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.

In addition to the questions below, you may find the question generator here helpful. It has questions on propositional Logic, set theory, big-oh notation, summation and more.

Babylonians

Assuming the Babylonain cubit is 518.6 mm long and the Euphrates River is 2,800 km long, if we expressed the length of the Euphrates in cubits (rounded to the nearest cubit), how many bablyonian digits (including spaces in place of a 0 digit) would you need to write it down?

A Baseic Puzzle?

Let v be a number written in binary. It has the following digits. Digits written with a ? are either 0 or 1.

\[v = 1????0??1?????1_2\]

It can also be written with three digits in another power-of-two base: Digits written with a ? could be any number.

\[v = ???_{2^?}\]

Use the following equations to help you recover the two values that v could originally have taken.

\[ v = x * b^2 + y * b + z\]

\[x = y * 2 = z * 4\]

Fractional values in binary

To store fractional values in a positional notation system, we use negative powers. For example, in base n:

n^2	n^1	n^0	.	n^-1	n^-2
\(n*n\)	\(n\)	\(1\)	.	\(1/n\)	\(1/(n*n)\)

Solve \(101.10_2 + 52.41_8 + AFF.3F_{16}\)

With n bits to the right of the decimal place, what is the smallest number you can store?

Vigenère Cypher

The Vigenère was first described by Giovan Battista Bellaso in 1553, but was named after Blaise de Vigenère for some reason.

A Vigenere cypher can be expressed algebraicly as:

\[ C_i = E_K(P_i) = (P_i + K_i) \bmod 26 \]

\[ P_i = D_K(C_i) = (C_i - K_i + 26) \bmod 26 \]

Where \( P = P_1 ... P_n \) is the plaintext, \(C = C_1 ... C_n\) is the cyphertext, and \(K = K_1 ... K_n\) is the key. If the key is shorter than n, it is repeated as needed. For example if the key word `LEMON` was used to encrypt a message 8 characters long, the key would be `LEMONLEM`

The following text has been encrypted using a key that is 3 letters long.

evip tzpegfvs wmsd rce mb els omgnsjpvm zj hcyhs fie mb els diocgv qsf tx

Crack the encryption to find the key and the plain text.

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Introducing Mathematics