Now I will explain how this whole game played out.

First, an introduction: The plot for this mafia is loosely based on

Dying of the Light, a novel by George R R Martin. I was originally going to make a Mountain Climbing Simulator game in Unity, and maybe I will in the future, but "Comet Mafia" is the result of me just wanting to write the story.

List of Teams:NEO & VALKAMA:

The two of you are Team Fawkes and your alignment is Underground. You are free to PM each other, but include me in all your PMs.

Every night, choose one player to convert to Underground alignment. That player will join Team Fawkes and they can be PM'ed at night from then onwards.

You win if there are more Underground players than non-Underground players by the end of Night 7.

You lose if all Underground players die, if everyone dies by the end of Night 7 (total loss condition), or if another alignment's victory condition is achieved by the end of Night 7.

SILVER & JOOGER:

The two of you are Team Gautama and your alignment is Town. You are free to PM each other, but include me in all your PMs.

Throughout the game, certain players may "vanish." These "vanished" players continue playing, but they can no longer interact with anyone else or vote.

Every night, you have a chance of receiving a task from the Village Elder player. If you get a task, work together to solve it. You have until the beginning of the next day. If the task is successfully completed, then players will not vanish that night.

You win if this team is alive at the end of Night 7. Just one living member is enough.

You lose otherwise.

IGN & FAITH:

The two of you are Team Magellan and your alignment is Town. You are free to PM each other, but include me in all your PMs.

Throughout the game, certain players may "vanish." These "vanished" players continue playing, but they can no longer interact with anyone else or vote.

Every night, you have a chance of receiving a task from the Village Elder player. If you get a task, work together to solve it. You have until the beginning of the next day. If the task is successfully completed, then before the start of the next day, you may choose a Vanished player to revive and they will become a full player again.

You win if this team is alive at the end of Night 7. Just one living member is enough.

You lose otherwise.

SIBARE & DERF:

The two of you are Team Socrates and your alignment is Town. You are free to PM each other, but include me in all your PMs.

Every night, you have a chance of receiving a task from the Village Elder player. If you get a task, work together to solve it. You have until the beginning of the next day. If the task is successfully completed, then on the next day, if a non-mafia player is lynched, the player will not die (their role will also be revealed, proving their alignment).

You win if this team is alive at the end of Night 7. Just one living member is enough.

You lose otherwise.

LAYTONKK & FAGNAROK:

The two of you are Team Newton and your alignment is Town. You are free to PM each other, but include me in all your PMs.

Every night, you have a chance of receiving a task from the Village Elder player. If you get a task, work together to solve it. You have until the beginning of the next day.

You win if you solve four tasks by the end of Night 7.

You lose otherwise. That means you also lose if mafia kills the two of you (thus ending your team).

BERRY PUNCH:

Your role name is Village Elder, and your alignment is Town.

The four groups in town who have proposed solutions to the impending crisis are Team Newton, Team Socrates, Team Magellan, and Team Gautama. Every day, pick one team to assign a task to. A randomnly generated task will be sent to players in that team the following night. If the team players solve the problem, a certain benefit will occur the next day. You will not know if they completed the task or not.

You win if you survive at the end of Night 7.

You lose if you do not survive at the end of Night 7.

GAME MECHANICS:-Town wins if Team Newton solves four task before the end of Night 7. The other teams solving tasks do have positive benefits for town, but are essentially red herrings. By default, every town player dies at the end of Night 7 from the cold.

-Team Fawkes will go underground, sleep in the pods, and repeat the cycle. This results in Underground Win, achieved if there are more Team Fawkes players than Town players by the end of Night 7.

-Total Loss occurs if (1) Team Fawkes is lynched and Team Newton does not solve four tasks before the end of Night 7; (2)There are less Team Fawkes players than town players and Team Newton does not solve four tasks before the end of Night 7. This is the most depressing ending, as everyone dies.

-When a town player

~~vanishes~~, they join Team Fawkes and gain the same win/loss conditions as Underground. Vanished players cannot post or vote. From Day 3 onwards, they gain the ability to escape. If they escape during the day, there is a 50% chance they succeed and a 50% chance they die. If they escape during the night, they must solve a difficult task, and if they succeed, they escape, and if they fail, they die. Escaped players do not join any team, but can post and vote again to essentially rat out their former members of Team Fawkes.

-On Night 3 onwards, every town-sided player except for the town elder receives the special task where they have to figure out the meaning of the game. The first person who succeeds learns the victory condition and can speak with the town elder at night.

-Town Elder is immune to abduction and thus cannot vanish. On Day 3, Town Elder will begin the journey up the mountain. He gains the ability to assign 2 tasks that night. On Day 4 onwards, Town Elder gains more abilities: he can (1) Assign a task to two different teams; (2) Ask 1 team what their benefit is if they solve the task.

-If Team Fawkes is fully lynched, democratic voting for team task assignment begins. This happened starting day 5. Elder loses the ability to assign tasks to two teams at night but gains: (1) During the day, ask 1 team what their benefit is if they solve the task; (2) At night, transfer players across teams. If a task is assigned to a team with no players left, the Elder and anyone that he can PM with must solve that task.

-After Team Fawkes is lynched, the last Team Fawkes player (in this case, Neo), becomes

~~vanished~~ and gains a nasty ability:

Team Fawkes has now been eliminated. You are "dead" and can no longer post, vote, or win the game. However, I will now reveal to you the details of the "total loss" condition I mentioned earlier.

Town's victory condition in this game is not to lynch all mafia. There is a puzzle that they must solve by night 7. If they do not solve this puzzle, then everyone dies. Every night, I will send you a list of players who are not working on tasks. You may choose to kill one of them.

Unlike Valkama, you were not buried. Shortly after you drank the poison, members of Team Fawkes (not players of the game) stole your body and brought you underground, where you were resuscitated with advanced technology. Most of your body was paralyzed by the poison and you can no longer speak. However, you can still use your right hand to write and issue a kill order every night.

Since underground can't win, is there a point to any of this? Not really. It's up to you if you want to kill town members or not. You can start killing tonight.

-On Night 6, Team Fawkes gains the ability to abduct the Elder if the team is still alive, or kill the Elder if the team is dead (in this case, the team was dead). If Team Fawkes takes the bait, then Day 7 becomes Open Task day, like what happened in this game. If they spare the Elder, then voting for team task assignment continues.

-On Night 7, which was never reached, the game becomes fully open. Any player can PM any other player to solve the final tasks.

Grand List of Tasks:

I'm not intelligent enough to have invented every problem in this game. Most of them were taken off

CEMC Past Contests, a website operated by my university which hosts math contests to high school students applying to my school. I reworded every problem to fit the story of this game. I did, however, create several of the more unique tasks in the game, including Team Newton's final problem.

Each task in this game has 4 difficulty levels: Easy, Medium, Hard, and Brutal.

**NIGHT 0**Difficulty: Easy

Assigned: Team Newton

*This is the only task which had

*zero relevance* to the story and was more of a warm up problem*

Some shape has four edges. One edge has length 10 units, another edge has length 6 units. The remaining two edges are consecutive positive integers. If the perimeter of the shape is 23 units, what are the lengths of remaining two edges?

**NIGHT 1**Difficulty: Easy

Assigned: Team Magellan

Tonight, you are assembling navigation devices for your worldly voyages. At the moment, your crew is in need of sextants.

IGN assembles 3 sextants in the same amount of time that Faith assembles 2. The village elder will be helping you tonight. He assembles 3 sextants in the same amount of time for Faith to assemble 1. Assume the three of you start assembling at the same time. How many sextants in total can be assembled by the time IGN makes 9 sextants?

**NIGHT 2**Difficulty: Hard

Assigned: Team Socrates

You will be exercising your logical reasoning skills so that you can convince the town to your cause.

Suppose A, B, and C are statements. A statement is anything that can be either true or false. For example, statement A could be "I am a dog," and statement B could be "Cats are cute." Statement C, however, will be true or false based on the link words between statement A and B. Some examples of link words are: "and", "or", "but," "then", "even though", "because", etc. So, statement C could be "I am a dog and cats are cute." This means statement C will only be true if both statement A ("I am a dog") and statement B ("Cats are cute") are true. But I could also make statement C "I am a dog or cats are cute," in which case just one of A or B must be true and statement C will be true.

Consider some statements A, B, and C such that the following conditions are satisfied:

1. When A is true and B is true, C is true.

2. When A is true and B is false, C is false.

3. When A is false and B is true, C is true.

4. When A is false and B is false, C is true.

Your task is to make up statements A, B, and C so the above four conditions are satisfied. Remember, statement C must be some link between statements A and B.

**NIGHT 3a**Difficulty: Medium

Assigned: Team Magellan

Tomorrow, you will ascend the mountain and reach a cave where you can take shelter. The path leading to this cave comprises flat ground and some uphill and downhill slopes. On flat ground, you walk at 5 km/hour; on uphill, at 4 km/hour; on downhill, at 6 km/hour.

The town elder estimates that it will take 96 minutes to reach the cave. However, in the case that the cave leads to a dead end, you will need to walk back down again. The elder estimates it will take 99 minutes to walk the same way back down to where you are now.

2.5 km of the path from here to the cave will be on flat ground. What do you estimate the total distance from here to the cave to be?

You have about twenty four hours to answer this question before the night ends.

**Night 3b**Difficulty: Medium

Assigned: Team Gautama

There is limited food available for the town to use on their journey, and the elder entrusts you, Team Gautama, as food treasurer.

The three most important food items in town are: potatoes, bread, and fish. One loaf of bread is worth as much as two potatoes. A fish is worth as much as five potatoes. Currently, there are ten times more potatoes than loaves of bread. The total value of all the food combined is equal to twenty fish.

How many potatoes, bread, and fish does town have?

**NIGHT 3c**Difficulty: Hard

Assigned: Fagnarok (Escaping)

In the middle of the night, you get up and quietly try to leave the underground base of Team Fawkes. As you reach the exit, you notice that there are some guards posted at this area. They are vigilantly standing in front of the escape door, and all of them are armed with weapons.

To escape, you will need to sweet talk your way out of this one. Your task tonight will be to hone your verbal reasoning skills so you can convince the guards to let you leave.

Suppose A, B, and C are statements. A statement is anything that can be either true or false. For example, statement A could be "I am a dog," and statement B could be "Cats are cute." Statement C, however, will be true or false based on the link words between statement A and B. Some examples of link words are: "and", "or", "but," "then", "even though", "because", etc. So, statement C could be "I am a dog and cats are cute." This means statement C will only be true if both statement A ("I am a dog") and statement B ("Cats are cute") are true. But I could also make statement C "I am a dog or cats are cute," in which case just one of A or B must be true and statement C will be true.

Consider some statements A, B, and C such that the following conditions are satisfied:

1. When A is true and B is true, C is true.

2. When A is true and B is false, C is false.

3. When A is false and B is true, C is false.

4. When A is false and B is false, C is true.

Your task is to make up statements A, B, and C so the above four conditions are satisfied. Remember, statement C must be some link between statements A and B.

**NIGHT 3d**Difficulty: Hard

Assigned: Sibare (Escaping)

Three guards are posted at the door leading out of the underground base of Team Fawkes. You label them A, B, and C in your head and begin to observe their patrol. This is a 2D patrol. They can walk in directions other than left or right and do not necessarily stand in a straight line.

At the start of the patrol, Guard A and B are standing in the same spot. Guard C stands 10 meters away. Guard A chooses a random direction and walks in that direction until he is 10 m away from Guard B. Then they return to their original positions.

You calculate that your best chance of escape is when guard A is closer to guard C than guard B is to guard C after guard A has walked 10 m in a random direction. What is the probability of this?

**NIGHT 4a**Difficulty: Medium

Assigned: Team Newton

Tonight, you will be building a rocket engine that requires five batteries. Unfortunately, the batteries that you have were just recently developed and may be prone to malfunction. Each battery has a 20% chance of failure. After performing your calculations, you have determined that the rocket engine needs three or more working batteries in order to function.

What is the probability that the rocket engine will function with five batteries inside?

**NIGHT 4b**Difficulty: Medium

Assigned: Team Gautama

Impressed with Team Gautama's efforts as food treasurer, you are now assigned to create a new berry mix from a basket for tomorrow's journey.

Initially, the basket contains 10 kg of blueberries. 2 kg of blueberries are removed and 2 kg of raspberries are added and thoroughly mixed. Then 2 kg of this mixture is removed and 2 kg of raspberries are added and mixed again. What is the ratio of the mass of blueberries to the mass of raspberries in the final mixture?

**NIGHT 5a**Difficulty: Hard

Assigned: Team Socrates

To keep town entertained during the cold night, the elder has created a game involving cashews and a number line.

Suppose the player starts off with 'x' cashews, all at position '0' on the number line. On each turn, the player must do one of the following:

(1) Take one cashew from position 0 and eat it. Take a second cashew from position 0 and put it on position 1.

(2) Select an integer 'y' greater than or equal to 2. Remove a cashew from position (y - 2) and remove a second cashew from position (y - 1). Eat one of them. Put the other at position y.

When no more moves are possible, the player wins if there are at most three cashews remaining, each at a distinct position on the number line and no two at consecutive integer positions. For example, if x = 7, you win the game with the following moves:

(1), (1), (2): y = 2, (1), (2): y = 3

This causes the remaining cashews to be at positions 1 and 3.

Your task is to determine an integer 'x' such that, when the game is won, there is one cashew left at position 5 and no other cashews.

**NIGHT 5b**Difficulty: Hard

Assigned: Team Gautama

The elder has given you three medicinal bottles, each of which contains a mixture of water and Chemical R. This chemical was synthesized by Team Newton and is believed to help treat the cold. However, it is not too effective so far, and thus the elder assigns you the task of creating a new medicine.

Medicine A contains 40 g of mixture, of which 10% is chemical R and 90% is water.

Medicine B contains 50 g of mixture, of which 20% is chemical R and 80% is water.

Medicine C contains 50 g of mixture, of which 30% is chemical R and 70% is water.

You pour some of the mixture from each bottle into an empty bottle to create a new mixture of mass 60 g and 25% of it is chemical R. Then you mix together the remaining contents of Medicines A, B, and C into another empty bottle to create a second new mixture.

What percentage of this second new mixture is chemical R?

**NIGHT 6a**Difficulty: Hard

Assigned: Town Elder (No Team Newton)

Team Newton has calculated that the comet will soon pass very close by a planet which is in stable orbit around the Sun. Their plan is to put everyone on a rocket and launch it from the comet to this planet. However, they do not know the distance from the comet to the planet.

For simplicity, ignore astrophysics and suppose point A is the fixed launch point on the comet and point B is the fixed landing point on the planet. These points do not ever move. When being launched through the atmosphere of the comet, the rocket will travel an average of 63 km/s. While travelling through the vacuum of space, the rocket travels an average of 77 km/s. When being pulled in by the planet's gravity, the rocket travel an average of 99 km/s. Team Newton theorizes that the whole trip from A to B will take 3 hours and 40 minutes.

To calculate the distance between the comet and the rocket, Team Newton hypothesizes a trip back. If we assume that we can flip the values as follows: the rocket travels an of average 63 km/s when flying out of the planet, 77 km/s through the vacuum of space, and 99 km/s when pulled in by the comet's gravity, Team Newton hypothesizes the return trip from B to A to take 4 hours and 20 minutes.

What is the distance between the comet and the rocket?

**NIGHT 6b**Difficulty: Medium

Assigned: Team Socrates

No one in town wants Team Fawkes around, especially with the town severely running out of food. The remaining members of Team Fawkes are sleeping in a straight row at the back of the cave. Suppose you number these members from 1 to n in the row sequentially.

Starting from one end of the row, you walk towards the other end and silently murder every second member of Team Fawkes. Once you've reached the other end, you head back and murder every second living member of Team Fawkes once again. Repeat this until only one member of Team Fawkes remains. This last member is at position x in the row.

If there are 50 members of team Fawkes (n = 50), what is position x, the number that survives?

**DAY 7a**Difficulty: Brutal

Assigned to: Team Newton

The energy source for a rocket that Team Newton is trying to build is a strange device consisting of three shapes. There is a cube of side length 7 meters. Inscribed inside the cube is a sphere that fits the cube perfectly - that is, if the diameter of the sphere were increased by any amount, it would not longer fit in the cube.

Inscribed inside this sphere is another cube which also fits the sphere perfectly. If any side length of this cube increased, it would not fit inside the sphere. This cube is the power source of the device. To calculate how much energy it can generate, you must find out the volume of this innermost cube.

**DAY 7b**Difficulty: Brutal+ (I believe this is the toughest question in the game)

Assigned to: Team Socrates

There is some money left in everyone's pockets. It is, of course, completely useless by this point. Team Socrates makes a game out of it.

Player 1 tosses three coins and Player 2 tosses two coins. Whoever has more heads wins the game. But if both players obtain the same number of heads, they continue playing until someone obtains more heads in one round. What is player 1's probability of winning?

**DAY 7c**Difficulty: Brutal

Assigned to: Team Magellan

Team Magellan heads towards the mountain summit at 4 km/h. At 8 km before reaching the summit, a bird lands directly beside them. It then immediately flies through the heavy snowstorm towards the summit at 10 km/h. As soon as it reaches the summit, the bird flies back and stops when it reaches Team Magellan again. Then it returns to the summit, and this process repeats until Team Magellan has reached the summit.

How far has the white bird travelled in total when Team Magellan reaches the summit?

**NIGHT 7d**Difficulty: Hard

Assigned to: Team Gautama

Town has saved two bags of food. The first bag contains 2 potatoes and 2 loaves of bread. The second bag contains 2 potatoes, 2 loaves of bread, and some discrete quantity of fish.

The food cannot be evenly distributed, and so each person takes to their own need from the bags. After the food is distributed, Team Gautama notices that the probability of drawing two of the same food items in two draws from the same bag is equal for both bags. How much fish was left?

~

TASKS SCORERS:

Neo: 0 (Team Fawkes has no tasks)

Berry: 1 (Night 6a)

Fagnarok: 0

LaytonKK: 2 (Night 0, 4a)

IGN: 0

Faith: 0

Silver: 3 (Night 3b, 7a, 7d)

Valkama: 0 (Team Fawkes has no tasks)

Sibare: 1 (Night 3d)

Joog: 0

Darf: 3 (Night 2, 5a, 6b)

**BRONZE:** Berry, Sibare

**SILVER:** LaytonKK

**GOLD:** Derf, Silver

Congratulations!