There are balance scales without weights and 3 identical-looking coins, one of which is counterfeit: it is lighter than the genuine ones (the 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.
When searching for a counterfeit coin among three coins, try placing one coin on each pan of the scales. Among four coins, place two on each pan, and among nine coins, place three on each pan.

## Solution

If we have 3 coins, one weighing is enough. Place one coin on each pan of the scales. If one of the pans is lighter, the counterfeit coin is on it. If the scales are in balance, the counterfeit coin is the one that was not placed on the scales. If we have 4 coins, two weighings will be required: during the first weighing, place 2 coins on each pan. During the second weighing, take the 2 coins that were lighter and place them one on each pan. The coin that is lighter is the counterfeit. If we have 9 coins, two weighings will be required again. Divide the coins into three groups of 3 coins and place two of these groups on the two pans of the scales. If the scales are in balance, consider the 3 coins that were not placed on the scales. If the scales are not in balance, consider the 3 coins that are lighter. Now the problem has been reduced to the very first one: "there are 3 coins, one of which is counterfeit." As we already know, in this case, only one weighing is required to determine the counterfeit coin.

## Answer

$1 ; 2 ; 2$.