Math Forum Discussions

Sam @ The NIMP Team

Posts: 56
Registered: 12/8/04

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

Plain Text

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 re-indexation)

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.
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