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I have an small ball of gold. When I was carrying it for home I accidentally put it in a bag with 15 other balls of the same size and the same color. To discover the true golden ball I have only one pan balance. What is the smallest number of weighings necessary to guarantee that I can separate the true golden ball from the others?

Note:
1. The golden ball is heavier than the others.
2. All the others balls have the same weight

2 3 4 5

2 solutions

Sebastian Hazzan
Jul 22, 2014

(to Kenny Lau) it says "smallest to guarantee" so it is asked correctly

Kenny Lau
Jul 9, 2014

The correct answer should be 3, and the question should be asking for the largest number of weighings, or else it would be 1.

I name those balls a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p.

Firstly, put a,b,c,d,e on the left hand side and f,g,h,i,j on the right hand side.

Case (1) is left hand side is heavier, case (2) is right hand side is heavier, case (3) is neither.

Case (1)+(2):

Name the five heavier balls q,r,s,t,u.

Put q,r on the left hand side and s,t on the right hand side.

If either side is heavier, the ball of gold is on either side, and it takes one more step to determine, leaving a total of $\fbox3$ steps.

If neither side is heavier, then the ball of gold is u, leaving a total of $\fbox2$ steps.

Case (3): The golden ball is one of k,l,m,n,o,p.

Put k,l on the left hand side and m,n on the right hand side.

If either side is heavier, the ball of gold is on either side, and it takes one more step to determine, leaving a total of $\fbox3$ steps.

If neither side is heavier, then the ball of gold is one of o,p, and it takes one more step to determine, leaving a total of $\fbox3$ steps.

Therefore the correct answer should be 3.

Thanks. I have updated the answer to 3.

Calvin Lin Staff - 6 years, 11 months ago

Eureka

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