Maths Problems

Fibonacci Without Recursion

void fib(int N) {
    int a = 0, b = 1;
    for(int i = 0; i < N; i++) {
        cout << a << " ";
        int temp = a + b;
        a = b;
        b = temp;
    }
}

Calculate Ceil (x/y)

// (x + y - 1) / y calculates ceil(x / y):

// Exact Multiple Case:
int x1 = 6, y1 = 3;
int result1 = (x1 + y1 - 1) / y1;  // result1 = 2 (same as ceil(6 / 3))

// Non-Multiple Case:
int x2 = 7, y2 = 3;
int result2 = (x2 + y2 - 1) / y2;  // result2 = 3 (same as ceil(7 / 3))

//Note: the method `(x + y - 1) / y` only works for integer division.
//It does not handle floating-point numbers or decimals directly.
// (6.1 + 3 - 1)/3 = 6/3 != ceil(6/3)

ceil(6/3) = (3 + 6 - 1 )/3 = (1 + 2 - 1/3) = 1 + 2 -1 = 2 ceil(7/3) = (3 + 7 -1)/3 = (1 + 3 -1/3) = 1 + 3 - 1 = 3

Find Combination nCr formula

int nCr(int n, int r){
  long long res =1;
  for(i=0; i<r; i++){
  res = res * (n-1); *10*9*8
  res = res/(i+1); 1*2*3
  }
  return res;
}
// nCr = n!/((n-r)!*(r)!)
// 10C3 = 10*9*8/(1*2*3)

Find if a Number is Perfect Square

//check if

#include <cmath>

bool isPerfectSquare(int num) {
    if (num < 0) {
        return false;
    }
    int sqrtNum = static_cast<int>(sqrt(num));
    return (sqrtNum * sqrtNum == num)
}

// or

bool isPerfectSquare(int num) {
    if (num < 0) {
        return false;
    }
    double sqrtNum = sqrt(num);
    return (sqrtNum == static_cast<int>(sqrtNum)); // if double and integer are equal
}

Swap Function

int x=5, y=10;

//Pass by Reference : Call swap(x,y)
void swap(int &a, int &b){
  int temp = a;
  a = b;
  b = temp;
}

// Using pointer Arguments : Call swap(&x, &y)
void swap(int *a, int *b){
  int temp = *a;
  *a = *b;
  *b = temp;
}

// Using XOR ( Bitwise Swap) ⭐
void swap(int &a, int &b){
  a = a ^ b;
  b = a ^ b;
  a = a ^ b;
}

// Using Arithmentic Operation
void swap(int &a, int &b){
  a = a + b;
  b = a - b;
  a = a - b;
}

// a = 6, b = 4

// swapping value of a & b with `temp` variable (Better TC)
temp = a // temp = 6
a = b // a = 4
b = temp // b = 6

// swapping value of a & b without `temp` variable (Better SC)
a = a + b // a = (6+4) = 10
b = a - b // b = (10-4) = 6
a = a - b // a = (10-6) = 4