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Imagine 101 coins in front of you. All of them look the same, but it is known that among them there is a defect one, a coin that doesn't have the same mass as his friends. What is the smallest number of measurements on scales without weights that should be carried out in order to determine whether the coin has a higher or lower weight than others?

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  • $\begingroup$ Do you have to find the coin or just whether it is light/heavy? $\endgroup$
    – Henry
    Oct 26, 2015 at 19:49
  • $\begingroup$ You should tell us what you mean by "scale." There are two types of scales possible in such problems. The most ccommon - where you weight two piles of coins against each other, and the less common (but more common in the real world,) where you put coins on the scale and get back a weight number. $\endgroup$ Oct 26, 2015 at 19:55
  • $\begingroup$ The first one :) $\endgroup$
    – McLinux
    Oct 26, 2015 at 20:07

1 Answer 1

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$2$? Weigh $33$ coins against another $33$.

Case I. they match. Then the odd one is one of the missing $35$ (and we have $66$ normals). So just weigh $35$ normals against the rest.

Case II. they don't match. Then the other $35$ are all normal. So weigh the lighter $33$ against $33$ normals. If they match then the odd coin is heavy. If they don't match the odd coin is light.

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    $\begingroup$ +1. Note also that 1 weighing can never be sufficient, since whatever the result we will not know whether the bad coin is heavy (on the heavy side) or light (on the light side). If they balance we still don't know if the bad one is heavy or light. $\endgroup$
    – vadim123
    Oct 26, 2015 at 20:13
  • $\begingroup$ @vadim123 not to mention that in a single weighing we either have unequal numbers on the two sides (no point at all) or we omit at least one. $\endgroup$
    – lulu
    Oct 26, 2015 at 20:14

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