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Re: 12 billiard ball problem
Posted:
Nov 18, 1997 2:01 PM


Jonathan Pearce wrote: > Does anybody know of an elegant solution to the twelve billiard ball > problem? > (Twelve apparently identical billiard balls, one of which is slightly > heavier than the rest, determine which is the heavier one making only three > weighings.)
Are you sure it is 'heavier' ? The original problem only states the odd ball is lighter or heavier?
Anyway here is my solution:
Let's call the marbles 123456789ABC. I'll call X a normal marble when we know there are some and it is possible to choose one.
I put a '*' when we know which marble is the odd one AND if it is lighter or heavier. Note that we don't know it only if it is marble C. *** note: another solution would fix that, but do you need it ? ******* Begin of hideous solution ********
1. If 1234=5678
we know that the odd one is in 9ABC
1.1. If 9A=XX
we know that the odd one is in BC
1.1.1. If B=X
the odd one is C
1.1.2. If B<>X
the odd one is B *
1.2. If 9A<>XX
we know that the odd one is in 9A
1.2.1. If 9=X
the odd one is A *
1.2.2. If 9<>X
the odd one is 9 *
2. If 1234<5678
then let's invert 1234 and 5678 and let's say 1234>5678 (just a reindexation)
3. If 1234>5678
then the odd one is in 12345678 and 9ABC are normal
3.1. If 1235=4XXX (my special trick)
then the odd one is not in 12354 then it is in 678 so we know it is lighter since 1234>5678
3.1.1. If 6=7
then the odd one is 8 *
3.1.2. If 6<7
then the odd one is 6 *
3.1.3. If 6>7
then the odd one is 7 *
3.2. If 1235<4XXX
then if the odd one is lighter it is in 1235 and 5678 > it's 5 and if it is heavier it is in 4XXX and 1234 > it's 4 so it is in 45
3.2.2. If 4=X
then the odd one is 5 *
3.2.2. If 4<X
then there is a bug somewhere :)
3.2.3. If 4>X
then the odd one is 4 *
3.3. If 1235>4XXX
then if it is heavier it is in 1235 and in 1234 so it's in 123 if it's lighter it's in 4XXX and in 6789: impossible so it's heavier and it's in 123
3.3.1. If 1=2
then it's 3 *
3.3.2. If 1<2
then it's 2 *
3.3.3. If 1>2
then it's 1 *
******* End of hideous solution ********
Hope it helps...
Sam.  \vvv/ (o O) With best wishes from Sammy _ oOO(_)OOooOO(_)OOo Samuel HOCEVAR  ÃÂÃÂ©lÃÂÃÂ¨ve ECP promo 2ooo (O o) http://www.chez.com/sammy/ mailto:sammy@chez.com /^^^



