Problem 13 - Discrete Mathematics with Applications

Question

Among seven identical coins lies a heavier counterfeit coin. Write an algorithm to identify the false coin using an equal-arm balance and minimum weighings.

Step-by-Step Solution

Answer

Divide the seven coins into three groups: Group A (3 coins), Group B (3 coins), and Group C (1 coin). Weigh Group A against Group B. If both sides are equal, the counterfeit coin is in Group C. If one side is heavier, the counterfeit coin is in the heavier group. Take the heavier group and divide it into three new groups: Group D (1 coin), Group E (1 coin), and Group F (1 coin). Weigh Group D against Group E. If both sides are equal, the counterfeit coin is in Group F. If one side is heavier, the heavier coin is the counterfeit coin.

Step 1: Divide the coins into groups

Divide the seven coins into three groups: Group A consists of 3 coins, Group B consists of 3 coins, and Group C consists of 1 coin.

Step 2: Weigh Group A and Group B

Put Group A on one side of the equal-arm balance and Group B on the other side. Observe the result of the first weighing.

Step 3: Evaluate the result of the first weighing

There are two possible outcomes: Outcome 1 - If both sides are equal, it means that the counterfeit coin is in Group C. Proceed to Step 4. Outcome 2 - If one side is heavier than the other, it means that the counterfeit coin is in the heavier group. Proceed to Step 5.

Step 4: The counterfeit coin is in Group C

Since Group C only has one coin, it is the heavier counterfeit coin. The algorithm is finished.

Step 5: The counterfeit coin is in the heavier group

Take the heavier group, and divide the 3 coins into three new groups: Group D with 1 coin, Group E with 1 coin, and Group F with 1 coin.

Step 6: Weigh Group D and Group E

Put Group D on one side of the equal-arm balance and Group E on the other side. Observe the result of the second weighing.

Step 7: Evaluate the result of the second weighing

There are two possible outcomes: Outcome 1 - If both sides are equal, it means that the counterfeit coin is in Group F. Group F's coin is the heavier counterfeit coin and the algorithm is finished. Outcome 2 - If one side is heavier than the other, it means that the counterfeit coin is in the heavier group. The heavier coin is the counterfeit coin and the algorithm is finished.

Key Concepts

Counterfeit Coin Problem

The Counterfeit Coin Problem is a classic puzzle that involves identifying a single counterfeit coin from a group of coins, using the fewest number of weighings. In this scenario, the counterfeit coin is heavier than the rest. The challenge lies in strategically using a limited number of weighings to determine which coin is the counterfeit. The problem tests deductive reasoning and logical thinking. It is often encountered in computer science and mathematics courses, as it helps illustrate principles of problem-solving and optimization. The goal is to identify the counterfeit coin using an algorithmic approach, efficiently narrowing down possibilities with each weighing.

Equal-arm Balance Weighing

An equal-arm balance is a critical tool in the Counterfeit Coin Problem. It consists of a beam balanced between two plates where weights can be placed to measure equivalency. Unlike modern electronic scales, equal-arm balances provide relative weight comparisons between two groups. This tool allows for a process of elimination by comparing sets of coins against each other.

• If the weights are equal, it informs that neither side contains the heavier coin.
• If one side is heavier, it identifies which group of coins contains the counterfeit.

The balance operates without giving precise weight values, aiding decisions based solely on heavier or lighter outcomes.

Problem Solving Steps

The problem-solving steps are methodical actions designed to pinpoint the counterfeit coin efficiently. The first step involves dividing coins into groups to facilitate manageable comparisons:

• Step 1: Divide into smaller groups. This simplifies the weighing process by creating manageable comparisons.
• Step 2: Compare groups using the balance. This involves weighing two groups of coins against each other to observe the weight difference.

These observations lead to subsequent steps where groups are further divided based on previous comparisons until the counterfeit coin is isolated. The algorithm ensures a minimal number of weighings by continuously narrowing down possibilities. Such strategic decision-making forms the crux of algorithm design, offering a logical path towards the solution.

Discrete Mathematics

Discrete Mathematics plays a foundational role in understanding problems like the Counterfeit Coin Problem. This branch of mathematics deals with distinct and separate values, facilitating logical analysis and computer science applications. It employs techniques like combinatorics, logic, and algorithm design to solve complex problems efficiently.

• Logic is used to form hypotheses and deduce conclusions based on observable outcomes.
• Combinatorics helps determine the most efficient way to arrange and group coins for weighings.

Understanding such mathematical frameworks aids in constructing effective algorithms, making discrete mathematics essential for solving optimization and decision-making problems presented in algorithmic puzzles like the counterfeit coin scenario.