A better solution to the 12 identical pool balls problem

cnaumann · 2005-05-10 14:04:17

A friend of mine told me this problem, and I worked out a solution. Then I looked up the solution on this website. I like my solution much better!

The Problem:

You have 12 balls identical in size and appearance but 1 is an odd weight (could be either light or heavy).

You have a set scales (balance) which will give 3 possible readings: Left = Right, Left > Right or Left < Right (ie Left and Right have equal weight, Left is Heavier, or Left is Lighter).

You have only 3 chances to weigh the balls in any combination using the scales. Determine which ball is the odd one and if it's heavier or lighter than the rest. How do you do it?

My Solution:

Label the balls 1-12

First Weighing:
Left: 1 2 3 4
Right: 5 6 7 8
Off: 9 10 11 12
Record the heavier side (L, R, or B)

Second Weighing:
Left: 1 2 5 9
Right: 3 4 10 11
Off: 6 7 8 12
Record the heavier side (L, R or B)

Third Weighing:
Left: 3 7 9 10
Right: 1 4 6 12
Off: 2 5 8 11
Record the heavier side (L, R, B)

There are 27 (3^3) possible combination of scale readings. A complete sorted list of the scale reading appears below. Note that only 24 of the 27 readings should be possible given the original problem statement. The algorithm was designed so that if all three scale readings are the same, an error is flagged indicating that the scale is stuck.

BBB Error! There is not a single light or heavy ball (or scale is stuck).
BBL Ball #12 is light
BBR Ball #12 is heavy
BLB Ball #11 is light
BLL Ball #9 is heavy
BLR Ball #10 is light
BRB Ball #11 is heavy
BRL Ball #10 is heavy
BRR Ball #9 is light
LBB Ball #8 is light
LBL Ball #6 is light
LBR Ball #7 is light
LLL Error! Scale is stuck!
LLB Ball #2 is heavy
LLR Ball #1 is heavy
LRB Ball #5 is light
LRL Ball #3 is heavy
LRR Ball #4 is heavy
RBB Ball #8 is heavy
RBL Ball #7 is heavy
RBR Ball #6 is heavy
RLB Ball #5 is heavy
RLL Ball #4 is light
RLR Ball #3 is light
RRB Ball #2 is light
RRL Ball #1 is light
RRR Error! Scale is stuck!

MathsIsFun · 2005-05-10 17:35:32

What a neat approach ... I haven't checked your answer, but the approach looks sound !

I will put it as part of official solution when I get a chance. Should I credit "cnaumann" ?

cnaumann · 2005-05-14 08:30:00

Thanks! I am glad you liked my solution. You can credit Charles Naumann.

MathsIsFun · 2005-05-15 19:34:36

I have added your solution here.

Mr T · 2005-05-31 00:16:28

wow ur famous

tt · 2005-07-04 11:17:00

Hello. I'm a new guy in here. Mr. Cnaumann 's solution is so impressive and systematic. Bravo !
I figured out a solution several months ago, although less nimble but...I tried my best :

Weigh (1st time) 4 balls against other 4 (choose randomly).
IF THE SCALE NOT BALANCED: the last 4 balls (not on the scale) are normal ones.
Name the 4 balls on heavier side A,B,C,D. If the odd ball is in group ABCD, it must be heavier than normal one.
Name the 4 balls on lighter side E,F,G,H. If the odd ball is in group EFGH, it must be lighter than normal one.
Weigh (2nd time) : (N is a normal ball)

A B N ~ C D E

1a/ If ABN = CDE : the odd ball is in group F,G,H.
Weigh F against G (3rd time):
If not balanced, the lighter one is the odd ball.
If balanced, H is the odd ball (and is lighter).
1b/ If ABN ≠ CDE :
1b1/ If the scale tipped towards A,B,N side (heavier): the odd ball is in group A,B,E.
Weigh A against B (3rd time) :
If not balanced, the heavier one is the odd ball.
If balanced, E is the odd ball and lighter.
1b2/ If the scale tipped towards C,D,E side : the odd ball is in group C,D.
Weigh C against D. The heavier one is the odd ball.

IF THE SCALE BALANCED (at the 1st weighing) : these 8 balls are normal ones.
Name the last 4 balls A,B,C,D.
Weigh (2nd time) :

A N ~ C D

1a/ If not balanced (AN≠ CD) : the odd ball is in group A,C,D.
1a1/ If the scale tipped towards A,N side :
If the odd ball is A, it must be heavier than normal one. If the odd ball is C or D, it must be lighter than normal one.
Weigh C against D (3rd time):
If not balanced, the lighter one is the odd ball.
If balanced, A is the odd ball and heavier.
1a2/ If the scale tipped towards C,D side :
If the odd ball is A, it must be lighter than normal one. If the odd ball is C or D, it must be heavier than normal one.
Weigh C against D (3rd time) :
If not balanced, the heavier one is the odd ball.
If balanced, A is the odd bal and lighter.
1b/ If balanced (AN=CD) : B is the odd ball.
Weigh B against a normal ball to know if it is lighter or heavier than normal one .

Last edited by tt (2005-07-04 11:33:46)

MathsIsFun · 2005-07-04 19:28:52

Hi tt, and welcome to the forum

That is a good solution, and would look good on TV, I reckon.

Hero says "you are going to weigh each one? Harumph!" *throws four on one side, four on the other* *switches a few balls around* "switches one more time* "Here- this one!"

tt · 2005-07-05 17:49:12

It helps if you have skills with balls. (j/k)

MathsIsFun · 2005-07-05 19:53:22

Oh yes, I love billiards (ahem)

justlookingforthemoment · 2005-07-05 20:04:48

What does tt stand for, tt?

Or isn't it meant to stand for anything?

tt · 2005-07-15 18:20:39

It's short for my email address.

Roraborealis · 2005-07-15 21:29:06

Aha, I see. Heeheeheeheeheeheehee.......

Zach · 2005-07-16 10:52:48

"tt" stands for "Too true", actually.

rszopa · 2005-10-04 10:01:57

hello. i'm sorry to say it, but the neat answer isn't as new as you would expect... it has been published in at least five books (however, it isn't very popular in the Internet, if it makes you feel proud ). just take a look at http://www.cut-the-knot.com/blue/OddballProblem1.shtml. however, i'm not accusing anybody of plagiarism--this solution had been found many times independently (quite a good exapmle of convegence, you would say).

the problem with this solution is the rules of inference cannot be observed so clearly as in the ``classical'' solution. it would be quite good if someone (the author?) explained what is the rule of putting the balls on the left and on the right throughout the 3 weightings.

Last edited by rszopa (2005-10-04 10:09:28)

MathsIsFun · 2005-10-04 11:03:59

Mathematics is like that ... Pythagoras was not the first to discover the "Pythagoras" Theorem, for example!

wcy · 2005-10-05 01:40:55

who can juggle 12 balls?

MathsIsFun · 2005-10-05 11:05:57

I can, if they are all placed in a bag!

justlookingforthemoment · 2005-10-05 18:28:01

Twelve!? I can't even juggle three properly. Can you juggle, wcy?

koundinya17 · 2011-09-07 06:37:42

Instead of dividing it into 3 groups...why not into 4 groups of 3 each
(1) 3
(2)3
(3)3
(4) 3

1) (1) vs (2) - if equal weight
(1) vs (3) - from this attemp you will find which has the odd one and you will find in which group that odd one is
Now you got one group having 3 balls..in which the odd ball is
now 1 ball vs 1 ball ...solution found in 3 attempts

bobbym · 2011-09-07 08:27:25

Hi koundinya17;

Welcome to the forum! That is a solution!

Math Is Fun Forum

#1 2005-05-10 14:04:17

A better solution to the 12 identical pool balls problem

#2 2005-05-10 17:35:32

Re: A better solution to the 12 identical pool balls problem

#3 2005-05-14 08:30:00

Re: A better solution to the 12 identical pool balls problem

#4 2005-05-15 19:34:36

Re: A better solution to the 12 identical pool balls problem

#5 2005-05-31 00:16:28

Re: A better solution to the 12 identical pool balls problem

#6 2005-07-04 11:17:00

Re: A better solution to the 12 identical pool balls problem

#7 2005-07-04 19:28:52

Re: A better solution to the 12 identical pool balls problem

#8 2005-07-05 17:49:12

Re: A better solution to the 12 identical pool balls problem

#9 2005-07-05 19:53:22

Re: A better solution to the 12 identical pool balls problem

#10 2005-07-05 20:04:48

Re: A better solution to the 12 identical pool balls problem

#11 2005-07-15 18:20:39

Re: A better solution to the 12 identical pool balls problem

#12 2005-07-15 21:29:06

Re: A better solution to the 12 identical pool balls problem

#13 2005-07-16 10:52:48

Re: A better solution to the 12 identical pool balls problem

#14 2005-10-04 10:01:57

Re: A better solution to the 12 identical pool balls problem

#15 2005-10-04 11:03:59

Re: A better solution to the 12 identical pool balls problem

#16 2005-10-05 01:40:55

Re: A better solution to the 12 identical pool balls problem

#17 2005-10-05 11:05:57

Re: A better solution to the 12 identical pool balls problem

#18 2005-10-05 18:28:01

Re: A better solution to the 12 identical pool balls problem

#19 2011-09-07 06:37:42

Re: A better solution to the 12 identical pool balls problem

#20 2011-09-07 08:27:25

Re: A better solution to the 12 identical pool balls problem

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