Unsolvable? You have 12 balls.... Word. Problem /riddle

Question

Unsolvable? You have 12 balls, all looking the same. One is lighter OR heavier. You have a balance scale to discern weight . You are allowed to weigh 3 times, after which you have to name the ball. Help?

Answer 1

SOLVABLE!  What is often missed with using a balance scale, is that there are three possible outcomes.  In Computer Science, we represent this as a node in a tree structure with three branches.  At the second and third weighing levels of the tree, the nodes also have three branches.  Thus after one weighing, 3 results; after 2 weighings, 3²=9 results; after 3 weighings, 3³=27 results.  This is enough to determine one of 24 possibilities (which of 12 balls and whether heavier or lighter).

 

Since this is a very standard problem, I copied the text of a solution as follows:

 

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 of balance scales which will give 3 possible readings:

Left = Right,
Left > Right, or
Left < Right
(i.e. 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.

GOAL:  Determine which ball is the odd one and if it's heavier or lighter than the rest.

How do you do it?

 

SOLUTION:

Number the balls 1, 2, 3, ... 10, 11, 12

Start off with them in 3 groups: [1, 2, 3 and 4], [5, 6, 7 and 8] and [9,10,11 and 12]

Weigh 1, 2, 3 and 4 vs 5, 6, 7 and 8 with 3 possible outcomes:

1. If they balance then 9,10,11,12 have the odd ball, so weigh 6,7,8 vs 9,10,11 with 3 possible outcomes:
 
1a
If 6,7,8 vs 9,10,11 balances, 12 is the odd ball. Weigh it against any other ball to determine if heavy or light.
 
1b
If 9,10,11 is heavy then they contain a heavy ball. Weigh 9 vs 10, if balanced then 11 is the odd heavy ball, else the heavier of 9 or 10 is the odd heavy ball.
 
1b
If 9,10,11 is light then they contain a light ball. Weigh 9 vs 10, if balanced then 11 is the odd light ball, else the lighter of 9 or 10 is the odd light ball.
 
 
 
2. If 5,6,7,8 > 1,2,3,4 then either 5,6,7,8 contains a heavy ball or 1,2,3,4 contains a light ball so weigh 1,2,5 vs 3,6,12 with 3 possible outcomes:
 
2a
If 1,2,5 vs 3,6,12 balances, then either 4 is the odd light ball or 7 or 8 is the odd heavy ball. Weigh 7 vs 8, if they balance then 4 is the odd light ball, or the heaviest of 7 vs 8 is the odd heavy ball.
 
2b
If 3,6,12 is heavy then either 6 is the odd heavy ball or 1 or 2 is the odd light ball. Weigh 1 vs 2, if balanced then 6 is the odd heavy ball, or the lightest of 1 vs 2 is the odd light ball.
 
2c
If 3,6,12 is light then either 3 is light or 5 is heavy. Weigh 3 against any other ball, if balanced then 5 is the odd heavy ball else 3 is the odd light ball.
 
 
 
3. If 1,2,3,4 > 5,6,7,8 then either 1,2,3,4 contains a heavy ball or 5,6,7,8 contains a light ball so weigh 5,6,1 vs 7,2,12 with 3 possible outcomes:
 
3a
If 5,6,1 vs 7,2,12 balances, then either 8 is the odd light ball or 3 or 4 is the odd heavy ball. Weigh 3 vs 4, if they balance then 8 is the odd light ball, or the heaviest of 3 vs 4 is the odd heavy ball.
 
3b
If 7,2,12 is heavy then either 2 is the odd heavy ball or 5 or 6 is the odd light ball. Weigh 5 vs 6, if balanced then 2 is the odd heavy ball, or the lightest of 5 vs 6 is the odd light ball.
 
3c
If 7,2,12 is light then either 7 is light or 1 is heavy. Weigh 7 against any other ball, if balanced then 1 is the odd heavy ball else 7 is the odd light ball.

Answer 2

solvable !

divide the 12 balls into 4 groups of 3 balls each

put 3 balls on one side and 3 balls on the other side

if the scale balances the "odd" ball is in one of the other two groups

so far only 1 weighing

put the other two groups of 3 balls on the scale, one group on one side and the other group on the other side

the scale will not balance; one side will go up and one side will go down

if the "odd" ball is lighter take the 3 balls that go up

if the "odd" ball is heavier take the 3 balls that go down

so far only 2 weighings

you have 3 balls and one is "odd"

take any 2 and put one on one side and one on the other side

if the scale balances the "odd" ball is the ball you did not weigh

if the scale does not balance the "odd" ball went down if it's heavier and went up if it's lighter, thus you have the "odd" heavier or lighter ball and this is the third weighing and the problem is solved

 

Answer 3

Separate the balls into two groups of 6. 

Put the two groups on the scale. Select the heavier group of six

Separate that group into two groups of three.

Put the two groups on the scale. Select the heavier group of three.

Select two of the three balls. Place on the scale. If they balance the heavier one is the one you did not select.

If they do not balance you know which one is heavier.