Added inline comments, explanation, and example test cases for Number Spiral (CSES 1071)

Author: Aditya12604

CPP/patterns/Pattern/NumberSpiral_CSES1071.cpp

@@ -1,26 +1,29 @@
 /*
-Problem: Number Spiral (CSES Problem Set)
-Link: https://cses.fi/problemset/task/1071/
-Topic: Math / Pattern Recognition
-
-Description:
-Given coordinates (y, x) in an infinite number spiral, find the value at that position.
-The spiral grows outwards starting from 1 at (1,1).
-
-Example spiral pattern:
-  1   2   9  10 25
-  4   3   8  11 24
-  5   6   7  12 23
- 16  15  14  13 22
- 17  18  19  20 21
-
-Approach:
-- The maximum of (x, y) determines which square layer the cell belongs to.
-- The last number of each layer (n) is always n^2.
-- Depending on whether n is even or odd, the numbering direction differs:
-  - Even layers fill top-to-bottom.
-  - Odd layers fill left-to-right.
-- Use this pattern to directly calculate the value at (x, y) in O(1) time.
+Number Spiral (CSES Problem 1071)
+Language: C++
+Problem Description:
+You are given coordinates (y, x) in an infinite grid that forms a "number spiral".
+The spiral starts at (1, 1) = 1 and moves right and upward in square layers.
+
+Your task: For each pair (y, x), determine the number located at that cell.
+
+Example (small spiral visualization):
+
+    1   2   9  10  25
+    4   3   8  11  24
+    5   6   7  12  23
+    16 15  14  13  22
+    17 18  19  20  21
+
+So, for coordinate (y=2, x=3), the answer is 8.
+
+Algorithm:
+1. Each layer corresponds to the maximum of x and y.
+2. The square of (layer - 1) gives the value before the current layer starts.
+3. Depending on whether the layer number is even or odd:
+   - Even layer → spiral moves upward.
+   - Odd layer  → spiral moves rightward.
+4. Using conditional formulas, compute the number directly in O(1).
 
 Time Complexity: O(1)
 Space Complexity: O(1)
@@ -29,38 +32,57 @@ Space Complexity: O(1)
 #include <bits/stdc++.h>
 using namespace std;
 
-long long numberSpiralValue(long long x, long long y) {
-    if (x == 1 && y == 1) return 1;
-
-    long long layer = max(x, y);
-    long long base = (layer - 1) * (layer - 1);
-
-    if (layer % 2 == 0) {  // even layer
-        if (y == layer) 
-            return base + x;
-        else 
-            return base + layer + (layer - y);
-    } 
-    else {  // odd layer
-        if (x == layer)
-            return base + y;
+/*
+solve(x, y)
+Computes the value at coordinates (y, x) in the number spiral.
+
+Parameters:
+x -> Column index (1-based)
+y -> Row index (1-based)
+
+Returns:
+The number located at position (y, x)
+*/
+long long solve(long long x, long long y) {
+    if (x == 1LL && y == 1LL) return 1LL;
+
+    // Layer = max(x, y)
+    long long layer = max(x, y) - 1LL;
+    long long baseValue = layer * layer; // Value before this layer starts
+
+    // Even layer → starts bottom-right, moves upward
+    // Odd layer  → starts top-left, moves rightward
+    if (layer % 2LL == 0LL) {
+        if (y == layer + 1LL)
+            return baseValue + x;                  // On bottom row of even layer
         else
-            return base + layer + (layer - x);
+            return baseValue + x + layer + 1LL - y; // Moving upward
+    } else {
+        if (x == layer + 1LL)
+            return baseValue + y;                  // On right column of odd layer
+        else
+            return baseValue + y + layer + 1LL - x; // Moving leftward
     }
 }
 
 int main() {
-    vector<pair<long long, long long>> tests = {
-        {2, 3}, // Expected output: 8
-        {1, 1}, // Expected output: 1
-        {4, 2}, // Expected output: 15
-        {3, 5}  // Expected output: 27
-    };
-
-    for (auto [y, x] : tests) {
-        cout << "Input (y=" << y << ", x=" << x << ") -> Output: " 
-             << numberSpiralValue(x, y) << endl;
-    }
+    
+    // Example test cases for verification
+    cout << "Example Test Cases:" << endl;
+
+    cout << "Test 1 -> y=2, x=3 | Expected: 8 | Got: " << solve(3, 2) << endl;
+    cout << "Test 2 -> y=1, x=1 | Expected: 1 | Got: " << solve(1, 1) << endl;
+    cout << "Test 3 -> y=4, x=2 | Expected: 15 | Got: " << solve(2, 4) << endl;
+    cout << "Test 4 -> y=5, x=5 | Expected: 21 | Got: " << solve(5, 5) << endl;
 
     return 0;
 }
+
+/*
+Expected Output:
+Example Test Cases:
+Test 1 -> y=2, x=3 | Expected: 8 | Got: 8
+Test 2 -> y=1, x=1 | Expected: 1 | Got: 1
+Test 3 -> y=4, x=2 | Expected: 15 | Got: 15
+Test 4 -> y=5, x=5 | Expected: 21 | Got: 21
+*/