Program to find winner of a rower breaking game in Python

Suppose we have an array height. There are n different towers with different height. Amal and Bimal are playing a game. The game rules are like below

• Amal always plays first

• During each move, the current player selects a tower of height X and break it down into Y different towers of height Z each. [Y*Z = X; X and Y > 1]

• Whoever has no move will lose the game

We have to find the winner's name.

So, if the input is like height = [3,1,2], then the output will be Bimal, because the initial heights are {3,1,2}. If Amal breaks the height of tower 2 to two towers of height 1, then new height array will be {3,1,1,1}, Bimal can break tower with height 3 and make three towers of height 1, so Amal has no move hence Bimal wins.

To solve this, we will follow these steps −

• Define a function util() . This will take limit, initial limit value is 10^3+5
• result := an array of size limit and fill with 0
• for i in range 2 to limit - 1, do
  • s := a new set
  • for j in range 1 to floor of square root of i, do
    • d := quotient of i/j, r := remainder of i/j
    • if r is same as 0, then
      • if j is odd, then
        • insert result[d]) into s
      • if d is odd, then
        • insert result[j] into s
  • j := 0
  • while j is present in s, do
    • j := j + 1
    • result[i] := j
• return result
• g := util()
• From the main method, do the following −
• r := 0
• for each i in height, do
  • r := r XOR g[i]
• if r is non-zero, then
  • return "Amal"
• otherwise,
  • return "Bimal"

Example

Let us see the following implementation to get better understanding −

def util(limit=10**3+5):
   result = [0] * limit

   for i in range(2, limit):
      s = set()
      for j in range(1, int(i**0.5)+1):
         d, r = divmod(i, j)

         if r == 0:
            if j & 1:
               s.add(result[d])
            if d & 1:
               s.add(result[j])

      j = 0
      while j in s: j += 1
      result[i] = j

   return result

g = util()

def solve(height):
   r = 0

   for i in height:
      r ^= g[i]

   if r:
      return "Amal"
   else:
      return "Bimal"

height = [3,1,2]
print(solve(height))

Input

[3,1,2]

Output

Bimal