Problem 9 - Discrete Mathematics with Applications

Question

Four coins, \(a\) through \(d,\) in a plate look identical, but one is counterfeit and heavier. Using an equal-arm balance and minimum weighings, identify the counterfeit coin and determine if it is lighter or heavier. Display your analysis in a decision tree.

Step-by-Step Solution

Answer

In order to identify the counterfeit coin and determine if it's lighter or heavier using an equal-arm balance, divide the coins into two groups: {a, b} and {c, d}. The first weighing compares these two groups and has three outcomes: they balance, {a, b} is heavier, or {a, b} is lighter. If they balance, the counterfeit coin is found within 1 weighing. For the other two outcomes, perform a second weighing by comparing coin "a" with a genuine coin (c). This allows us to find the counterfeit coin and determine if it's lighter or heavier in just 2 weighings, as shown in the decision tree.

Step 1: Start the Decision Tree

In order to find the counterfeit coin with minimum weighings, let's start by dividing the coins into two groups: {a, b} and {c, d}.

Step 2: First Weighing

For the first weighing, compare {a, b} with {c, d} on the equal-arm balance. Based on the result of the first weighing, there are three possible outcomes: 1. The groups balance (a, b) = (c, d). 2. Group {a, b} is heavier (a, b) > (c, d). 3. Group {a, b} is lighter (a, b) < (c, d). We will analyze each of the outcomes.

Step 3: Outcome 1 - The groups balance

If the two groups balance, then we know that the counterfeit coin is one of the coins we didn't weigh which are now eliminated. Since in this case, all remaining coins are identical, no further weighing is needed, and we've found the counterfeit coin with just 1 weighing.

Step 4: Outcome 2 - Group {a, b} is heavier

If group {a, b} is heavier, we know that either a or b is the counterfeit coin and it's heavier. In order to identify the counterfeit coin, perform a second weighing: Weigh coin "a" against coin "c" (which we know is a genuine coin). There are two possible results: 1. (a) = (c) - They balance, so coin "b" is the counterfeit and it's heavier. 2. (a) > (c) - Coin "a" is heavier, so it is the counterfeit coin. In this case, we found the counterfeit coin in 2 weighings.

Step 5: Outcome 3 - Group {a, b} is lighter

If group {a, b} is lighter, we know that either a or b is the counterfeit coin and it's lighter. We will perform the second weighing similarly to outcome 2: Weigh coin "a" against coin "c" (which we know is a genuine coin). There are two possible results: 1. (a) = (c) - They balance, so coin "b" is the counterfeit and it's lighter. 2. (a) < (c) - Coin "a" is lighter, so it is the counterfeit coin. In this case, we also found the counterfeit coin in 2 weighings.

Step 6: Analyzing the Decision Tree and Final Result

We can observe that with each weighing, we were able to eliminate half of the remaining possibilities. This ensures that we found the counterfeit coin with the minimum number of weighings. The decision tree will have: - The first weighing decision with 3 branches for each outcome. - In outcomes 2 and 3, there is a second weighing decision with 2 branches for each of its outcomes. In conclusion, it's possible to identify the counterfeit coin and whether it is lighter or heavier in just 2 weighings, using an equal-arm balance and following the steps outlined in the decision tree.

Key Concepts

Decision Tree

A decision tree is an essential tool in problem-solving because it simplifies complex decisions by illustrating choices and their possible consequences.
It visually represents decisions and outcomes, helping to eliminate options systematically.
In the counterfeit coin problem, the decision tree tracks each step of the weighing process and the various possible outcomes.

• At each level of the tree, you make a decision based on the options available, represented as branches.
• Each branch leads to a new decision, guiding us to eliminate options until the solution is found.

The decision tree for the problem begins by considering the possible outcomes of the first weighing:

• If the two groups weigh equally, you immediately know the counterfeit coin is neither in group (a, b) nor in group (c, d).
• If one group is heavier or lighter, it points towards which set potentially holds the counterfeit, thus narrowing down your options.

Utilizing this approach ensures that fewer steps are needed to pinpoint the solution, making it efficient and effective.

Counterfeit Coin Problem

The counterfeit coin problem is a classic puzzle illustrating logical thinking and problem-solving strategies.
The goal is to identify which among the several possible coins is fake and whether it's heavier or lighter than the real ones.
In our specific scenario, we have four coins that look identical, one of which is counterfeit and heavier.

• The counterfeit coin problem challenges you to find the fake coin using the least number of weighings.
• Here, you aim to determine the counterfeit in as few steps as possible by strategically comparing coins.

This problem encourages critical thinking and the use of systematic methods to reduce possibilities at each step.
It serves as a fun way to improve logical reasoning and analytical skills.
By understanding the underpinning strategies, like using weighings to logically deduce incorrect possibilities, anyone can master the solution.

Weighing Techniques

Weighing techniques are the strategic methods used to determine differences among seemingly identical items.
For the counterfeit coin problem, you rely on an equal-arm balance to carry out these techniques.

• The first weighing divides the coins into two equal groups, allowing for an initial comparison.
• Following these outcomes gives clues as to which coins need further inspection.

When coins balance, no further action is needed for those already checked.
However, if they don't balance, you perform a second weighing to pinpoint the impostor.

• This process is efficient because it halves the number of potential counterfeit coins each time.
• Expected outcomes for each weighing guide the next steps, continuously narrowing down possibilities.

Ultimately, effective weighing techniques minimize steps and maximize accuracy in identifying the counterfeit coin.

Equal-Arm Balance

An equal-arm balance is a classic weighing device that provides a simple way to compare weights.
It uses two pans suspended on either side of a central pivot, offering equal measurement capability.

• The principle behind it is straightforward: when two items of equal mass are placed on either side, the beam stays level.
• If one item is heavier, the beam tips towards that side, indicating the difference in weight.

In the counterfeit coin problem, the equal-arm balance is crucial as it allows quick and accurate weighings.
Its straightforward design ensures reliability, essential in problem-solving scenarios involving detection of minute differences.

• The balance helps identify discrepancies in weight, pointing out the counterfeit item effectively.
• Students can appreciate the elegance and simplicity of this tool in logical deduction tasks.

By mastering the use of an equal-arm balance, students enhance their grasp of physics principles and problem-solving skills.