DIVGAME - Editorial



Author: Konstantin Sokol
Tester: Gerald Agapov
Editorialist: Tasnim Imran Sunny




Game Theory, Sieve of Erasthosenes


Two players are playing a game where they start with an integer N. At each move a player can decrease the integer by 1 or divide it by one of it’s divisor except itself and 1. The players have turns to give one move alternately. The player who cannot make any move ( the integer becomes 1) loses the game. Determine who would win the game if they start with N.


If you are not familiar with “Game theory” problems, read this tutorial before.

The terminal position ( here it is N=1) is a losing position. A position is winning if it is possible to move to a losing position in one move. And a player is in a losing position if he can move to winning positions only.

The simplest way to solve this problem would be to notice the pattern by analyzing all the winning/losing positions between 2 and M (where M is a reasonable value say 10000) with a straightforward DP solution. The pattern is that all prime numbers except 2 and 17 are losing positions. All composite numbers except 16, 34 and 289 are winning positions. So the solution is just to check whether the number is a prime or a composite number.


Here are the positions for N up to 15.

 N          = 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 
 position[] = L  W  L  W  L  W  L  W  W  W  L  W  L  W  W

When N = 2n where n >= 1:
N = 2, 4 and 8 are winning because 1,3 and 7 are losing positions. But 16 is not winning since {2,4,8,15} are not losing positions. But for all other powers of 2 greater than 16, the player can make a move to 16 and thus winning, so all the other powers of 2 are winning.

When N = 17k where k >= 1:
N = 17 is winning since 16 is losing position.
N = 172 = 289 is losing since {17, 288} are winning positions.
For all other greater powers of 17, the player can make a move to losing position 289 and thus winning the game, so all other powers of 17 are winning.

*When N = 2n 17k where n >= 1 and k >= 1:
The smallest such N = 34 is losing, since {2, 17, 33} are all winning positions.
But for any other multiples of 34, the player can make a move to losing position 34 and thus winning, so any other multiple of 34 would be winning.

When N = Number having at least one prime divisor which is not 2 or 17:
It can be proven that all such prime numbers are losing positions and all such composite numbers are winning. Consider an algorithm similar to Sieve of Erasthoneses:

  • Find the smallest number N which is not determined yet whether winning or losing
  • Determine the state of that position
  • If it is losing then mark all it’s multiples as winning positions.Because then all multiples of N would have a losing divisor N, so those would be winning.

Let’s start with the first number 3 which is losing position (since 2 is winning), so all the multiples of 3 would be winning. The next undetermined number would be the next prime number 5 and that would be losing too and all it’s multiples would be winning. The next undetermined position would always be a prime number( without considering 17) P and that would be losing since the composite number P-1 would be marked losing by one of it’s losing prime divisor. All composite numbers having prime factor other than 2 or 17, would always be winning since it would be marked winning by one of it’s losing prime divisor.


Author’s solution can be found here.
Tester’s solution can be found here.


Very nicely written editorial.


I am wondering how did you figure out you only have to consider 2 and 17 as a factor of N? How are they connected?