LeetCode #780 — HARD

Reaching Points

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

Given four integers sx, sy, tx, and ty, return true if it is possible to convert the point (sx, sy) to the point (tx, ty) through some operations, or false otherwise.

The allowed operation on some point (x, y) is to convert it to either (x, x + y) or (x + y, y).

Example 1:

Input: sx = 1, sy = 1, tx = 3, ty = 5
Output: true
Explanation:
One series of moves that transforms the starting point to the target is:
(1, 1) -> (1, 2)
(1, 2) -> (3, 2)
(3, 2) -> (3, 5)

Example 2:

Input: sx = 1, sy = 1, tx = 2, ty = 2
Output: false

Example 3:

Input: sx = 1, sy = 1, tx = 1, ty = 1
Output: true

Constraints:

1 <= sx, sy, tx, ty <= 10⁹

Step 01

Brute Force Baseline

Problem summary: Given four integers sx, sy, tx, and ty, return true if it is possible to convert the point (sx, sy) to the point (tx, ty) through some operations, or false otherwise. The allowed operation on some point (x, y) is to convert it to either (x, x + y) or (x + y, y).

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Math

Example 1

Example 2

Example 3

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Largest constraint values

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #780: Reaching Points
class Solution {
    public boolean reachingPoints(int sx, int sy, int tx, int ty) {
        while (tx > sx && ty > sy && tx != ty) {
            if (tx > ty) {
                tx %= ty;
            } else {
                ty %= tx;
            }
        }
        if (tx == sx && ty == sy) {
            return true;
        }
        if (tx == sx) {
            return ty > sy && (ty - sy) % tx == 0;
        }
        if (ty == sy) {
            return tx > sx && (tx - sx) % ty == 0;
        }
        return false;
    }
}

// Accepted solution for LeetCode #780: Reaching Points
func reachingPoints(sx int, sy int, tx int, ty int) bool {
	for tx > sx && ty > sy && tx != ty {
		if tx > ty {
			tx %= ty
		} else {
			ty %= tx
		}
	}
	if tx == sx && ty == sy {
		return true
	}
	if tx == sx {
		return ty > sy && (ty-sy)%tx == 0
	}
	if ty == sy {
		return tx > sx && (tx-sx)%ty == 0
	}
	return false
}

# Accepted solution for LeetCode #780: Reaching Points
class Solution:
    def reachingPoints(self, sx: int, sy: int, tx: int, ty: int) -> bool:
        while tx > sx and ty > sy and tx != ty:
            if tx > ty:
                tx %= ty
            else:
                ty %= tx
        if tx == sx and ty == sy:
            return True
        if tx == sx:
            return ty > sy and (ty - sy) % tx == 0
        if ty == sy:
            return tx > sx and (tx - sx) % ty == 0
        return False

// Accepted solution for LeetCode #780: Reaching Points
struct Solution;

impl Solution {
    fn reaching_points(sx: i32, sy: i32, mut tx: i32, mut ty: i32) -> bool {
        while sx < tx && sy < ty {
            if tx < ty {
                ty %= tx;
            } else {
                tx %= ty;
            }
        }
        sx == tx && sy <= ty && (ty - sy) % sx == 0 || sy == ty && sx <= tx && (tx - sx) % sy == 0
    }
}

#[test]
fn test() {
    let sx = 1;
    let sy = 1;
    let tx = 3;
    let ty = 5;
    let res = true;
    assert_eq!(Solution::reaching_points(sx, sy, tx, ty), res);
    let sx = 1;
    let sy = 1;
    let tx = 2;
    let ty = 2;
    let res = false;
    assert_eq!(Solution::reaching_points(sx, sy, tx, ty), res);
    let sx = 1;
    let sy = 1;
    let tx = 1;
    let ty = 1;
    let res = true;
    assert_eq!(Solution::reaching_points(sx, sy, tx, ty), res);
}

// Accepted solution for LeetCode #780: Reaching Points
// Auto-generated TypeScript example from java.
function exampleSolution(): void {
}
// Reference (java):
// // Accepted solution for LeetCode #780: Reaching Points
// class Solution {
//     public boolean reachingPoints(int sx, int sy, int tx, int ty) {
//         while (tx > sx && ty > sy && tx != ty) {
//             if (tx > ty) {
//                 tx %= ty;
//             } else {
//                 ty %= tx;
//             }
//         }
//         if (tx == sx && ty == sy) {
//             return true;
//         }
//         if (tx == sx) {
//             return ty > sy && (ty - sy) % tx == 0;
//         }
//         if (ty == sy) {
//             return tx > sx && (tx - sx) % ty == 0;
//         }
//         return false;
//     }
// }

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(log n)

Space

O(1)

Approach Breakdown

ITERATIVE

O(n) time

O(1) space

Simulate the process step by step — multiply n times, check each number up to n, or iterate through all possibilities. Each step is O(1), but doing it n times gives O(n). No extra space needed since we just track running state.

MATH INSIGHT

O(log n) time

O(1) space

Math problems often have a closed-form or O(log n) solution hidden behind an O(n) simulation. Modular arithmetic, fast exponentiation (repeated squaring), GCD (Euclidean algorithm), and number theory properties can dramatically reduce complexity.

Shortcut: Look for mathematical properties that eliminate iteration. Repeated squaring → O(log n). Modular arithmetic avoids overflow.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.