Sylver Coinage

The emperor decides to establish a new system of currency with a new set of coin denominations. Each coin denomination must have an integer value, otherwise it would be too difficult to use. The emperor decides to give the project to his two greatest mathematicians, James and John. They would take turns coming up with new coin denominations for the workers to mint. If a mathematician comes up with a coin value that can be arrived at with already minted coins, they would be executed. Eventually, one of the James decide to create a coin of value one. This made all the previously ordered to be minted coins obsolete and all the workers commissioned with making those coins were fired. Even though James decreed a legitimate new coin value, the workers were very angry about this and decided to execute him anyways.

The above introduction is written in reference to John H. Conway's introduction of the game in Winning Ways for Your Mathematical Plays. The game was actually named after James Joseph Sylvester who came up with the equation of a Frobenius number for a numerical semigroup with embedding dimension two.
The rules of the game are as follows:

• Each player takes their turning naming an integer that cannot be made as a linear combination of previously named integers
• The first player to name the number 1 loses
• For the benefit of the players, this version of the game allows you to make an illegal move (i.e. name an integer that CAN be made as a linear combination of previously named integers) up to 2 times. On the third time making this mistake, you automatically choose 1 and lose.

Select the bot you want to play against (2 games): Test Bot (hard) Test Bot 2 (medium) Test Bot 3 (easy) Test Bot 4 (unknowable) Uploaded Bot 1