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2018 NSW: Maths Puzzle 6

6th puzzle by Dr Harry Wiggins

You are given nine coins of the same denomination and you know that one of them is counterfeit and that it is lighter than the others. You have a pan balance which means you can put any number of coins on each side and the balance will tell you which side is heavier, but not how much heavier. Explain how you can find the COUNTERFEIT COIN in exactly two weighings.

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8/3/2018 11:24 AM
The first round you put 3 coins on the left, and 3 coins on the right. From this you will know which group of 3 coins contains the counterfeit coin. (Left, right or the excluded coins if balanced). From the counterfeit group you place 1 coin on the left and 1 on the right. From this you will know which coin is the counterfeit coin (left, right or the excluded coin if balanced).
8/3/2018 4:40 PM
Step 1
Separate the coins into three groups, having three coins each. Compare two of these groups. If one is lighter it has the counterfeit coin. If they have the same weight, the counterfeit coin is in the group that was not weighed.
Step 2
Take the group with the counterfeit coin, and weigh two of the coins in that group. If one is lighter, it is the counterfeit coin. If they have the same weight, then the coin that was not weighed is the counterfeit coin.
Sachin Reddy Northwood School
8/3/2018 5:01 PM
Dividing n coins into 4 or more groups, would need 2 steps or more to find the group with the counterfeit coin, while dividing the coins into three groups, needs only one step to find the group with the counterfeit coin.
Since the most efficient procedure for any amount of coins is to divide it into three groups of equivalent size, then divide the the one with the counterfeit coin, into three groups of equivalent size, and so on. If you have 3^n coins, then the least number of weighings needed to find the counterfeit coin is n. For an integer k, such that 3^(a)<k<3^(a+1), the least number of weighings, is a+1. Thus for any n coins, the least number of weighings needed to determine the counterfeit coin is equal to the exponent, of the smallest power of three greater than or equal to n, or ceiling(log3(n)).
Sachin Reddy Northwood School.
8/4/2018 5:32 AM
Well done Keegan and Sachin. Also do Sachin for changing 9 to k and solving the general case. Ie, if you have 2018 coins, how would you identify the lighter or counterfeit coin.
8/5/2018 4:32 PM
separate into 3 groups: group 1 has 4 coins , group 2 has 4 coins and group 3 has 1 coin. way group 1 and 2 if they are the same then group 3 has the counterfeit , but if not devide the ligter group (group 1 or 2) into 2 groups and way them the lither group will have the counterfeit
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