Introduction Link to heading

Normally, the factorial of a positive integer n is the product of all positive integers less than or equal to n. For example, factorial(10) = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1.

We instead make a clumsy factorial: using the integers in decreasing order, we swap out the multiply operations for a fixed rotation of operations: multiply (*), divide (/), add (+) and subtract (-) in this order.

For example, clumsy(10) = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1. However, these operations are still applied using the usual order of operations of arithmetic: we do all multiplication and division steps before any addition or subtraction steps, and multiplication and division steps are processed left to right.

Additionally, the division that we use is floor division such that 10 * 9 / 8 equals 11. This guarantees the result is an integer.

Implement the clumsy function as defined above: given an integer N, it returns the clumsy factorial of N.

Example 1:

Input: 4
Output: 7
Explanation: 7 = 4 * 3 / 2 + 1

Example 2:

Input: 10
Output: 12
Explanation: 12 = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1

Note:

1 <= N <= 10000
-2^31 <= answer <= 2^31 - 1  (The answer is guaranteed to fit within a 32-bit integer.)

Solution Link to heading

State Machine solution:

func clumsy(N int) int {
    if N == 0 {
        return 0
    }
    sum := 0
    val := N
    N--
    for state := 0; N > 0; N, state = N - 1, state + 1 {
        state %= 4
        switch state {
            case 0:
                // mul
                val *= N 
            case 1:
                // divide
                val /= N
            case 2:
                // add
                sum += val
                val = N
            case 3:
                // subtract
                sum += val
                val = -N
        }
    }
    sum += val
    return sum
}

Mathematics solution:

func clumsy(N int) int {
    precompute := []int{1, 2, 2, -1, 0, 0, 3, 3}
    if N > 4 {
      return N + precompute[N % 4]
    }
    return N + precompute[N + 3]
}

Explanation Link to heading

Lets create a simple state machine that will have 4 states:

  • 0 for multiply
  • 1 for divide
  • 2 for add
  • 3 for subtract

Each step we have two values: sum and val.

  • sum is using for low-priority operations like add and subtract
  • val is using gor high-priority operations like mul and devide

Clumsy starts with positive val, but later it will be always negative, because of subtract operation.

Another way to solve this problem is to use mathemetics and convert this problem to O(1) formula.