Reference no: EM133124988

Mathematics Assessment Workbook

Problem 1
Gygax has generated an event in the game where a magician appears in a tavern and asks the player to play a game. The game consists of a series of rounds. In each round, the player bets a set number of coins and specifies either heads or tails for all coins that are bet. Through some magical operation, the coins will either disappear and be lost to the player, or the coins will be converted into coins of greater value, e.g. if a player places silver coins, they will be converted to gold coins if they win!

Behind the scenes, Gygax  determines the win or loss outcome by using a series of logic gates. Gygax  uses the coins as the input into the series, and converts any coins placed as heads to a true value and any coins placed as tails as a false value. A final value of true results in a win for the player, and a final value of false results in a loss of the player's coins.

Before playing, the magician makes the following statements to the player:

Players may only play once.
Every round has a chance of winning and a chance of losing.
No two rounds will provide equivalent results (i.e. providing the same heads and tails configuration for all options will not result in the exact same outcomes-some options may be equivalent but not all options).
All coins will affect the outcome (i.e. all coins should have an influence on the final outcome-if one or more of the coins doesn't influence the output then it breaks this rule).
You test the game a few times and realise the statements that the magician made to the player are not true in some cases!

The following series of logic gates are used for the rounds of the game. For each round, you should:
Verify the round adheres to the statements that the magician made and provide your proof! If not, provide details of which rules are broken and how.
If the round does not adhere to the statements that the magician made, provide a fix to the logic gates that will allow it to adhere to the magician's statements. Try to make as few changes as possible to fix the logic gates.

After identifying all of the logic gates that do not adhere to the magician's rules, you submit your results to the programming team who are able to use the information to identify bugs in their Gygax's  code and fix the event. Now players are able to complete the event without hitting any bugs!

Problem 2

Gygax  has generated an event in a dungeon in which the player finds themselves encountering an incredible powerful and ancient knight  that will attack the player if they fail the proceeding challenges.

Gygax seems a little too excited about the prospect of the player's demise -- wouldn't you agree?

On the ground of the dungeon floor are numerous circles overlapping each other. The ancient knight  challenges the player by telling them to stand in certain positions, which both use the circles and other symbols that appear on the floor.

When play testing the challenges, you realise that some of the challenges made by the ancient knight  seem entirely impossible!

For each question you should:
Prove that the challenge is possible by providing all valid solutions to the challenge, or prove that it is impossible. You should provide visuals and supporting logical reasoning in your proof. Remember: your proof should contain all potential valid answers (not just one)!
If the challenge is impossible, you should provide a fix that would allow the player to solve the challenge by altering it in as few ways as possible. (There may be many appropriate solutions)

The ancient knight  poses the following challenges:
2.1.1 Stand in an area that indicates that you are crazy and are a pirate (reminder: this is regardless of whether you have wings or are an automaton).

2.1.2 Stand in an area that indicates that you are a pirate or an automaton, and in either case you have wings.

2.1.3 Stand in an area that indicates that you are an automaton and a pirate, but do not have wings and are not crazy.

2.1.4 Stand in an area that indicates that you are not crazy and are not an automaton, but you do have wings or you are a pirate but you aren't a pirate with wings.

Two new circles appear and the ancient knight  states that the following circles represents the treasure you can find in the dungeon. The ancient knight  continues by posing the following challenges:
2.2.1 Speak aloud the number of pieces of treasure in the dungeon that are magical or are weapons.

2.2.2 Speak aloud how many pieces of treasure in the dungeon are not both magical and weapons.

2.2.3 Speak aloud how many pieces of treasure in the dungeon are magical and are ancient but are not weapons.

2.2.4 Speak aloud how many pieces of treasure are magical or are not weapons.

You identify all the ancient knight  challenges that are impossible, and provide the fixes for those to the programming team who are able to apply those solutions to fix the existing challenges. They will look into the more fundamental programming issues within Gygax's  code but for now at least the game can be released without any buggy challenges

Problem 3

After the player successfully completes all the ancient knight  a gate opens leading to the end of the dungeon. The ancient knight  bids the player farewell and congratulations on receiving their reward, however they seem to have a cheeky smile about their face. I wonder what is in lies ahead for the player...

At the very end of the dungeon, the player is presented with four rooms containing three chests each as a reward.

Each chest contains either gold or silver; never both. However, there is a catch to receiving this wonderfully valuable treasure!

Written on a piece of parchment, in each room containing the chests, Gygax  has created a riddle that the player must solve to gain access to the treasure within. To solve each riddle, the player must indicate which chests contain gold and which contain silver. If they are wrong, then the chest will be locked forever.

Upon solving one of the riddles, you realise that it is flawed! It provides no way to unlock it! You decide to test the remaining three rooms to validate which rooms contain broken riddles.

For each riddle, you should:
Solve it, or prove that the riddle is unsolvable. You should prove your solution by using Formal Logic with supporting Truth Tables, if required. Try to avoid solving the riddle using large Truth Tables, as that takes a long time to create and read!
If the riddle is indeed unsolvable, you should provide changes to the riddle that make it possible to solve. Try to make as few changes as possible to fix the riddle.

After you identify the rooms that contain chests riddles that are unsolvable, you remove those from the game and request the level designers to craft new riddles to replace those. Another task has been added to the programmers list, to figure out and solve the bug in Gygax's  code that is causing these riddles to be generated with no possible solution!
At Least for now the player is actually able to receive their rewards for braving the ancient knight  and the riddles!

Problem 4

The player scoops up their rewards from the chests, featured in the previous problem, but suddenly the floor disappears and the player plunges down into the darkness and eventually lands in a maze beneath the original dungeon! You hear the cackling laughter of the ancient knight  from a distance as you fall into the maze below.

Inhabiting this maze are pirates that will attack the player on sight. The artificial intelligence for the pirates seems to be functional but it definitely has some shortcomings and bugs.

Gygax  generated the artificial intelligence for the pirates using an industry-standard pattern called a Finite-State Machine, which is encoded as a graph with directed edges, and it is visualised on the next page.

The vertices of the graph are referred to as states, e.g. Attack is a state that will have the pirate swing their weapon at the player.
The entry point to the graph is the vertex that has an edge that originates at no vertex.

A vertex with two incoming edges indicates that all vertices connect to it. This does not indicate the entry point and saves having to draw a lot of arrows!

For the pirate to transition between states, the propositions labelled above the edge must be true. These propositions are called transition conditions.

The following propositions are used for the pirate's AI:
A: pirate has a weapon
B: pirate sees an enemy
C: pirate is in range for an attack
D: pirate has low health
E: pirate has no health
F: pirate sees a place to hide
G: pirate has high health

4.1 When a pirate enters Chase from Patrol, it yells out "Is that a player!? Attack them!", which works well until the pirate enters the Flee state and then quickly recovers its health, and then turns around and yells the same phrase at the player again. It seems weird that the pirate says that phrase when they were just battling the player.

A solution would be to add a new edge from the Flee to Chase which would not trigger the pirate yelling out that phrase again, as the pirate only yells that phrase when entering Chase from Patrol.

To represent this change, draw the updated Finite-State Machine graph with a new edge from Flee to Chase with the correct transition conditions between those states.

4.2 A designer has added new, special Armour that has a chance to break the enemy's weapon when the player is hit with an attack. Looking at the implementation for the Attack state, the code assumes that the pirate will have a weapon-if the pirate doesn't have a weapon, it will cause an error and crash the game. Will the pirate's AI handle this case? If not, draw the Finite-State Machine with a provided fix.

4.3 There seems to be a pretty big bug where the pirates will attack only once and then not do anything unless the player attacks the pirate until they are low health.

Find the bug and draw the Finite-State Machine graph with a fix to the bug.

4.4 While walking around the maze, you turn a corner and are confronted by a horde of five mean looking pirates! They manage to surround you and reduce your health to 0, resulting in the player's avatar dying. Later, you return to this area, determined to take vengeance on those pirates; however, they seem frozen in place, just like in the case of 4.3.

Find the bug and draw the Finite-State Machine graph with a fix to the bug.

Tip: You may have to add a new proposition; be sure to record that!

4.5 Finally, you determine the AI is pretty good with your new fixes; however, you notice one last issue. When the pirate enters the Flee state, it sometimes seems to return to Patrol immediately or sometimes it never leaves the Flee state.

Find the bug and draw the Finite-State Machine graph with a fix to the bug.

Problem 4
You take your final Finite-State Machine and override Gygax's generated logic. The programmers recently updated Gygax's  machine learning algorithm and by looking at your logic it is able to add a bunch of more behaviours to various artificially intelligent characters throughout the game -- and hopefully fix some bugs too!

Problem 5

Gygax  has procedurally generated the world that the players explore during their adventures within the game. Procedural generation is a common strategy in modern games and simulations as it reduces the cost of building large worlds.

As you navigate around the world, you begin to realise that there are many issues with it. Specifically, some of the cities are impossible to get to, some of the rivers run uphill, and some cities are impossible to leave!

5.1 You tell management that Gygax  has named the game world Gygaxaria and you think that perhaps the AI has become self-aware and has developed an ego. Management ignores that point and wants you to propose a new name for the fantasy world; what would you suggest is a better name?

Here is a visualization of the world that Gygax has generated where the points represent cities:

5.2 Cities in Gygaxaria, or whatever you are calling this world, are connected by three different transport methods. Those are Roads, Rivers, and  Teleporters. Below are matrices that represent the transport methods between the cities.

Provide a visual representation, in the form of a graph, of the cities and the transport connections between them, where the cities are nodes and the transport connections are directional edges. Different transport methods should be represented by different colours or strokes so that they are recognisable. Your graph should have a heading at the top that specifies ‘Map of <the name you chose in 5.1>' and a legend to allow viewers to differentiate between the transport methods

Tip: Here is an incomplete visualisation to get you started. You may find it helpful to label each city, such as A->I or 1->9.

5.3 The designers of the world specified the following rules that transport methods should follow.

Rules
 All roads should be bidirectional.
Rivers are unidirectional and only run from north to south as the altitude of the world generally decreases from north to south.
 Teleporters are unidirectional, but two pairs of teleporters can be used to create bidirectional travel between two cities.
All transport methods should be between two cities, i.e. no transport method should be from one city to itself.

5.4 You've also noticed that at least one city does not have any entering transport methods and at least one city does not have any exiting transport methods! This means players may get stuck in a city and not enter every city in the world. This is certainly a bug!

To provide a fix for these issues and the ones you found in 5.2, you are tasked with creating three new matrices that represent the transport options between cities in the world, where each city should have at least one entering and exiting transport method and all transport methods should adhere to the rules that the designers specified in 5.3.