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# yet another balance puzzle

## Question

i bleive this challange may have been posted before, never the less here goes.

you have 3 wieghing scales, a normal scale that gives the correct result, a reverse scale that gives the oppisite result, and a random scale that gives, well you guessed it, a random result.

you have 4 normal coins and 4 heavy coins. your task however is to simply identify which scale is which with the fewest weighings, and you don't intially know which coins are which either.

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I place one coin on one side of each scale (or, if both sides must be engaged, I place 7 of the coins on one side and one coin on the other side). Obviously, the heavier side of each of the scales is the one with the coin on it. Any scale that answers correctly is not the reverse scale; any scale that answers incorrectly is not the normal scale. So, at this point, with one weighing on each scale we can positively identify one of them, either the normal or reversed, whichever one the random scale did not imitate. However, I can't see a method of bounding the number of additional weighings necessary to definitely isolate the random scale. If it randomly imitates the normal scale for 1000 trials in a row, it would take at least as many weighings to isolate, it seems...

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I don't see how this is possible. Theoretically the random scale has a small possibility of replicating either of the constant scales indefinitely just by random chance. If it randomly gives the correct answer the first ten times, any observation on it will be the same as if it were the truth telling scale; same with if it's a hundred or infinite times.

You can determine the identity of a single scale in 3 weights. Weigh any one scale with a single coin on one side and no coins on the other side. The correct one will always weigh it the correct way, the false one will always weigh it the wrong way, and the random one can be either. Whichever one is unique to the other two can be determined (e.g. 2 correct answers and 1 incorrect means the incorrect is definitely the bad scale).

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I think you are both on the right track.

either approach will reveal one of the "constant" scales [normal or reverse] as the scale that is different from the others. This will take 1 weighing of either combination described previously [1 coin vs no coins or 1 coins vs 7 coins]. [at this point you have weighed the items once on each scale]

You no longer have to do any more weighing on the scale that is different on the first weighing.

It will take at least 1 more weighing to determine which of the remaining scales is random.

you could simply reverse the loads [or re-weigh them on the same sides] and weigh them again and again.

when the results diverge, you will have determined which one is which.

The hard part is to try to estimate how long the random scale and the remaining "constant" scale track each other.

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will the reverse scare read correctly if they are equal?

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yes. the random scale may or may not. i tend to agree. could ptontially be infinite wieghings.

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To make it solvable, change the random scale into an alternating scale.

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To make it solvable, change the random scale into an alternating scale.

With an alternating scale it becomes simple; after your third weighing, use either other scale once. If the scale gives a different result than before, it's alternating.

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