## Assignment Question

The goal is to Determine the equation that the machine does. Hint: We want to know what the card will be when you input cards a and b into the machine. There is an equation that we can use. You should be able to get this from the 2 examples that are given. How many operations will the machine do before we are left with 1 card? This is a pretty doable question once we think about it appropriately. Answer the following 2 iterations of the question. Given 100 cards, how many operations will the machine do before we are left with 1 card? Given n cards, how many operations will the machine do before we are left with 1 card? 2. what will be the final card? We would like to answer this question for n=100 (n being the last card that we start with). Before we do this, are we sure that we will always get the same “final card”? Or will it depend on the order in which we input the cards into the machine? Let’s look into this first and come up with an answer to whether the final card will be unique or different along with justification.

# Answer

**Introduction**

In the realm of mathematical puzzles and conundrums, there exists a captivating enigma that has piqued the curiosity of many: the Card Machine. This intriguing machine, shrouded in mathematical mystique, holds the power to reduce a deck of cards into a solitary card through a series of seemingly simple operations. Our quest in this paper is to decipher the underlying equation governing the Card Machine’s actions and to uncover the final card that remains when faced with different initial conditions.

To determine the equation that the machine follows and the number of operations it performs before being left with 1 card, we can first analyze the given examples and then generalize for any value of n.

**Example 1:** Given 100 cards, we want to find out how many operations the machine will do before we are left with 1 card. Let’s break this down step by step:

- Start with 100 cards: [1, 2, 3, …, 100]
- Discard every second card, leaving only the odd-numbered cards: [1, 3, 5, …, 99]
- Repeat the process: Discard every second card again: [1, 5, 9, …, 97]
- Continue this process until only one card remains.

**Example 2:** Now, let’s consider a more general case with n cards:

- Start with n cards: [1, 2, 3, …, n]
- Discard every second card, leaving only the odd-numbered cards: [1, 3, 5, …, n-1] (n becomes n/2 if n is even, or (n-1)/2 if n is odd)
- Repeat the process: Discard every second card again: [1, 5, 9, …, n-3] (n/2 becomes n/4 if n/2 is even, or (n-1)/4 if n/2 is odd)
- Continue this process until only one card remains.

Now, let’s address your questions:

**Question 1:** How many operations will the machine do before we are left with 1 card, given 100 cards?

To find the number of operations, we need to determine how many times we can repeat steps 2 and 3 until we are left with only one card.

For 100 cards:

- After the first operation, we have 50 cards.
- After the second operation, we have 25 cards.
- After the third operation, we have 13 cards (rounding up from 12.5).
- After the fourth operation, we have 7 cards.
- After the fifth operation, we have 4 cards.
- After the sixth operation, we have 2 cards.
- After the seventh operation, we have 1 card.

So, it takes 7 operations to be left with 1 card when starting with 100 cards.

**Question 2:** What will be the final card when n = 100?

To find the final card, we need to follow the process described earlier:

- Start with 100 cards: [1, 2, 3, …, 100]
- Discard every second card: [1, 3, 5, …, 99]
- Repeat the process until only one card remains.

As we found earlier, it takes 7 operations to be left with 1 card. So, after 7 operations, the final card will be 1.

**Question 3:** Will the final card be unique, or will it depend on the order in which we input the cards into the machine?

The final card will always be the same, regardless of the order in which you input the cards into the machine. This is because the process of discarding every second card is a deterministic operation. Whether you start with the cards in ascending order or descending order, the algorithm will always discard the same cards in the same order, leading to the same final card.

In summary, for any positive integer n, the machine will take log₂(n) operations to be left with 1 card, and the final card will always be 1, regardless of the initial order of the cards.

**Frequently Asked Questions (FAQs)**

**Q1: What is the Card Machine, and why is it of interest to mathematicians and puzzle enthusiasts?**

*A1:* The Card Machine is a mathematical puzzle that involves reducing a deck of numbered cards to a single card through a series of operations. It is of interest to mathematicians and puzzle enthusiasts because it offers a simple yet intriguing problem with a surprising mathematical solution. Understanding the behavior of the Card Machine can shed light on mathematical concepts such as logarithms and deterministic processes.

**Q2: How does the Card Machine work in practice?**

*A2:* The Card Machine operates by starting with a deck of numbered cards, typically ranging from 1 to n. In each operation, it discards every second card, leaving behind the remaining cards. This process is repeated until only one card remains.

**Q3: What is the equation governing the number of operations required by the Card Machine?**

*A3:* The equation that governs the number of operations is log₂(n), where n is the number of cards initially present. This equation represents the logarithmic relationship between the number of cards and the number of operations needed to reduce them to one.

**Q4: Does the initial order of the cards matter in determining the final card?**

*A4:* No, the initial order of the cards does not matter. The Card Machine’s operation is deterministic, meaning it follows a specific pattern of discarding every second card, which remains the same regardless of the initial order. Thus, the final card is constant for a given n, regardless of the card order.

**Q5: Can the Card Machine be extended to scenarios with a different number of cards, such as non-integer values or negative numbers?**

*A5:* The Card Machine’s basic operation is defined for positive integers. Extending it to non-integer values or negative numbers would require redefining the problem, which may lead to different patterns and outcomes. For simplicity, the standard Card Machine puzzle focuses on positive integer values of n.

**Q6: Are there real-world applications or implications for the Card Machine puzzle?**

*A6:* The Card Machine is primarily a recreational mathematical puzzle and may not have direct practical applications. However, its underlying mathematical concepts, such as logarithms and deterministic processes, have broader relevance in various fields of science and engineering.

**Q7: Are there variations of the Card Machine puzzle, or is it always the same process?**

*A7:* The basic Card Machine puzzle involves the same process of discarding every second card until one card remains. While variations may exist, the core concept remains consistent. Some variations may involve different starting conditions or rules, but they still typically lead to interesting mathematical explorations.