Tuesday, April 5, 2005

24, the arithmetic game

There's an arithmetic game called 24. The idea is to take four
integers and using the four basic arithmetic operations plus grouping,
produce the number 24. A friend recently mentioned this game again
and mentioned two of the more difficult sets: "3 3 7 7" and "1 3 4 6".
They're good ones. Some simpler examples: "1 2 3 4" is solved as
1*2*3*4. "3 4 5 6" is solved as (5+3-4)*6. "5 5 9 9" is solved as
(5*5)-(9/9). For low integers, roughly 80% of combinations work.

Of course, if you try to solve one of those 20%, you're just going to
be frustrated. My friend wrote a program to simply spit out a yes/no
answer as to whether a given combo works. I though it was a good idea
so I wrote one too. Knowing something is solvable without knowing the
solution makes for a good puzzle.

Other ways the puzzle is played is to not limit yourself to 24, but to
try to combine a set of numbers to produce 0, then 1, then 2, then 3,
etc. Some sets of numbers will let you get quite high. "1 2 5 6" will
produce every integer up to 43, and 60 distinct integers. "2 5 9 10"
will produce 79 of the numbers from 0 to 100 and 124 distinct values
overall. It's first miss is 41.

Some values are "easier" to hit than 24, which is to say an even
higher fraction of sets can be combined to produce it. "2" is the
most universal result for low numbers. All but 2 combinations of
numbers 8 and under can be combined to equal 2. Limiting yourself to
numbers 7 and under, you can always produce 0. All of the single
digit results are more frequent than 24, as are 10, 12, 14, 15, and


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