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Among 100 coins, there are four counterfeit ones. All genuine coins weigh the same, and the counterfeit ones also weigh the same, but a counterfeit coin is lighter than a genuine one.
How can you find at least one genuine coin with two weighings on a balance scale without weights?
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Let's divide 100 coins into two groups (No.1 and No.2) of 33 coins each and one group (No.3) of 34 coins. For the first weighing, we will place groups 1 and 2 on the scales. If one of the pans is heavier than the other, then it contains no more than one counterfeit coin. Then, for the second weighing, we can compare any two coins from this group: if one is heavier, it is genuine, and if both are the same, both are genuine. If, however, the first weighing shows that the two groups of 33 coins weigh the same, this means that the counterfeit coins are distributed among the three groups in one of the following ways: (0,0,4), (1,1,2), or (2,2,0). For the second weighing, we do the following: add one coin from the other pan to one of the groups, and remove all the other coins from the other pan, replacing them with the 34 coins from group No.3. This weighing has three possible outcomes.
1) 1+3334. This means that either the case (0,0,4) or the case (1,1,2) applies, in which the moved coin is not counterfeit. In both cases, the moved coin is genuine.
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