Add Number Spiral (CSES 1071) solution with explanation and test cases

CPP/patterns/Pattern/NumberSpiral_CSES1071.cpp

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+/*
+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)
+*/
+
+#include <bits/stdc++.h>
+using namespace std;
+
+/*
+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 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() {
+
+    // 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
+*/