Mystery Cosign

Mathematics DM: Mystery Cosign

In Mystery Cosign, players discover properties of several mystery operations, then try to come up with values that satisfy as many equations as possible.

There are 12 operations in this game, but you do not know what they are. They are lettered from A to L. Each operation takes in two nonnegative integers and uniquely returns a third integer.

Here are several sample operations:

42 X 6 = 252: The operation is A*B. 8 Y 5 = 9: The operation is the sum of the lengths of the numbers, written out as English words. 25 Z 12 = 1337: The operation is the concatenation of abs(A-B) and (A+B). (The concatenation of two strings is the number formed by joining the strings together. abs represents the absolute value function.)

This game has two major phases: A Collaboration Phase, in which both players try to gleam information about the operations, and a Shootout Phase, in which players use this information to try to earn points. Each of these happen three times.

In the first Collaboration/Shootout Phases, only Operations A-D are active. In the next pair, Operations E-H become active. In the final one, all operations are active.

The Collaboration Phase

In the Collaboration Phase, players take turns deciding on numbers. There are five Collaboration Rounds per Phase.

Each Collaboration Round has a Player 1 and Player 2. These roles alternate every round. A single round goes as follows:

1) First, Player 1 provides a number P. This must be a nonnegative integer between 0 and 999. 2) Second, Player 2 provides a number Q. The same restrictions apply as above. 3) Then, both players choose some of the operations (as a reminder, these are letters from A to L). You may only choose operations that are active at a given time (so only A-D in Collab 1, for instance). In Collab 1, each player chooses two operations; in Collab 2, each player chooses three, in Collab 3, each player chooses four.

After this, the pair of numbers will be evaluated under all chosen operations and outputted separately. For instance, if the numbers were 9 and 10, and Player 1 picked AB while player 2 picked AC, then 9A10, 9B10, and 9C10 would all be revealed.

The Shootout Phase

In the Shootout Phase, players are given target values, and must come up with inputs to meet their chosen target value.

In a single Shootout, players are given three integers that act as target values. Players choose one of these as their own target. These selections are independent and private; it is okay for both players to choose the same target.

After this, players must come up with values P and Q such that out of the values {P A Q, P B Q, P C Q...}, as many as possible are equal to the target value. (Only active operations are considered.)

A player earns points equal to the number of values that hit the target. Players are only told how many points they received.

There are three Shootouts per Phase. A player's scores are summed among these.

Gameflow and Endgame

The game alternates between Collaboration Phases and Shootout Phases. So, it goes Collaboration -> Shootout -> Collab -> Shootout -> Collab -> Shootout. After the third Shootout Phase, the game is over and the player with more points overall at that time wins. If it is a tie, we consider the most recent time that the score was not tied; the player winning then is the winner. If such a time never happened, the player with the Advantage wins.

Timing

Players have 30 seconds for every Collaboration Phase move. There is no time between different Collaboration Rounds.

If a player fails to decide on a number, their number is randomly selected between 1 and 9. If they fail to decide on a set of operations, operations are randomly selected.

Players have 3 minutes for every Shootout Round. Failure to submit will give you zero points.

Miscellaneous Details

Operations are bounded by some level of mathematical complexity (something like calculus is far outside of the scope). I'm not going to write an entire list of specifics; however, you may ask me a topic and I will tell you if it could feasibly show up. These clarifications are private.

In some cases, it is possible for a calculation to be undefined. If this occurs in a Collaboration Round, undefined is returned; in a Shootout Round, an undefined result cannot earn points.

A simple calculator is allowed to do calculations you would not want to do by hand. Windows's built-in calculator, for instance, is okay. Do not use a graphing calculator or Wolfram-Alpha or anything like that. This is honor rule basis but please do follow it. (Please ask if there's anything else you'd like to use.)

For the sake of sanity, submission numbers (for both rounds) are bounded by 999.

Advantage

The player with the Advantage decides who starts as Player 1 and Player 2; in addition, they win in the event of an unbreakable tie in endgame.

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Clarification(s) and a slight rule change: In the Collaboration Phase, the operations each player chooses are publicly revealed. In the Shootout Phase, the number of points each player received in each shootout is revealed publicly, but no more than that.

Returned values are guaranteed to be nonnegative.

In Shootout Rounds, you send a single (private) submission containing your chosen target number and the P and Q you'd like to use, within three minutes.

If an operation is undefined, I'll post -1 instead of undefined.