Puzzle Question:
There are 12 stones with identical shape and size.
11 stones have the same weight, 1 stone has different weight and we don’t know which one.
Let’s call this special stone as “the anomaly”.
This scale will not show us how many kg or grams each stone, it can only compare the weight of its left and right hands.
How do we find the different stone using the scale as shown on the left and we can only do maximum 3 times of weighing?
We also need to know whether the anomaly stone is lighter or heavier.
Solution : (select the text to reveal solution)
Comparison 1:
Divide the stones into 3 groups, 4 stones each. Let’s call them group A, group B and group C. The stones inside each group are A1, A2, A3, A4; B1, B2, B3, B4 and C1, C2, C3, C4 respectively.
Now put the stones in group A into the left side of the scale and the stones in group B into the right side. Compare their weight.
There are 2 possibilities here, whether the scale is equal or one side is heavier.
If the scale is equal, then the anomaly stone is one of the stone in group C.
Comparison 2A if the result of comparison 1 was equal:
Put C1, C2 and C3 on the left side of the scale. Put A1, A2 and A3 on the right side (this can be any stones from group A or group B that we already know have normal weight).
If the scale is equal, then C4 is the anomaly.
Remember that we put normal stones on right side of the scale.
If the left side is heavier, then the anomaly stone is heavier then the other stones.
If the right side is heavier, then the anomaly stone is heavier then the other stones.
Comparison 3A if the result of comparison 1 was equal and the result of comparison 2A was also equal:
Now put C4 on the left side of the scale and any other stone on the right side.
If the left side is heavier, then C4 is the anomaly stone and it is heavier then the other stones.
If the left side is lighter, then C4 is the anomaly stone and it is lighter then the other stones.
Comparison 3B if the result of comparison 1 was equal and the result of comparison 2A was not equal:
From comparison 2A, we already knew the type of the anomaly stone (heavier or lighter) but still don’t know which one.
Put C1 on the left side and C2 on the right side.
If the weight of C1 and C2 are equal, then C3 is the anomaly stone (we already knew heavier or lighter from comparison 2).
If C1 is heavier than C2 and the comparison 2 showed that the anomaly is heavier, then the anomaly stone is C1.
If C1 is heavier than C2 and the comparison 2 showed that the anomaly is lighter, then the anomaly stone is C2.
If C1 is lighter than C2 and the comparison 2 showed that the anomaly is heavier, then the anomaly stone is C2.
If C1 is lighter than C2 and the comparison 2 showed that the anomaly is lighter, then the anomaly stone is C1.
Comparison 2B if the result of comparison 1 was not equal:
If group A and group B have different weights, then the anomaly stone can be in either group A or group B.
Let’s rename the stones. Stones in the heavier group are called H1, H2, H3, H4. Stones in lighter group are called L1, L2, L3, L4.
Put H1, H2, L1 on the left side of the scale. Put H3, L2 and any stone from group C (normal weight) on the right side.
If the left side is heavier, then the possibilities are: H1 or H2 could be the anomaly stone (heavier) or L2 could be the anomaly stone (lighter).
If the right side is heavier, then the possibilities are: H3 could be the anomaly stone (heavier) or L1 could be the anomaly stone (lighter).
If the scale shows equal weight, then the possibilities are: H4 could be the anomaly stone (heavier) or L3 or L4 could be the anomaly stone (lighter).
Comparison 3C if the result of comparison 1 was not equal and the result of comparison 2B was heavier left side:
Put H1 and L2 on the left side of the scale. Put C1 and C2 on the right side (they’re stones with normal weight).
If left side is heavier, then H1 is the anomaly stone (heavier).
If right side is heavier, then L2 is the anomaly stone (lighter).
If the scale shows equal weight, then H2 is the anomaly stone (heavier).
Comparison 3D if the result of comparison 1 was not equal and the result of comparison 2B was heavier right side:
Put H3 and L1 on the left side of the scale. Put C1 and C2 on the right side (they’re stones with normal weight).
Logically, it is not possible that the scale will show equal weight for this comparison.
If left side is heavier, then H3 is the anomaly stone (heavier).
If right side is heavier, then L1 is the anomaly stone (lighter).
Comparison 3E if the result of comparison 1 was not equal and the result of comparison 2B was equal:
Put H4 and L4 on the left side of the scale. Put C1 and C2 on the right side (they’re stones with normal weight).
If left side is heavier, then H4 is the anomaly stone (heavier).
If right side is heavier, then L4 is the anomaly stone (lighter).
If the scale shows equal weight, then H3 is the anomaly stone (heavier).
There are some different forms of this puzzle, like saying they are balls instead of stones. Still basically the same puzzle.
There are also multiple ways of solving this puzzle. The steps above are just one of the possible solutions.
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