KRYPTO - speed arithmetic card game Available from Dale Seymour Publications, PO Box 10888, Palo Alto CA 94303. Call 1-800-872-1100 from the USA. Cost $8.95 in September 1995. Cards are identical on one side; the other side shows a number. (Black for numbers through 10, Red for 11...17, and Blue for 18...25.) The distribution of values in my deck (52 cards) is 1..10: 3 each (30 cards) 11..17: 2 each (14 cards) 18..25: 1 each ( 8 cards) A friend's deck, also fresh from the factory, includes an additional 7, 8, 9, and 10 (56 cards total), but I think that's a shipping error. In this directory is a scan of the instruction pages; I've adjusted the formatting a bit. The instruction sheet estimates "3 million games". If you treat all 52 cards as distinct, there are clearly 122151120 ( =(52 choose 5) * 47 ) possible games, each consisting of a set of 5 cards plus one objective card from among the remaining 47. Observing that the cards are _not_ distinct, I get just over 2million distinct games. In how many ways can you play a round? The attempt to combine the 5 cards arithmetically requires: shuffling the cards into one of 120 orders; interleaving arithmetic operations into the four spaces between cards (this can be done in 256 ways); and then deciding on placement of parentheses. I note that for each set of 5 cards if you wish to compute all possible ways of combining them, you can consider casting all the computations into RPN (i.e. "Hewlet Packard format"). Your final string is a list of 5 numbers with 4 operations interspersed, such that when reading left to right there are always more numbers than operations. There are only 14 possible such sequences: NNNNNOOOO, i.e., (N1 op4 (N2 op3 (N3 op2 (N4 op1 N5)))) NNNNONOOO NNNNOONOO NNNNOOONO NNNONNOOO NNNONONOO NNNONOONO NNNOONNOO NNNOONONO NNONNNOOO NNONNONOO NNONNOONO NNONONNOO NNONONONO, i.e., ((((N1 op1 N2) op2 N3) op3 N4) op4 N5). Many of these are the same when the operations are + or * because of commutativity, but distinct when using - or / . Thus given a 5-card hand and an objective card, there are at most 14 x (4^4 sequences of 4 operations) x (5! permutations of the hand)=430920 possible ways to combine them (many of which will certainly yield the same value), making it quite feasible to run an exhaustive check to solve any particular hand. A BASIC program to do this is included. In its present form it can decide in a minute or less whether a hand is playable in a minute or so. Actually this enumeration will include the same calculations more than once. Certainly some formulas -- e.g. (((N1+N2)+N3)+N4)+N5 -- will give the same values if the N_i are permuted. Sometimes it is also true that the values produced with one formula may be computed with another, too, possibly by permuting the N_i. For example, formulas which involve (A/B)/C may be rewritten A/(B*C). Among the 256*14=3584 formulae to consider, I compute that there is a subset of only 606 such that all elements of the first set are equivalent to an element of the second set using the transformations (A/B)/C --> A/(B*C), A/(B/C) -> (A*C)/B, and A*B -> B*A when B is a longer string of symbols than A as well as the corresponding formulae with + and - replacing * and /. The number 606 is not minimal, as these transformations do not allow us to declare as equivalent such formulae as (A+B)*(C-D) and (A-B)*(C+D), (A*B)*C and (A*C)*B, or A-(B-C)*D and A+(C-B)*D (although the latter two should _not_ be considered equivalent for our purposes since in KRYPTO one must keep all intermediate stack entries non-negative and integral.) There certainly are unwinnable rounds (e.g. {25 | 1, 1, 1, 2, 2}) but they are quite rare. I had the computer look randomly for hands to play. Typically it can run for several hundred rounds before stumbling over unwinnable hands. I have not yet computed how many hands are unplayable; the package insert suggests they number in the low thousands. A list of some unplayable hands found is in the other summary file. dave rusin@math.niu.edu http://www.math.niu.edu/~rusin