First, I started with 1 biscuit in the first jar and 2 biscuits in the second jar (starting configuration: 1,2).
Let's see what will happen in all possible scenarios:
1) If the 1st person takes one biscuit from the first jar, the 2nd person can take 2 from the second jar and win.
2) If the 1st person takes 1 from the second jar, the 2nd person can take 1 from each jar and win.
3) If the 1st person takes 2 from the second jar, the 2nd person can take 1 from the first jar and win.
4) If the 1st person takes 1 from each jar, the 2nd person can take the remaining 1 biscuit from the second jar to win.
In every possible scenario, the second player wins. Therefore, if you start with the configuration (1,2), no matter what the 1st person plays, the 2nd person will always win.
From the above conclusion, we can get some winning strategies:
a) If there are n biscuits in one jar and n+1 in the second, you can take n-1 biscuits from each jar to make the new amount of biscuits (1,2). Using what we found out previously, no matter what the other person plays after this, you can take all biscuits in your next turn and win.
b) If one of the original jars starts with 1 biscuit, you can take the required amount of biscuits from the other jar to make the other jar have 2 biscuits. Now the new configuration is (1,2) and therefore you win.
c) Similarly, if one of the original jars starts with 2 biscuits, you can take the required amount of biscuits from the other jar to make the other jar have 1 biscuit. Now the new configuration is (1,2) and therefore you win.
If we continue using the method of checking through all possible scenarios and determining which player ends up winning as we did with the configuration (2,1), you will see that the following configurations end up in the 2nd person winning:
(1,2) - we have proven this
(3,5)
(4,7)
(6,10)
If you look at these pairs of numbers, you can see that there is a pattern in them.
The pattern is that each time, the difference between the numbers of biscuits in each jar increases by 1.
For example:
2 - 1 = 1
5 - 3 = 2
7 - 4 = 3 etc.
We can also see that no number is ever repeated in more than one pair. For example, after (4,7) the next pair was not (5,9), because the number 5 was already used in (3,5). So we skip (5,9) and go straight to (6,10).
We can predict that the next few pairs after (6,10) will be:
(8,13)
(9,15)
(11,18)
(12,20)
(14,23)
In conclusion, any starting configuration with the pattern we found above will guarantee the 2nd player to win, while all other starting configurations guarantee the 1st (starting) player to win. Furthermore, a good strategy to win is to take biscuits from the jars so that they turn into one of the above configurations that were mentioned ((1,2) (3,5) (4,7) etc.). This way, you will be guaranteed to win.