Problem Analysis:
The problem involves a sequence of operations on a set of numbers, where each operation is defined by a specific rule. The goal is to determine the final state of the sequence after applying these operations iteratively.
#### Key Points from the Problem:
1.
Initial Setup:
- There are \( n \) numbers: \( a_1, a_2, \ldots, a_n \).
- These numbers are arranged in a circle.
- Each number \( a_i \) is associated with an index \( i \).
2.
Operation Definition:
- For each pair of adjacent numbers \( (a_i, a_{i+1}) \), compute the difference \( |a_i - a_{i+1}| \).
- Replace the original pair \( (a_i, a_{i+1}) \) with the new pair \( (|a_i - a_{i+1}|, |a_i - a_{i+1}|) \).
3.
Iterative Process:
- Repeat the operation until all numbers in the sequence become equal.
4.
Objective:
- Determine the final value of all numbers in the sequence after the process terminates.
---
Solution Approach:
To solve this problem, we need to analyze the behavior of the sequence under the given operation and identify any patterns or invariants that can help us determine the final state.
#### Step 1: Understanding the Operation
- The operation replaces each pair \( (a_i, a_{i+1}) \) with \( (d, d) \), where \( d = |a_i - a_{i+1}| \).
- This means that after one iteration, every pair of adjacent numbers becomes equal to the absolute difference of the original pair.
#### Step 2: Observing the Effect on the Sequence
- After the first iteration, the sequence will consist of repeated values of the form \( |a_i - a_{i+1}| \).
- In subsequent iterations, the same operation is applied to the new sequence. Since all pairs are now identical, the differences between adjacent numbers will remain the same.
#### Step 3: Identifying the Invariant
- Notice that the operation does not change the
greatest common divisor (GCD) of the sequence. This is because the GCD of two numbers \( a \) and \( b \) is the same as the GCD of \( |a - b| \) and \( b \). Mathematically:
\[
\gcd(a, b) = \gcd(|a - b|, b)
\]
- Therefore, the GCD of the sequence remains invariant throughout the process.
#### Step 4: Convergence to a Single Value
- Since the operation repeatedly applies the same transformation, the sequence will eventually converge to a single value. This value must be the GCD of the initial sequence \( a_1, a_2, \ldots, a_n \).
- This is because the GCD is the smallest positive integer that divides all numbers in the sequence, and the operation ensures that all numbers in the sequence will eventually become equal to this GCD.
#### Step 5: Final Answer
- The final value of all numbers in the sequence after the process terminates is the GCD of the initial sequence \( a_1, a_2, \ldots, a_n \).
---
Final Answer:
\[
\boxed{\gcd(a_1, a_2, \ldots, a_n)}
\]
Parent Tip: Review the logic above to help your child master the concept of free printable dot to dot puzzles hard.
