Problem 5 - Discrete Mathematics with Applications

Question

Among the \(N\) coins in a collection plate, one is counterfeit and heavier. Using an equal-arm balance, find the minimum number of weighings needed to ascertain the counterfeit, for each value of \(N.\) $$12$$

Step-by-Step Solution

Answer

In the worst case, it takes a minimum of \(2\) weighings to find the counterfeit coin among \(12\) coins using an equal-arm balance. To do this, first divide the coins into three groups and perform the first weighing. Based on the outcome, we identify the group with the counterfeit coin. Then, do a second weighing in that group to determine the counterfeit coin.

Step 1: First Weighing

Divide the 12 coins into three groups: two groups with 4 coins each (Group A and Group B) to be weighed, and a third group with 4 coins (Group C) not being weighed. Place Group A on the left scale and Group B on the right scale. Perform the first weighing.

Step 2: Analyzing the Outcome of the First Weighing

After the first weighing, there are three possible outcomes: 1. The left side (Group A) is heavier: This means the counterfeit coin is in Group A. 2. The right side (Group B) is heavier: This means the counterfeit coin is in Group B. 3. The scale is balanced: This means the counterfeit coin is in Group C, which was not weighed.

Step 3: Second Weighing

Based on the outcome in Step 2, we now know which group (A, B, or C) contains the counterfeit coin. In the next weighing, take three coins from the suspected group, and compare two of them. If one coin is heavier, we have found the counterfeit coin, and we stop. Otherwise, the third coin not on the balance is the counterfeit one. In the worst case, we require a maximum of two weighings to find the counterfeit coin among the 12 coins.

Key Concepts

Discrete Mathematics

Discrete Mathematics involves studying objects that can be counted, distinct in nature, and includes concepts such as sets, graphs, logic, and numbers. One of the intriguing challenges in discrete mathematics is the Counterfeit Coin Problem. This problem is an excellent example of how discrete mathematics can help us solve puzzles efficiently.

In this problem, you have a specific number of coins and need to determine which one of them is counterfeit and heavier than the others. The key is to use logic and mathematical reasoning rather than trial and error. This solution requires a methodical approach involving concepts like divisibility and equality. By dividing the coins into distinct groups and weighing them, you can mathematically deduce which group contains the counterfeit coin by leveraging the principles of discrete structures.

Discrete mathematics extends beyond simple arithmetic; it provides tools for problem-solving and algorithmic thinking that prove crucial in fields such as computer science and cryptography.

Problem Solving

Problem solving is a core component of mathematics and logic that involves discovering solutions to practical challenges. The Counterfeit Coin Problem is a perfect demonstration of effective problem solving. It requires understanding constraints and systematically breaking down the issue into smaller, manageable parts.

Here, the problem begins with a set of 12 coins, one of which is unique by being heavier. To determine the odd one out using a minimum number of weighings involves tactical planning:

• First, we strategize by dividing the coins into equal groups.
• Then, we employ the strategy of using an equal-arm balance to identify actionable insights.

By the first weighing, enough information can often be gathered to significantly narrow down the possibilities. The key is evaluating the scale's result to guide further actions.

This systematic approach is what lies at the heart of effective problem solving, ensuring that each step leads us closer to the solution in the most efficient manner.

Logical Reasoning

Logical reasoning involves deducing the best course of action through systematic analysis and is foundational in solving problems like the Counterfeit Coin Problem. This problem requires *if-then* logical techniques to distinguish the counterfeit coin among an arbitrary number of regular coins.

The core strategy lies in drawing meaningful conclusions from each weighing:

• Determine which group is heavier or if they are equal.
• Formulate logical deductions about the whereabouts of the counterfeit coin based on scale outcomes.

For example, if during the first weighing, one side of the balance is heavier, it instantly implies that the counterfeit coin exists within that group. Otherwise, if the scales balance, it must lie within the unweighed group.

Logical reasoning simplifies complex choices by predicting possible outcomes and preparing an appropriate response to each. This makes it an invaluable skill in mathematics, science, and decision-making processes overall.

Combinatorial Weighting

Combinatorial weighting is an approach used to carefully analyze scenarios involving limited set arrangements, making it vital to solving the Counterfeit Coin Problem efficiently. By cleverly splitting a set of elements, we can determine unknowns with fewer steps.

In the counterfeit coin scenario, the arrangements of coins are essential:

• Initially divide the coins into three equal groups.
• Use a balance to compare groups and deduce the location of the counterfeit.

This method entails understanding permutations of balancing options and assessing fewest steps to find a solution. Unlike checking each coin individually, combinatorial weighting allows solving the problem with as few as two weighings for 12 coins.

It showcases the power of combinatorial techniques to resolve problems efficiently through optimal arrangement and assessment within defined logical constraints, highlighting practical applications in both theoretical and real-world settings.