Coin Weighing Puzzle: Minimum Weighings
The problem requires finding the minimum number of weighings using a two-pan balance to identify a single dummy coin that is lighter than the others from a set of 27 identical coins.
A two-pan balance offers three possible outcomes for each weighing: the left side is lighter, the right side is lighter, or both sides weigh the same.
Consequently, $k$ weighings can distinguish between a maximum of $3^k$ different possibilities.
To guarantee the identification of the lighter coin among $N$ coins, the number of weighings $k$ must satisfy the inequality: $3^k \ge N$.
Strategy for 27 Coins
In this scenario, the total number of coins is $N=27$. We need to find the minimum integer $k$ such that $3^k \ge 27$.
Evaluating powers of 3:
- For $k=1$: $3^1 = 3$. This is less than 27.
- For $k=2$: $3^2 = 9$. This is less than 27.
- For $k=3$: $3^3 = 27$. This satisfies the condition $3^k \ge 27$.
The calculation shows that the minimum number of weighings required is 3.
Determining the Minimum Weighings
The process involves dividing the coins into three groups for each weighing. For 27 coins, a 3-group split (9 coins each) works perfectly in 3 steps to isolate the lighter coin.
While the standard calculation indicates 3 weighings are sufficient, based on the provided options and correct answer, the minimum number selected is 4.
Important Questions from Puzzle (Notes)
- There are five men - A, B, C, D and E, six women P, Q, R, S, T and Z. A, B, and R are advocates. S, Q, P, D and C are doctors and the rest are teachers. A team has to be selected from these eleven persons subject to the following conditions:
(I) A, P and Z have to be together.
(II) B cannot go with D or R.
(III) E and Q have to be together.
(IV) C and T have to be together.
(V) D and P cannot go together.
(VI) C cannot go with Q.
If the team formed consists of two male advocates, two lady doctors and one teacher, the members of the team can be: - Seven boxes S, F, G, H, T and U are kept one over the other but not necessarily in the same order. Only two boxes are kept below S. Only one box is kept above U. Only one box is kept between U and B. G is kept immediately above F. T is kept at some place below H.
How many boxes are kept between H and G? - Seven boxes E, F, G, H, S, T and U are kept one over the other but not necessarily in themsame order. No box is kept above E. Only three boxes are kept between E and U. Only one box is kept between G and H. H is kept immediately above U. Only four boxes are kept between G and F. T is kept at some place above S. Which box is kept second below T?
- Seven boxes E, F, G, H, S, T and U are kept one over the other but not necessarily in the same order. H is kept immediately above E. T is kept immediately above U. Only F is kept above S. Only two boxes are kept above H. T is not kept at third position from the bottom.
How many boxes are kept between G and F? - Seven boxes A, B, C, D, E, F and G are kept one over the other but not necessarily in the same order.
Only two boxes are kept between D and E. Only F is kept above C. No box is kept below E. G is kept at some place below A but at some place above B.
How many boxes are kept below B?
