Applying dynamic programming to a simple two-person game of perfect information A natural number n represents the initial position in the game. When it is a players turn he/she is allowed to 

I)  Subtract 2 from n
II) Subtract 3 from n
III)    Subtract 5 from n


We call the player who begin the game Adam and the other player Berta. The players alternate by applying on of the three rules to the number 0 or a negative number his/her opponent. If a player manages to produce the number 0 or a negative number he/she wins the game.

Here is an example of a game played by Adam and Berta (for n=15)

15 is given to Adam. He decides to subtract 5 leaving 15-5 = 10 to Berta
10 is given to Berta. She decides to subtract 3 leaving 11-3=8 to Beta
8 is given to Adam. He decides to subtract 2 leaving 8-2=6 to Berta
6 is given to Berta. She decides to subtract 2 leaving 6-2=4 to Adam
4 is given to Adam. He decides to subtract 5 producing -1 a negative number, Adam wins!


b) we define a one dimensional array X(1), X(2),X(3),..,X(n)

i)  X(j) =1 if Adam has a method to win when given the number j
ii) X(j)=0 if Adam has no method that guarantees that he wins when the given the number k


Calculate X(1),X(2),X(3)….,X(23),X(25)

What is X(8), X(13) and X(24)?     Answer should be of the form
boolean boolean boolean  so if X(8)=0 , X(13)=1 and X(24)=1 the correct answer is 011
Thus the correct answer is one of the following 000 001 010 011 100 101 110 111

My attempt is 

n=8
Adam: 8-5=3
Berta: 3-3 =0
Berta wins  0

n=13 
Adam=13-5=8
Berta: 8-3=5
Adam 5- 5
Adam wins 1

I get really stuck with 24, so far I have 01 


Is there a method for this type of problem, i have been stuck on it for ages now. 
Thanks in advance
Since you want to calculate $X(1), X(2), X(3)$,... and not just some $X(i)$, you can start from $X(1)$. Let's denote the case where there is a winning strategy for first player by $1$ and the other case by $0$.

Clearly, value of $X(i)$ for $i \in \{1,2,3,4,5\}$ is $1$. Let's store the values of $X(i)$ that we computed in an array. For $i&gt;5$, if at least one of the values of $X(i-2), X(i-3), X(i-5)$ is $0$, then the value of $X(i)$ will be $1$. If none of them is $0$, then the value of $X(i)$ will be $0$.