**π‘ Problem Formulation:** This article tackles the challenge of determining whether a specific numerical goal can be achieved by repeatedly adding any two given lengths. The problem involves calculating the possibility to land exactly on a target number when starting from zero and only making jumps that are either of length *a* or length *b*. For example, given lengths 4 and 6, can one reach the number 14 through a combination of jumps?

## Method 1: Iterative Check Using Modulo

The first method involves an iterative check using the modulo operation to determine if the target number can be expressed as a multiple of the greatest common divisor (GCD) of the two jump lengths. This approach is efficient and mathematically sound because if the GCD of the jump lengths does not divide the target number, it is not possible to reach the target using those jumps.

Here’s an example:

import math def can_reach_target(target, a, b): gcd_ab = math.gcd(a, b) return target % gcd_ab == 0 print(can_reach_target(14, 4, 6))

Output: `True`

This snippet defines a function `can_reach_target()`

that calculates the GCD of the two given jump lengths and then uses modulo to check if the target can be reached. The function returns `True`

if the target number is a multiple of the GCD, demonstrating that it is reachable using jumps of length `a`

and `b`

.

## Method 2: Using Linear Combination

In this method, we use the principle that any number that can be reached by jumps of lengths *a* and *b* can be expressed as a linear combination of *a* and *b*, where coefficients are non-negative integers. We can iteratively adjust the coefficients to see if the target number can be obtained.

Here’s an example:

def can_reach_target_linear(target, a, b): for x in range(target//a + 1): if (target - x*a) % b == 0: return True return False print(can_reach_target_linear(14, 4, 6))

Output: `True`

The function `can_reach_target_linear()`

iterates through possible coefficients for length `a`

, checking if the remaining difference is divisible by length `b`

. The function returns `True`

when a valid combination of coefficients is found, indicating that the target is reachable.

## Method 3: Recursive Solution

This recursive method explores all possible combinations of jumps. It subtracts jump lengths from the target recursively and checks for a base case where the target is reduced to zero. It is not the most efficient, but it showcases how recursion can be used to model the decision-making process.

Here’s an example:

def can_reach_target_recursive(target, a, b): if target < 0: return False if target == 0: return True return can_reach_target_recursive(target - a, a, b) or can_reach_target_recursive(target - b, a, b) print(can_reach_target_recursive(14, 4, 6))

Output: `True`

The function `can_reach_target_recursive()`

keeps subtracting the jump lengths from the target and returning `True`

once it exactly hits zero. It uses the logical OR to return `True`

if either recursive path confirms that the target can be reached.

## Method 4: Dynamic Programming

Dynamic programming can be used to solve this problem by breaking it down into subproblems. We maintain a boolean array where each index represents the possibility to reach that number. This method ensures that we do not perform redundant calculations and is efficient for larger numbers.

Here’s an example:

def can_reach_target_dp(target, a, b): dp = [False] * (target + 1) dp[0] = True for i in range(target + 1): if dp[i]: if i + a <= target: dp[i + a] = True if i + b <= target: dp[i + b] = True return dp[target] print(can_reach_target_dp(14, 4, 6))

Output: `True`

The function `can_reach_target_dp()`

uses dynamic programming to build up an array of reachable states. If any state `i`

is reachable, it marks states `i+a`

and `i+b`

as reachable as well, finally returning whether the target is reachable.

## Bonus One-Liner Method 5: Advanced Mathematical Approach

For those with a strong mathematics background, the Frobenius Coin Problem gives us a one-liner solution by providing a condition for the largest number that cannot be formed using two co-prime numbers. This method rests on advanced mathematical theorems.

Here’s an example:

print(math.gcd(4, 6) == 1 and 14 >= (4 - 1) * (6 - 1))

Output: `False`

This line checks if 4 and 6 are coprime and then uses the theorem from the Frobenius Coin Problem to deduce the reachability. It suggests that the target number is not within the unreachable bounds, thus can be reached.

## Summary/Discussion

**Method 1: Iterative Check Using Modulo.**Straightforward and efficient. It fails only when the numbers are not divisible by the GCD of the jump lengths.**Method 2: Using Linear Combination.**Makes use of the numerical representation of the problem. It may become less efficient for larger target numbers with large differences between jump lengths.**Method 3: Recursive Solution.**Offers a clear conceptual understanding of the decision process but is not practical for large numbers due to recursion depth and performance issues.**Method 4: Dynamic Programming.**An optimized approach for larger target numbers. Requires additional memory to store intermediate results.**Bonus One-Liner Method 5: Advanced Mathematical Approach.**A single condition check based on a mathematical theorem. Effective when the inputs are co-prime, limited by the theorem’s restrictions.