Numerical Systems

Introduction Positional Notation Converting Between Bases Binary Addition Binary Multiplication Challenges (Python) Challenges (Java)

Modular Arithmatic

Introduction The Modulus Operator Cyphers Modular Arithmetic Modular Inverse Challenges (Java)

Algorithms

Introduction Algorithms Time Functions Asymptotic Complexity Summation Challenges

Propositional Logic

Introduction Propositional Logic Conjunction and Disjunction Implication and Equivalence Bitwise Operators Challenges

Set Theory

Introduction Sets Subsets and Supersets Set Operations Challenges

Graph Theory

Introduction Graphs Directed Graphs Structure of Language Search Trees Neural networks Game Object Trees Challenges Graph Question Generator

Proof

Introduction Proof Proof by Cases Proof by Contradiction Proof by Induction Challenges

Maths to Code

Maths to Code Challenges Question Generator

Descriptive Statistics

Introduction Statistics Measures of Central Tendency Measures of Spread Data Visualisation Correlation Challenges

Probability

Introduction Probability Basics Combinatorics Probability Trees Law of Total Probability Bayes Rule Challenges

Inferential Statistics

Introduction Probability Distributions Samples and Populations Hypothesis Testing Z Scores Student's T-Test Challenges

Challenges

We will be working on these challenges in the lab session for this topic.

You do not have to complete these in advance. However you may choose to get started on them in advance if you wish. You still need to attend the lab even if you have completed the challenges! If you do not complete them in the lab it is recommended that you finish them in your own time.

Binary Flags

Background

In programming, a flag is a value that is true or false. Often the term ‘flag’ is used because the values are not stored as boolean values (true or false), but as part of a binary number. For example, I might write a program which stores 4 flags as a number. Flag A is set if my number contains a 1 if written in binary, flag B if it contains a 2, flag C if it contains a 4, and so on.

This can be very memory efficient in certain situations. For example, imagine that I have to store 8 boolean variables. If I created a class and stored each flag as a separate boolean value in this class, it would take 8 bytes to store one of these objects (ignoring any possive overhead from the class itself). This is because boolean variables are usually stored as 1 byte, even though they only store true or false. If instead I stored the value as 8 binary bits, I could encode it into a 1 byte number or char in Java. This might not seem like much, but imagine I had to store a 1000x1000x1000 array of these objects: I’ve just reduced my memory footprint from 8GB to 1GB.

Task

Imagine that I’m making a voxel-3D game, a bit like Minecraft. I have to keep lot of these blocks in memory. For every block, I want to store some values. Rather than store each block as an object defined by a class, I’ve decided it would be more efficient to store each block as a number.

Part 1

Update: The model answer is available here

Lets say I want to store 8 binary values per block.

Write functions to read and write each of these flags from a given number. One of these a readFlag(char a, int n) function takes a char (numerical value between 0-255) a and an integer n between 0 and 7 and returns the value of the nth flag on a. The other function is setFlag(char a, int n, boolean v), which sets the value of flag n on a to v, which is either true or false, and then returns the updated value of a.

Part 2

I now want to store the following information about each block. Fork the Replit project below and write a set of Java functions to read and write all of the values below, only using a single char (= 1 byte) to store a block’s data.

I want to differentiate 16 different “types” of block as possible: e.g. sand, stone, wood, earth, lava, etc. 4 of these are “flowing types” (e.g. water).
I want damage to persist over time, so I want to store 4 different damage states, from not damaged to very damaged. Flowing-type blocks cannot be damaged.
For flowing-type, I want to store whether that block is a “source” (this is a boolean property).
If a flowing-type block is a source I want to store whether it is a natural feature or placed by a player.
I want to store how many resorces a block yeilds when harvested: either none, low or high.

Part 3

Write functions that implement the following propositional logic operators in a bitwise fasion: ⇒, ⇔ (Hint: see if you can use existing bitwise operators to do this for you)

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Getting Started

Introducing Mathematics