waiting to be cleaned for posting, so if you have a related question, just ask.

Other pages at my own web site you may find of interest:

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Last modified 2006/07/13 by Dave Rusin,