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The 8 rings logical question.? There are 8 rings but one is fake. But we do not know if the fake rock is heavier or lighter. We also have a scale with two hands. You can use the scale only twice to find the fake rock.

The 8 rings logical question.?

 
There are 8 rings but one is fake. But we do not know if the fake rock is heavier or lighter. We also have a scale with two hands. You can use the scale only twice to find the fake rock.

Grade:12

2 Answers

Vikas TU
14149 Points
9 years ago
You can't do the problem as stated. You are doing two measurements, each of which has three possible results (left side heavier, right side heavier, sides equal). So you can distinguish at most 3^2 = 9 cases. But you have 16 possible cases (#1 is heavier, #1 is lighter, #2 is heavier, #2 is lighter, etc.) Admittedly, we don't care about the distinction between some of these, but the first weighing will not eliminate possibilities in a way that takes advantage of this. 

If you do know whether the fake is heavier or lighter, then you can do it in the manner shibashis suggests. Weighing 3 against 3 will give you a group of 2 or 3 which you know contains the fake, then weigh one against one from this group to see which is the fake. 

If you don't know whether the fake is heavier or lighter, we should be able to do it with four rings, because then we have eight cases. In this case, weigh rings 1 and 2 against each other; then weigh rings 1 and 3 against each other. If the first pair match then you know either 3 or 4 is the fake, and the second weighing will tell you which (if they match, 4 is the fake, if not, 3 is the fake). If the first pair doesn't match, you know either 1 or 2 is the fake, and the second weighing will tell you which (if they match, 2 is the fake, if not, 1 is the fake). 

Alternatively, we can do it with eight rings if we are allowed three weighings, since now we can distinguish 3^3 = 27 cases which is more than enough. First weigh rings 1, 2, 3 against rings 4, 5, 6. 
If rings 1, 2, 3 are lighter than rings 4, 5, 6: Either one of 1, 2, 3 is light, or one of 4, 5, 6 is heavy. Rings 7 and 8 are normal. Weigh rings 1, 2, 4 against 3, 7, 8. If 1, 2, 4 are lighter, then ring 1 or 2 is light. Weigh one of these two against ring 7 - if it doesn't match, it's the fake, else the other one is. If rings 1, 2, 4 are heavier than rings 3, 7, 8, then either ring 3 is light or ring 4 is heavy. Weigh one of these against ring 7 to find out which is the fake. If rings 1, 2, 4 match rings 3, 7, 8, then ring 5 or 6 must be heavy. Weigh one of these against ring 7 to determine which. 

If rings 1, 2, 3 are heavier than rings 4, 5, 6: same as above but swap light and heavy around. 

If rings 1, 2, 3 match rings 4, 5, 6: ring 7 or 8 must be the fake. and rings 1-6 must all be good. Weigh ring 7 against one of the good rings to see if they match. If they do, ring 8 is the fake (and you can do one more weighing to see whether it is heavy or light, if you wish); if not, ring 7 is the fake.
Guru
10 Points
9 years ago
thnx!

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