Why do two approaches to counting paths give different results?

The first approach is incorrect because it assumes that the task can be divided into independent subtasks (from 1 to 9 and from 11 to 30), and the results can be multiplied. However, this is wrong because there is actually no path from 9 to 11 that bypasses 10 - the graph does not allow such transitions, so the approach simply counts an empty set of paths.

Contents

• Main Problem with the First Approach
• Analysis of the First Code
• Analysis of the Second Code
• Mathematical Justification of Differences
• Correct Solution to the Problem
• Path Examples for Clarity
• Conclusion

Main Problem with the First Approach

The key error of the first approach lies in the incorrect assumption about the divisibility of the task. The developer assumed that one could simply count paths from 1 to 9, then from 11 to 30, and multiply the results. However, this is fundamentally incorrect because:

• No continuous paths: In reality, paths must be continuous sequences from the starting to the ending number
• Missing transition 9 → 11: To transition from 9 to 11, one must pass through 10 (9 + 1 = 10, then 10 + 1 = 11)
• Violation of graph structure: The operations “add 1” and “multiply by 2” create a specific graph structure where certain transitions are impossible

Analysis of the First Code

Let’s examine in detail what the first approach does:

def f(st, end):
    a={st: 1}
    for i in range(st + 1, end + 1):
        a[i]=0
        if i - 1 in a:
            a[i]+=a[i - 1]
        if i % 2 == 0 and int(i//2) in a:
            a[i]+=a[int(i//2)]
    return a[end]

print(f(1,9)*f(11,30))

What happens:

1. f(1,9) counts the number of paths from 1 to 9
2. f(11,30) counts the number of paths from 11 to 30
3. The results are multiplied

Problem: This approach does not account for the fact that there are no paths from 9 to 11 that bypass 10. As a result, the approach simply multiplies two numbers, but does not consider that these paths cannot actually be combined into a single path from 1 to 30.

Important: In the graph of “add 1” and “multiply by 2” operations, there is no edge from 9 to 11. The only way to get from 9 to 11 is to pass through 10, which is prohibited by the condition.

Analysis of the Second Code

The second approach implements the correct strategy:

def f(st, end, propusk):
    a={st: 1}
    for i in range(st + 1, end + 1):
        a[i]=0
        a[propusk]=0
        if i - 1 in a:
            a[i]+=a[i - 1]
        if i % 2 == 0 and int(i//2) in a:
            a[i]+=a[int(i//2)]
    return a[end]

print(f(1,30, 10))

Key differences:

1. Unified computation: Counts all paths from 1 to 30 in a single pass
2. Explicit exclusion: Explicitly sets a[10] = 0 at each iteration
3. Correct constraint handling: Dynamic programming correctly accounts for the prohibition of passing through 10

How it works:

• The algorithm builds a table a, where a[i] is the number of paths from 1 to i
• At each step, it explicitly prohibits the use of number 10
• It correctly considers only those paths that never pass through 10

Mathematical Justification of Differences

Let:

• P(S, E) be the set of paths from S to E
• P'(S, E) be the set of paths from S to E that bypass 10

The first approach calculates: |P(1, 9)| × |P(11, 30)|

The second approach calculates: |P'(1, 30)|

Mathematical error:

$$|P'(1, 30)| \neq |P(1, 9)| \times |P(11, 30)|$$

Reason: The set P'(1, 30) requires that the path be a continuous sequence from 1 to 30. Splitting into P(1, 9) and P(11, 30) violates this requirement, as there is no continuation from 9 to 11.

Correct relationship:

$$|P'(1, 30)| = |P(1, 30)| - \sum_{k=1}^{9} |P(1, k)| \times |P(k, 30)|$$

where the sum only considers paths that pass through 10.

Correct Solution to the Problem

The correct approach should:

1. Count all paths from 1 to 30 without restrictions
2. Subtract paths that pass through 10
3. Or explicitly exclude 10 during dynamic programming

Alternative correct code:

def count_paths(start, end, forbidden):
    # First count all paths without restrictions
    all_paths = dynamic_programming(start, end)
    
    # Then count paths that pass through forbidden
    paths_through_forbidden = 0
    for k in range(start, end + 1):
        if k == forbidden:
            continue
        paths_through_forbidden += dynamic_programming(start, k) * dynamic_programming(k, end)
    
    # Subtract the forbidden paths
    return all_paths - paths_through_forbidden

def dynamic_programming(start, end):
    dp = {}
    dp[start] = 1
    for i in range(start + 1, end + 1):
        dp[i] = 0
        if i - 1 in dp:
            dp[i] += dp[i - 1]
        if i % 2 == 0 and (i // 2) in dp:
            dp[i] += dp[i // 2]
    return dp[end]

Path Examples for Clarity

Example paths from 1 to 9:

• 1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9
• 1 → 2 → 4 → 5 → 6 → 7 → 8 → 9
• 1 → 2 → 4 → 8 → 9

Example paths from 11 to 30:

• 11 → 12 → 13 → 14 → 15 → 16 → 17 → 18 → 19 → 20 → 21 → 22 → 23 → 24 → 25 → 26 → 27 → 28 → 29 → 30
• 11 → 12 → 24 → 25 → 26 → 27 → 28 → 29 → 30

Why can’t they be combined?

• No path from 9 to 11: 9 + 1 = 10 (forbidden), 9 × 2 = 18
• Only option: 9 → 10 → 11, but 10 is forbidden

Conclusion

Main conclusions:

1. Incorrect decomposition: The first approach incorrectly breaks down the task into independent subtasks, although the operation graph does not allow such decompositions.

2. Lack of continuity: Paths must be continuous sequences, and the first approach violates this requirement.

3. Correct dynamic programming: The second approach correctly uses dynamic programming with explicit exclusion of the forbidden number.

4. Mathematical error: Multiplying the number of paths in independent segments does not equal the number of paths in a connected problem with constraints.

Recommendation: When solving path-counting problems with constraints, always use unified dynamic programming with explicit consideration of all constraints, rather than attempting to decompose the problem into independent subtasks.