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)
- Four individuals - P, Q, R and S - are suspects in a theft case. They make the following statements :
P says: "Q is guilty."
Q says: "R is guilty."
R says: "P is innocent."
S says: "I am innocent"
If it is known that exactly one of them is guilty, and only the guilty person lies (all innocent people tell the truth), then who is the guilty one? - 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: - Read the following informations and answer the question given below :
A total of nine cards consisting of four kings, four queens and one joker are distributed among three persons P, Q and R such as :
I. Q has two cards, P has three cards and R has four cards.
II. Person, who has maximum number of cards, does not have the joker.
III. Each one has atleast one king while Q has two kings.
Who has three queens ? Consider the following information regarding a 3-digit PIN.
The correct PIN is- A boatman on a bank of a river has to carry a dog, a cat and a container full of milk to the other bank in a boat, one at a time. The cat cannot be left alone with the milk or with the dog on either bank. What is the minimum number of times the boatman has to cross the river to carry all to the other bank?
