From: wang@math.psu.edu (Feng Wang)
Newsgroups: soc.culture.china
Subject: A little research on 24 pt card game
Date: 18 Dec 92 03:58:59 GMT
Reply-To: wang@math.psu.edu
Organization: Dept. of Mathematics, Penn State University


The 24-point card game is really a simple game.  Using a set of playing cards,
the number of combinations of 4 cards is well within 2000.  Some netter has
already pointed out that exhaustive search is possible.  But I don't like to
actually do that---after that the game would have become completely dull!

I also like to share with netters some research I did on this game a long
time ago.  Thinking that some people still think any such things would spoil
the fun of the game, let's have a SPOILER ALERT--------

Skip this article if you don't like to spoil the fun




I was interested in the methods used and put them into different categories.
Some categories are easier and some are difficult to think of.  It's also
interesting to think that why people can't find such simple methods.
I categorize them by the final stages of the calculations, list in the
ascending order (I think) of difficulty:

(1) A*B type (e.g. 1*24, 2*12, 3*8, 4*6)
    This is the easiest type, since we have a lot of freedom here: both A and
    B could be made up by 1 number, 2 numbers or 3 numbers.  No wonder it is
    the most commonly encountered type.  Many combinations have more than one
    non-equivalent (e.g. not by changing order of multiplication) solutions.
    [Side topic: finding multiple solutions is also interesting]

(2) A*B+-C type
    This is a large variety.  It could be 5*5-1, 2*7+10, 2*10+4, 4*4+8, 3*5+9,
    5*7-11, 2*9+6, 3*11-9 ......
    We have less freedom, since only one of A, B or C could be made up by 2
    numbers.  Some of the combinations also have category (1) solutions, but
    some have only unique solutions in this type.

(3) Distributive law type: A*B+-C*B
    All 4 numbers are used.  Some combinations are easier than category (2),
    for example 4*4+4*2; and some are not so obvious.  
    Examples: <12, 12, 12, 10>, <8, 8, 8, 5>, <10, 6, 6, 6>, <10, 4, 4, 4>,
              <9, 6, 6, 5>, ......

(4) Summation type:
    It is surprising that many people just don't see it when simply adding
    up the 4 numbers gives 24.
    Examples: <4, 5, 6, 9>, <3, 4, 8, 9>, ......

The above are easier ones.  Most of the combinations in the categories below
are just unsolvable to a new hand.

(5) (A+-B/C)*C type:
    This type especially includes the fraction involving combinations.  Without
    fraction, it should be contained in category (1), an easy one.
    Examples: <1, 5, 5, 5>, <3, 3, 7, 7>, <2, 2, 11, 11>, <5, 5, 7, 11>,
              <2, 7, 7, 10>, <4, 4, 7, 7>, <2, 2, 13, 13>, <5, 7, 7, 11>
              [Any more?]

(6) (A*B+-C)/D type:
    The difficulty of this type maybe that it ends with a division of a big
    number by one of the numbers in the group.  But this is a huge category
    anyway.
    Examples: <7, 7, 2, 1>, <11, 11, 5, 1>, <13, 11, 6, 1>, <13, 13, 7, 1>,
              <10, 10, 4, 4>, <11, 7, 5, 3>, <13, 7, 5, 3>, <13, 12, 12, 6>,
              <13, 4, 4, 4>, <11, 8, 8, 4>, <13, 8, 8, 4>, ......

There could be other types, and more difficult problems.  The above is all
I have gathered.

[Side topics: solve the combinations which have certain patterns, for example,
<3, 3, 3, k>, 1<=k<=12; or <a, a+1, a+2, a+3>, ......]
Xref: sparky soc.culture.china:33189 soc.culture.taiwan:16341 soc.culture.hongkong:16527