Find Smallest and Largest Number of Children in Game Before Start

Suppose we have an array A with K number of elements. Consider, in a game there are N players and a game master is there. This game has K rounds. In ith round game master announces to form groups with A[i] number of children. Then the remaining children form as many groups of A[i] children as possible. One child cannot take part into multiple groups. Those who are left without a group leave the game. The others proceed to the next round. A round may have with no player loss. In the end, after the K-th round, there are exactly two children left, and they are declared the winners. We have to find the smallest and the largest possible number of children in the game before the start, or determine that no valid values of N exist.

So, if the input is like A = [3, 4, 3, 2], then the output will be [6, 8], because if game starts with 6 children, then it proceeds

• In round 1, 6 of them form two groups of 3 people

• They are forming two groups with 4 and 2 children

• Then a group with 1 child and another with 3, and that 1 will leave the game

• Three of them form a group of 1 and 2. and 1 will leave.

last 2 children are declared the winners.

Steps

To solve this, we will follow these steps −

n := size of A
Define a large array a, l, r, a of size: 100010.
l := 2, r = 2
for initialize i := 1, when i <= n, update (increase i by 1), do:
   a[i] := A[i - 1]
for initialize i := n, when i >= 1, update (decrease i by 1), do:
   x := a[i], L := (l + x - 1)
   if L > R, then:
      return -1, 0
   l := L, r = R + x - 1
return l, r

Example

Let us see the following implementation to get better understanding −

#include <bits/stdc++.h>
using namespace std;

void solve(vector<int> A){
   int n = A.size();
   int l, r, a[100010];
   l = 2, r = 2;
   for (int i = 1; i <= n; i++)
      a[i] = A[i - 1];
   for (int i = n; i >= 1; i--){
      int x = a[i], L = (l + x - 1) / x * x, R = r / x * x;
      if (L > R){
         cout << "-1, 0";
      }
      l = L, r = R + x - 1;
   }
   cout << l << ", " << r << endl;
   return;
}
int main(){
   vector<int> A = { 3, 4, 3, 2 };
   solve(A);
}

Input

{ 3, 4, 3, 2 }

Output

6, 8