Once upon a time, the innocent people of BDMO Forum did geometry using only synthetic techniques. Yes, sometimes they'd do some dushtu stuff and add in a page of trig or long-winded length chasing, but it was harmless fun. The great priests of the land, those blessed with the gift of projective, maintained law and order throughout the land. The children diligently carried out their angle chasing, hoping to one day match the power of one of the Legends of AoPS.

But within this prosperity the seed of something horrible was growing. Then, one day it all broke loose. Complex numbers and barycentric coordinates reigned in the once hallowed sanctuaries of Geometry, and rumors where heard of Cartesian monsters lurking in the villages. The people of the forum realized that this was the work of one of them, and thus began the most important mission in the lifetime of the forum: the hunt for the bashers.

Guide to mafia:

Once signups are over, I will send a PM to each player telling them their role in the game. This PM will have all you need to know, don't worry. And don't reveal the role information to anyone outside the game!

In this game, the Synthetic Solvers are the civilians and the Bashers are the mafia. Every day phase, the forum votes to decide which person is lynched - their copies are burned and their compasses are thrown into the sea. And every night, the bashers creep around. When they find a victim, they show him the horrendous calculations and overpower his mind - making it unfit for holding any geometric thought for years.

The Bashers want to overpower the synthetic solvers and establish a reign of bash on the forums. The synthetic solvers must lynch all the Bashers before it is too late.

Since the roles are changed a lot in this game, we won't be having the day lynch by a majority. Thus day phase will NOT end early. At the end of the day, the person with the most votes is lynched.

The possible roles you may receive (more will be added if we have more players):

$\text{Figure Scout}$ - Synthetic Solver

Oh look, that arbitrary cevian forms a cyclic quad with the mixtillinear and excircle touchpoints!

Every night, you may target a person. At the end of the night phase, learn if that person is a Basher or not.

$\text{Angle Chaser}$ - Synthetic Solver

So we have $\angle PDC = \angle ADC = \angle ACB$, and... uh.... uhhhh..... I'm stuck

When the voting totals are counted at the end of day phase, you have $-1$ vote total on you.

$\text{Projective Priest}$ - Synthetic Solver

Clearly $(A,G;D,H) = (A,C;X,N) = -1. Then by Desargues and Brianchon we are done

Every night, you may target a person. At the end of the night phase, you learn who targeted that person and whom they targeted.

$\text{Geo Hater}$ - Synthetic Solver

It takes me 4 hours to do a G2$\dots$

You are immune to all targetted actions during the night phase, except for the basher's nightkill and BashBaba's role (which may negate this ability for others).

$\text{Gonju}$ - Synthetic Solver

Let $\mathcal{K}$ be the circum-conic of $\triangle ABC$ passing through its Brocard points $\Omega_1$ and $\Omega_2$ and let $X$ be the 4th intersection of $\mathcal{K}$ and the circumcircle $\odot(ABC).$

During the day phase, your vote counts twice/has double power. (No one understands your awesome solutions fully, but they respect your words nonetheless)

$\text{Telv Cohl}$ - Synthetic Solver

[Generalization of Mixtilinear Incircle]

Every night, you may target a person. That person may not die during that night phase (they are soothed by awesome geometry solutions).

$\text{darij grinberg}$ - Synthetic Solver

Lemma 1: In triangle $\triangle ABC$, if $AB=AC$ then $\angle ABC = \angle ACB$

In any phase, you may target a player and show them a synthetic solution which starts from Euclid's axioms. Thus kills BOTH your target and yourself at the end of the phase.

$\text{Determinant Detector}$ - Basher

I once proved the existence of the Nine point circle... using Cartesian Coordinates!

Every night, you may target a person. At the end of the night phase, learn their role.

$\text{BashBaba}$ - Basher

So what if it's a G6? Conway Notation and strong EFFT kills it.

Every night, you may target person. Their role does not work for this night phase only.

The game will be played by discussing in this thread. During the day phase, you may post in this thread and talk about whatever you want, and try to hunt out who may be Mafia. Please don't discuss outside the thread, as then people might be left out of the conversation. Once you're ready to vote on someone, you can do so in the following format:

Code: Select all

`[b]VOTE: (username)[/b]`

If there is a tie at the end of the day phase, the votes are resolved in this order (The person who was voted on by the earliest role in this list is lynched): Bashbaba > Determinant Detector > darij grinberg > Telv Cohl > Gonju > Geo Hater > Projective Priest > Angle Chaser > Figure Scout

48 hours after the beginning of the day phase, it will end and the person with most votes will be lynched. That will begin the night phase.

During the night phase, you may NOT post in this thread. Mafia, and people with activated roles, may send me their actions through PM.

Role order: BashBaba > Geo Hater > Telv Cohl > Darij Grinberg > Nightkill > Determinant Detector > Figure Scoper > Projective Priest

Ask me if you're confused about anything!

Some final things to note:

*If you are still interested after 9 people have signed up, post here/PM me and I'll keep you as a substitute. If initial interest is very high, we may have a game with more than 9 players. Also, PM me if you're interested in being a spectator for the game (then you'll get to know who has which role).