Can anyone help me solve this puzzle?

Question

Suppose that you have been given 12 identical rocks,

all of which weigh the same amount except for one of the rocks. You

dont know which one has different weight. Show that by using a balance

scale∗ at most three times, you can determine exactly which rock has

different weight, and if it is heavier or lighter.

Answer 1

suppose one is lighter than the other 11

1. six each side, keep the lightest 6

2. 3 each side, keep the lightest 3.

3. 1 each side.  If one is lighter, that's the odd one out.  If they weigh the same, it's the third one.

if one was heavier

1.  six each side, keep the heaviest 6

2.  3 each side, keep the heaviest 3.

3. 1 each side.  If one is heavier, that's the odd one out.  If they weigh the same, it's the third one.

You don't need to know at the beginning whether it's lighter or heavier, but on the first weighing you owuld know whether to go with the 1st set or 2nd.

Answer 2

Number the stones 1 through 12.

Start with a sum of 0.

Weigh 1, 2, 4, 8 vs 5, 7, 10, 11

If the scale tips towards the 1, add 1.

If the scale balances, add 0.

If the scale tips towards the 5, add - 1.

Weigh 3, 4, 7, 11 vs 2, 5, 6, 12

If the scale tips towards the 3, add 3.

If the scale balances, add 0.

If the scale tips towards the 2, add - 3.

Weigh 5, 6, 9, 11 vs 7, 8, 10, 12

If the scale tips towards the 5, add 9.

If the scale balances, add 0.

If the scale tips towards the 7, add - 9.

The absolute value of the sum will give the stone, and the direction scale tips when that stone is involved tells whether it is lighter or heavier.

derivation:

+ 9 + 3 + 0 = 12, + 9 + 3 - 1 = 11, + 9 + 0 + 1 = 10

- 9 - 3 + 0 = - 12, - 9 - 3 + 1 = - 11, - 9 + 0 - 1 = -10

+ 9 + 0 + 0 = 9, + 9 + 0 - 1 = 8, + 9 - 3 + 1 = 7

- 9 + 0 + 0 = - 9, - 9 + 0 + 1 = - 8, - 9 + 3 - 1 = - 7

+ 9 - 3 + 0 = 6, + 9 - 3 - 1 = 5, 0 + 3 + 1 = 4

- 9 + 3 + 0 = - 6, - 9 + 3 + 1 = - 5, 0 - 3 - 1 = - 4

0 + 3 + 0 = 3, 0 + 3 - 1 = 2, 0 + 0 + 1 = 1

0 - 3 + 0 = - 3, 0 - 3 + 1 = - 2, 0 + 0 - 1 = - 1

1, 2, 4, 8 vs 5, 7, 10, 11

3, 4, 7, 11 vs 2, 5, 6, 12

5, 6, 9, 11 vs 7, 8, 10, 12