Suppose you are given 12 coins, one of which is counterfeit and weighs a little more or less than the real coins. Using a two-pan balance scale, what is the minimum number of weighings necessary to find the counterfeit coin?
A) 7 weighings
B) 3 weighings
C) 1 weighings
D) 4 weighings
E) 6 weighings

Question

Suppose you are given 12 coins, one of which is counterfeit and weighs a little more or less than the real coins. Using a two-pan balance scale, what is the minimum number of weighings necessary to find the counterfeit coin? ​
A) 7 weighings
B) 3 weighings
C) 1 weighings
D) 4 weighings
E) 6 weighings

Quizplus · Accepted Answer

The solution involves dividing the 12 coins into three groups of 4 coins each, and weighing two of the groups against each other. If the scales balance, the counterfeit coin must be in the third group. We can then take that third group and divide it into two groups of 2 coins each and weigh them against each other. If the scales balance, the counterfeit coin must be one of the other two coins, and we can weigh them against each other to find the counterfeit coin in a total of 3 weighings. If the scales do not balance the third group, we pick the heavier group, and then divide it into two groups of 2 coins each and weigh them against each other. If the scales balance, the counterfeit coin must be one of the other two coins in the remaining group, and we can weigh them against each other to find the counterfeit coin in a total of 3 weighings. If the scales do not balance, we pick the heavier group, and then weigh any two of the coins against each other. If the scales balance, the remaining coin is the counterfeit coin, and we find it in a total of 4 weighings. If the scales do not balance, we can determine which coin is the counterfeit by weighing any two of the three coins on one pan against any two of the three coins on the other pan. This will leave us with two possibilities for the counterfeit coin, and we can determine which one it is by weighing it against any real coin. This process takes a total of 3 weighings (one to isolate the third group, one to isolate the heavier subgroup, and one to isolate the counterfeit coin). Therefore, the minimum number of weighings necessary is 3.

Suppose You Are Given 12 Coins, One of Which Is