[ Weighing ]

There are balance scales without weights and 3 identical-looking coins. One of the coins is counterfeit, and it is unknown whether it is lighter or heavier than the genuine coins (genuine coins weigh the same). How many weighings are needed to determine the counterfeit coin? Solve the same problem in the cases where there are 4 coins and 9 coins.

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Note: It is required to identify the counterfeit coin, without specifying whether it is lighter or heavier than the genuine ones.

## Solution

| If we have 3 coins, two weighings are sufficient. Place one coin on each pan of the balance. If the balance is not in equilibrium, the coin that remains is genuine. Place it on the balance with any of the others, and immediately determine which one is counterfeit. If the balance is in equilibrium, the counterfeit coin is the one that remains, and the second weighing can even determine whether it is lighter or heavier than the genuine ones. If we have 4 coins, two weighings are sufficient again. Divide the coins into two piles of 2 coins each and place one of the piles on the balance - one coin on each pan. If the balance is in equilibrium, both coins on it are genuine. If the balance is not in equilibrium, both coins on the table are genuine. Thus, we now know which pile contains the counterfeit coin. Place one coin from the pile where both are genuine on one pan of the balance, and one coin from the pile where the counterfeit is on the other pan. If the balance is in equilibrium, the counterfeit coin is the one that remains on the table; if not in equilibrium, we have placed it on the balance (in this case, we will even know whether it is lighter or heavier). If we have 9 coins, three weighings are required. Divide the coins into three piles of 3 coins each and place two of these piles on the two pans of the balance. If the balance is in equilibrium, the counterfeit coin is in the remaining pile, and in two weighings (as shown in the first part of the problem), we can determine the counterfeit coin. Thus, we need a total of three weighings. Now, let the balance not be in equilibrium, meaning one of the piles on the balance contains the counterfeit coin, while the pile that remained contains only genuine ones. Place this pile on the balance and any of the first two. This way, we will not only find the pile with the counterfeit coin but also immediately determine whether it is lighter or heavier than the genuine ones. We have performed two weighings, but now we can determine the counterfeit coin with just one more weighing (as shown in the first part of the problem 49). Thus, we need a total of three weighings.

## Answer

$2 ; 2 ; 3 ; 4$.

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