## Enigma Machine (By Michelle So)

#exploremaths

1. What
is the flaw in enigma?

The
flaw in the enigma machine is that the machine cannot represent the letter that
is being typed into the enigma machine. For example, if the letter ‘c’ was
typed into the machine, it cannot generate the letter ‘c’ as the encoded
letter. This made it easier for the allies to crack the code as through trial
and error, the code can be obtained when there are no letters that are repeated
as itself.

2. How
did the allies crack it and get around this sophisticated machine?

Allies cracked this machine by using a machine
known as the ‘Bomb’. This machine operates like a search machine, searching for
the code. It generates a inter-electrical circuit. In addition, as the machine
cannot generate itself in the machine, the allies slide the code until there is
no repeated letters. Furthermore, the allies were able to get around the
machine easier as the operators in the Nazi tend to use the same setting to
send the message, therefore allowing it to be easier to crack the code.

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## A Fraction of Fractals (Nina Qiu)

Snow and ice comes to mind when the word “fractal” is
mentioned. But of course, they are indeed examples of fractals. What a fractal
is, is that it’s a shape or an on-going complex pattern that is self-similar
across different scales. Some of well-known fractals include the Sierpinski
triangle, the Mandelbrot set, the Koch snowflake, the Julia set, and the
Apollonian gasket. Some examples of fractals occurring in nature are river
networks, frost crystals, coastlines, Mountain Goat horns, and patterns on tree
leaves.

One of my favorite fractals is definitely the Apollonian
gasket, named after the Greek mathematician, Apollonius of Perga. It seems
quite simple, but it’s maths. The fractal is generated from triples of circles,
where each of them is tangent to the other two. And the Apollonian network is a
graph derived from finite subsets of the Apollonian gasket.

#exploremaths

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## Of Symmetry and Tessellation (Nina Qiu)

There are four types of symmetry mentioned: rotational
(point symmetry), reflectional (line symmetry), scale, and translational (tessellation)
symmetry. If you look closely into nature, you could find all kinds of
symmetry, especially tessellation.

Some examples include reptile scales, spider webs,
honeycombs, and pineapples. You can identify tessellation by looking at the
shapes that assembled together to form an object’s appearance, such as a snake’s
skin. You can see the fan shaped scales overlapping each other, thus translational.
It is quite amazing how nature has a way of forming mathematical patterns that
humans cannot fathom. It gives a certain aspect towards the question: whether
or not is math invented or discovered?

#exploremaths

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## Enigma Machine (by Sandeep Darapuneni)

What is the Flaw
in Enigma?

1.
The
Enigma Machine had one notable limitation, that a letter cannot be encrypted as
itself.

2.
It
could also be said that how the Germans used the machine allowed the British to
crack it more easily. The Germans sent many daily messages, and would normally
end their messages with ‘Heil Hitler’, meaning that the British could eliminate
many possibilities by knowing this.

What did the
allies do to crack it?

The British used the fact that a letter cannot be encrypted as itself and
created ‘The Bomb’, a machine that would go through all of the possibilities
until it decoded the message. This machine was used by the British to regularly
intercept Nazi messages, and it is estimated that the war would have went on
for around 3 more years if ‘The Bomb’ wasn’t created.

#exploremaths

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## Number Systems (by Sandeep Darapuneni)

Egyptian Number System

The Purpose and the Distinctive Features of this
Number System

Egyptians
used this number system like we use the Hindu-Arabic System today. But instead
of numbers, this system used everyday objects to represent the numbers.

Where and How this Number System is Used Today

The
Egyptian Number System is not used today.

This
number system made it easier for the Egyptians to carry out simple operations
such as addition and subtraction. But this system required you to draw out many
symbols, meaning that a lot of time and space would be used.

The Purpose and the Distinctive Features of this
Number System

is of base 16 and uses 0-9 for the first 10 characters and A, B, C, D, E for
the next 6 characters. Primarily, this number system was created to write
binary code shorter. For example, 1111 0011 1000 1010 in binary code can be
simply written as F38A in hexadecimal.

Where and How this Number System is Used Today

Computer
programmers use this language today to quickly read binary numbers.

is useful as it allows for the quick reading of binary. But one drawback of
hexadecimal is that there is a limit to the size of numbers that can be made,
with 65 535 being the largest number possible.

#exploremaths

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## Artwork Ideas (by Sandeep Darapuneni)

I
was flipping through my book for ideas for this assignment when I came across
the Koch Snowflake. I realised how both the Koch Snowflake and Sierpinski’s
Triangle both used triangles to form their basic outline and how combining
these 2 fractals to create ‘SierKochski’s Snow Triangle’ would be perfect for
this assignment. I don’t know the exact details of my artwork, but I am
definitely going to incorporate this into it.

#exploremaths

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## Video composition ideas (Michelle So, Hiya Ganju, Sophia Kim and Joyce Liu)

#exploremaths

After brainstorming through several concepts, our group decided to use the mathematical concept found in the novel “The Da Vinci Code”, written by Dan Brown. The novel inspired us as it contains the mathematical concepts of encryption and the golden ratio which we had discussed during one of our lessons in class. One of the most interesting topics we have learnt in class is encryption as codes and methods of cracking them are highly fascinating. In addition, we found that because this mathematical concept appealed to all the members of our group, meaning the entire class would engage with our video composition.

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## The Stigma of the Enigma [Adam Tan]

1. What is the flaw in the Enigma Machine?
In an Enigma coded message, a letter cannot appear as itself, nor the same corresponding letter twice (unless it appears 26+ times). There where also certain words or phrases (eg. Heil Hitler) that would appear often in a message. By recognising these phrases, codebreakers could eliminate/investigate the possibilities of certain letters.

2. What did the Allies do to crack it?

They created a device – a ‘Bombe’ – that would crunch through every Enigma possibility until it found a setting in which no letter would appear the same twice. When a letter was entered, it as aligned with 25 letters – all except for itself. Alan Turing was the mastermind behind this operation, and many German messages were decrypted using his ‘Bombe’.

#exploremaths

## Cryptanalysis of Enigma (Novin Noori)

What was the flaw Enigma Had?

The problem in which enigma had was that when a specific letter was typed into the machine, the letter would be matched up with 25 other letters except for itself. This was a huge flaw as it would eliminate the possibilities how the message was encrypted making enigma easier to crack when the flaw is recognised. The allies recognised this having ‘Alan Turing’ able to this and continued to do so until the war with the Germans was over.

What did the allies do to break Enigma?

The allies recognised this flaw having to create a mechanical machine known as ‘The Bomb’ which was constructed and made in Bletchley park. The mother of computers (The Bomb) would be tweaked towards specific setting having the letters be matched up with 25 letters except for itself. This huge flaw eliminated the very possibilities towards how the message was encrypted. Alan Turing was able to crack this and continued to use this information along with a device called ‘The Bomb’ which cracked the German plans leading to the allies victory. The making of the bomb, shortened the war by less than 2 to 4 years.

By Novin Noori

#exploremaths

## Substitute Cipher (Novin Noori)

What is the weakness of the substitution cipher?

The substitution cipher is weak in today’s standard, meaning that it is very easy to crack. Not only that but every letter is ALWAYS encoded by the same symbol, which makes frequency analysis a very effective tool.

Another problem: knowing context of the message is very useful (for instance, if you know that the text is about the types and number of airplanes in the enemy’s army, you should expect words “airplane”, “aircraft”, “weapon”, and so on)

Guesses also help: for instance, if you deciphered a part of the sentence “A cat drinks …”

you would guess that the last word is “milk” (although it could be “lemonade”)

Suggest a solution towards this?

One solution towards this is to make the code more complicated instead of being lazy having to add an specific encryption to the keys. It would take more time for the enemy to decipher it meaning more time to crack it. Using different languages or unusual languages  will also make cracking the code more difficult as it is harder for people to guess the start or end of the sentence.

By Novin Noori

#exploremaths