Is Magic Number
Logical Breakdown
- [ ] Subproblem Identification:
- [ ] Optimal Substructure:
- [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
- [ ] Optimization: (Matrix Exponentiation if 'A' is huge)
Mathematical Rigor
State Definition
Let \(dp[i]\) be the state for the \(i\)-th subproblem.
Recurrence Relation
Visualization
Complexity Analysis
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative DP | \(O(N)\) | \(O(1)\) |
Code Reference
package com.dsa.recursion;
import java.util.*;
/ * Problem: Is Magic Number * Group: 07. Recursion */ / * ============================================================================================ * PROBLEM: Is Magic? * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a number A, check if it is a "Magic Number". * * Definition: * A number is "Magic" if the recursive sum of its digits eventually equals 1. * This means we calculate the sum of digits of A, then calculate the sum of digits of the result, * repeating this process until we obtain a single-digit number. * If that single digit is 1, return 1; otherwise, return 0. * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * Let S(n) be a function that calculates the sum of the digits of n. * Let R(n) be the recursive reduction function. * * Recurrence Relations: * 1. Summing Digits: * S(n) = (n % 10) + S(n / 10) * * 2. Reducing to Single Digit: * R(n) = n if n < 10 * R(n) = R(S(n)) if n >= 10 * * Mathematical Shortcut (Digital Root): * This process is mathematically equivalent to finding the "Digital Root" of a number. * The digital root of n is congruent to n (mod 9). * - If n % 9 == 1, then the recursive sum will eventually be 1. * - Exception: If n is a multiple of 9, n % 9 == 0, but the digital root is 9. * * --------------------- * 3. Constraints & Complexity Analysis * --------------------- * Constraints: * 1 <= A <= 10^9 * * Time Complexity: * O(log A) - The number reduces drastically with every sum-of-digits operation. * Even for the max integer (approx 2 billion), the sum of digits is at most 9+9+... (approx 46). * The number of reduction steps is negligible (< 5). * * Space Complexity: * O(1) if iterative, or O(log A) stack depth if purely recursive for digit summing. * * --------------------- * 4. Input / Output Format * --------------------- * Input: * - A single integer A. * * Output: * - Return 1 if A is a Magic Number, else return 0. * * --------------------- * 5. Examples * --------------------- * Input 1: * A = 83557 * Output 1: * 1 * Explanation: * Sum(83557) = 8+3+5+5+7 = 28 * Sum(28) = 2+8 = 10 * Sum(10) = 1+0 = 1 * Result is 1 -> Magic. * * Input 2: * A = 1291 * Output 2: * 0 * Explanation: * Sum(1291) = 1+2+9+1 = 13 * Sum(13) = 1+3 = 4 * Result is 4 -> Not Magic. * * ============================================================================================ */ public class IsMagicNumber { public int solve(int A) { // TODO: Implement solution return 0; } }