To get the slingshot back I have to beat the guard at the stick game, but as far as I can tell you are programmed to always lose the stick game because you always start. An NPC mentioned I should go talk to the taverns assistant but when I try he has no dialog for me at all. Has this happened to anyone else?
After an intense study of the game, I understood that you have ONLY 1 possible way to win it, unless the "PC" don't make error.
In-depth:
In this game, you always start
first and you can only extract 1, 2 or 3 sticks at a time, out of 15 sticks. These are the basic rules.
Now... If watch it carefully, if you end up at 5 sticks left on the plate and it's your turn... You have lost. 5 sticks means:
If you extract 1: the pc will extract 3, leaving you the last one. You lose
If you extract 2: the pc will extract 2, leaving you the last one. You lose
If you extract 3: the pc will extract 1, leaving you the last one. You lose
From this point, we can say, if 5 sticks are on the plate and if its your turn, you lose.
This is true... Even for 9 sticks on the plate: if 9 sticks are down and it's your turn, you have lost. No matter what you do, if the PC doesn't do errors... It will bring you to 5 sticks which means, lose.
This is true.... Even for 13 sticks on the plate: if the pc doesn't make errors, it will bring you to 9 and then to 5 and finally, to 1 stick and your lose.
5 / 9 / 13 are ALL death numbers.
Now what about 14? Since you start with 15. The answer is simple... even here, if the PC doesn't do errors, you lose... As it will bring you to 13 (extracting just 1 stick) and to your lost. Now, why am I saying this? Because... (I'm not sure) I think the PC did an error on this spot and made me win. HOWEVER
13... is the sweet spot. If you START with 15, you need to extract 2 and bring your opponet to the death number of 13. From here, he cannot win UNLESS you are making an error.
TL;DR. if you start to 15, you need to extract 2 and bring your opponet to 13. After this, you need to bring rispectively to 9, 5 and then 1, making you win the game. IF you extrat 2 while you are a 15, UNLESS you are going to make an error, no matter what your opponet do... You are gonna win no matter what.
Good luck!