[USACO13DEC] The Bessie Shuffle S

**The Bessie Shuffle**

**Problem Description**

Bessie the cow has a large herd of cows, and she wants to shuffle them in a specific way. She has $n$ cows, numbered from1 to $n$, and each cow is initially standing at its own position on a line. The shuffling process consists of two steps:

1. **Step1**: Bessie takes the first cow (numbered1) and moves it to the end of the line.
2. **Step2**: For each remaining cow, she swaps it with the cow that is one position ahead of it.

Bessie wants to know how many cows will be standing at their original positions after this shuffling process.

**Problem Constraints**

* $1 leq n leq10^5$
* The input contains only positive integers**Solution Approach**

To solve this problem, we can use a simple mathematical approach. Let's analyze the situation:

After Step1, the first cow (numbered1) is moved to the end of the line. This means that all cows initially standing at positions $2$ to $n$ will be shifted one position back.

Now, let's consider what happens in Step2. For each remaining cow, Bessie swaps it with the cow that is one position ahead of it. This effectively reverses the order of the cows from positions $1$ to $n-1$. However, the last cow (numbered $n$) remains at its original position.

We can see that the shuffling process has a periodic nature. Specifically, if we start with an initial arrangement of cows and apply this shuffling process repeatedly, we will eventually return to the same arrangement after a certain number of iterations.

Let's denote the number of cows standing at their original positions as $x$. We want to find the value of $x$.

**Mathematical Analysis**

We can analyze the situation by considering the parity (even- or oddness) of the number of cows. If there are an even number of cows, then after each iteration of the shuffling process, exactly half of the cows will be standing at their original positions. This means that $x$ will be equal to half of the total number of cows.

On the other hand, if there are an odd number of cows, then after each iteration of the shuffling process, one more cow will be added to the group of cows standing at their original positions. This means that $x$ will be equal to half of the total number of cows plus one.

In both cases, we can see that $x$ is related to the total number of cows and its parity.

**Code Implementation**

Here's a simple code implementation in Python:

def bessie_shuffle(n):
 """
 Calculate the number of cows standing at their original positions after Bessie's shuffling process.
 Parameters:
 n (int): The total number of cows.
 Returns:
 int: The number of cows standing at their original positions.
 """
 # If there are an even number of cows, return half of the total number if n %2 ==0:
 return n //2 # If there are an odd number of cows, return half of the total number plus one else:
 return (n +1) //2# Test the function with some examplesprint(bessie_shuffle(10)) # Output:5print(bessie_shuffle(11)) # Output:6

This code defines a function `bessie_shuffle(n)` that takes an integer `n` as input and returns the number of cows standing at their original positions after Bessie's shuffling process. The function uses simple mathematical logic to determine this value based on whether the total number of cows is even or odd.

The test cases demonstrate how the function works with different inputs. For example, when there are10 cows (an even number), the function returns half of that number, which is5. When there are11 cows (an odd number), the function returns half of that number plus one, which is6.

This code provides a concise and efficient solution to the problem, making it suitable for large inputs.