## Arithmancy

written by Rosalina Milanette

If you need any clarification feel free to owl me!

##### Last Updated

05/31/21

##### Chapters

5

##### Reads

6,972

### Chapter 4

##### Chapter 4

Before we begin talking about sequences (which is our first topic here) you must know about a wonderful resource on the interbet, the Online Encyclopedia of Integer Sequences: https://oeis.org/

Fibonacci

The Fibonacci sequence is probably the most famous one out there. It is easy to understand, to remember, and it is seen everywhere, especially in nature. Here is the sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

The rule is that a certain term in the series is the sum of the previous two. So for instance, term number 15 would we the sum of the 14th and 13th term.

To write that mathematically, we write F(n)=F(n-1)+F(n-2). This is also known as a recursive sequence, but that is a whole other lesson for another time.

An interesting pattern found in the Fibonacci sequence is that the ratio of F(n) and F(n-1) converges to 1.618, or the Golden Ratio. For instance 13/8=1.625, but 55/34=1.6176 which is much closer to the Golden Ratio.

Shell with squares on it. Each square has the square of a Fibonacci number. The squares increase in the Fibonacci sequence

The Fibonacci sequence can be seen in nature. One of the most famed examples is a spiral shell. Yes, a regular shell on a beach has the beautiful Fibonacci sequence etched into it:

The way the Fibonacci sequence is in the spiral, is that each square has the side equal to the next number in the Fibonacci sequence. For instance, as seen above, we start with two squares of side lengths 1 and 1, then side length 2, then 3, then 5, etc. Nature is truly wondrous!

Finally, one of my favorite places to find the Fibonacci sequence in is the Pascal's triangle. The Pascal's triangle (seen below) is a triangle, starting with the number 1 on top. Every row below has ones on the sides, and the middle terms are the sum of the two numbers above. What is also cool about the Pascal triangle is that the sum of every row is equal to a power of 2, while the sum of every diagonal is....A FIBONACCI NUMBER!!

An image of the Pascal's triangle. Starting with one circle on top with 1. Two circles under with two 1. Three circles below with 121. etc...

Shows how each diagonal can be added up to be a Fibonacci number (in the order of the Fibonacci sequence)

Look and Say sequence

Let's now talk about a sequence that is a favorite of mine. The Look and Say sequence. Here are the beginning few terms in it:

1, 11, 21, 1211, 111221, 312211, 13112221...

So, what is the rule? If the name didn't clue you in, try speaking the above terms out loud by digit; one, one one, two one, etc.

If that STILL didn't clue you in guess I'll just have to explain. The sequence relies on the way we say things. The first term is 1 but the amount of ones is one. So that is "ONE ONE". Thus the second term is 11. Now there are two ones in the second term so "TWO ONE" or 21. That is "ONE TWO and ONE ONE" or 1211, etc.

Some interesting properties of this sequence is the fact that whatever number you start with, that number will stay the end digit for ALL terms. For instance if we started with a 5, then the sequence would be as follows:

5, 15, 1115, 3115, 132115, 1113122115

Clearly the last digit stays a 5 throughout.

This is the wonderful look and say sequence. Try and write it out for your friends, and see them try to guess the rule!

Knights and Knaves

The Knights and Knaves puzzle name was created by Raymond Smullyan in his book What is the Name of This Book? The puzzles all follow the same format; characters either always tell the truth or always lie. The setting is an island inhabited by "Knights" or truth-tellers and "Knaves" or liars.

Now, many puzzles have evolved into introducing "alternators" or characters who alternate between telling the truth or lying, and "normals" who can choose. This increases the level of the difficulty of the puzzles, creating some monstrous mind-benders.

Most puzzles ask to determine a certain fact based on who said what. However, some puzzles also allow the reader to "ask" a question to determine a certain fact. We will examine both today.

Type 1: Determine a certain fact based on who said what.

This puzzle can be solved by creating tables or charts and doing casework. The casework is as follows: choose one character to be the "control". You will first solve the puzzle if he is lying, then if he is telling the truth. Only one of these times should work without contradictions and give you the right answer.

Seems easy, right? The casework is usually the easiest part, however with harder puzzles you can get up to three "controls"!! The MOST important part here is choosing the control. We will now go over two examples so you can understand this better.

1: You see two of your friends, John and Bill walking down a path. John says to you, "We are both knaves." Who is who?

Since John was the only one who said anything we will make him the control. If he is a knight we get a contradiction because he must be telling the truth. Thus, he is a knave who lied about them BOTH being knaves. Thus, John is a knave and Bill is a knight.

2: You see two of your friends, James and Bart walking down a path. James says, "We are the same kind." but Bart says, "We are of different kinds." Who is who?

In this scenario they are making contradictory statements, and so one must be a knight and one must be a knave. Since that is exactly what Bill said, Bill must be the knight, and John is the knave.

Type 2: "ask" a question to determine a certain fact

This puzzle can be solved by trying to find holes in the argument and then using what you know about the problem to try different questions.

1: Wikipedia

2: Gold/Silver/Bronze coins.

Primes

Primes are numbers that are divisible only by themselves and one, in other words they have 2 divisors. Prime numbers have been a constant area of wonder and ideas for mathematicians and our knowledge about them is still expanding.

One thing that we know about prime numbers is that there's infinitely many of them. This was proved a long time ago by Euclid. However, over time we have learned more and more about primes.

Primes are even found in nature! For instance, a breed of cicadas live most of their lives underground. They only emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Prime numbers!! Scientists believe that this prime-numbered cycles have evolved so that predators cannot naturally synchronize with these cycles and thus the cicadas are more safe!

Another thing we know about primes came recently to the board. As stated earlier, no one has ever figured out a pattern for prime numbers. When they appear, how they show up, etc. However, Yitang Zhang in 2013 proved that there is some number n (less than 70 million) that there are infinitely many pairs of consecutive primes exactly n apart. This number was brought down to 246 by James Maynard, and a Polymath Project organised by Terence Tao. Some recent mathematical discoveries for you all!

Fibonacci

The Fibonacci sequence is probably the most famous one out there. It is easy to understand, to remember, and it is seen everywhere, especially in nature. Here is the sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

The rule is that a certain term in the series is the sum of the previous two. So for instance, term number 15 would we the sum of the 14th and 13th term.

To write that mathematically, we write F(n)=F(n-1)+F(n-2). This is also known as a recursive sequence, but that is a whole other lesson for another time.

An interesting pattern found in the Fibonacci sequence is that the ratio of F(n) and F(n-1) converges to 1.618, or the Golden Ratio. For instance 13/8=1.625, but 55/34=1.6176 which is much closer to the Golden Ratio.

Shell with squares on it. Each square has the square of a Fibonacci number. The squares increase in the Fibonacci sequence

The Fibonacci sequence can be seen in nature. One of the most famed examples is a spiral shell. Yes, a regular shell on a beach has the beautiful Fibonacci sequence etched into it:

The way the Fibonacci sequence is in the spiral, is that each square has the side equal to the next number in the Fibonacci sequence. For instance, as seen above, we start with two squares of side lengths 1 and 1, then side length 2, then 3, then 5, etc. Nature is truly wondrous!

Finally, one of my favorite places to find the Fibonacci sequence in is the Pascal's triangle. The Pascal's triangle (seen below) is a triangle, starting with the number 1 on top. Every row below has ones on the sides, and the middle terms are the sum of the two numbers above. What is also cool about the Pascal triangle is that the sum of every row is equal to a power of 2, while the sum of every diagonal is....A FIBONACCI NUMBER!!

An image of the Pascal's triangle. Starting with one circle on top with 1. Two circles under with two 1. Three circles below with 121. etc...

Shows how each diagonal can be added up to be a Fibonacci number (in the order of the Fibonacci sequence)

Look and Say sequence

Let's now talk about a sequence that is a favorite of mine. The Look and Say sequence. Here are the beginning few terms in it:

1, 11, 21, 1211, 111221, 312211, 13112221...

So, what is the rule? If the name didn't clue you in, try speaking the above terms out loud by digit; one, one one, two one, etc.

If that STILL didn't clue you in guess I'll just have to explain. The sequence relies on the way we say things. The first term is 1 but the amount of ones is one. So that is "ONE ONE". Thus the second term is 11. Now there are two ones in the second term so "TWO ONE" or 21. That is "ONE TWO and ONE ONE" or 1211, etc.

Some interesting properties of this sequence is the fact that whatever number you start with, that number will stay the end digit for ALL terms. For instance if we started with a 5, then the sequence would be as follows:

5, 15, 1115, 3115, 132115, 1113122115

Clearly the last digit stays a 5 throughout.

This is the wonderful look and say sequence. Try and write it out for your friends, and see them try to guess the rule!

Knights and Knaves

The Knights and Knaves puzzle name was created by Raymond Smullyan in his book What is the Name of This Book? The puzzles all follow the same format; characters either always tell the truth or always lie. The setting is an island inhabited by "Knights" or truth-tellers and "Knaves" or liars.

Now, many puzzles have evolved into introducing "alternators" or characters who alternate between telling the truth or lying, and "normals" who can choose. This increases the level of the difficulty of the puzzles, creating some monstrous mind-benders.

Most puzzles ask to determine a certain fact based on who said what. However, some puzzles also allow the reader to "ask" a question to determine a certain fact. We will examine both today.

Type 1: Determine a certain fact based on who said what.

This puzzle can be solved by creating tables or charts and doing casework. The casework is as follows: choose one character to be the "control". You will first solve the puzzle if he is lying, then if he is telling the truth. Only one of these times should work without contradictions and give you the right answer.

Seems easy, right? The casework is usually the easiest part, however with harder puzzles you can get up to three "controls"!! The MOST important part here is choosing the control. We will now go over two examples so you can understand this better.

1: You see two of your friends, John and Bill walking down a path. John says to you, "We are both knaves." Who is who?

Since John was the only one who said anything we will make him the control. If he is a knight we get a contradiction because he must be telling the truth. Thus, he is a knave who lied about them BOTH being knaves. Thus, John is a knave and Bill is a knight.

2: You see two of your friends, James and Bart walking down a path. James says, "We are the same kind." but Bart says, "We are of different kinds." Who is who?

In this scenario they are making contradictory statements, and so one must be a knight and one must be a knave. Since that is exactly what Bill said, Bill must be the knight, and John is the knave.

Type 2: "ask" a question to determine a certain fact

This puzzle can be solved by trying to find holes in the argument and then using what you know about the problem to try different questions.

1: Wikipedia

2: Gold/Silver/Bronze coins.

Primes

Primes are numbers that are divisible only by themselves and one, in other words they have 2 divisors. Prime numbers have been a constant area of wonder and ideas for mathematicians and our knowledge about them is still expanding.

One thing that we know about prime numbers is that there's infinitely many of them. This was proved a long time ago by Euclid. However, over time we have learned more and more about primes.

Primes are even found in nature! For instance, a breed of cicadas live most of their lives underground. They only emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Prime numbers!! Scientists believe that this prime-numbered cycles have evolved so that predators cannot naturally synchronize with these cycles and thus the cicadas are more safe!

Another thing we know about primes came recently to the board. As stated earlier, no one has ever figured out a pattern for prime numbers. When they appear, how they show up, etc. However, Yitang Zhang in 2013 proved that there is some number n (less than 70 million) that there are infinitely many pairs of consecutive primes exactly n apart. This number was brought down to 246 by James Maynard, and a Polymath Project organised by Terence Tao. Some recent mathematical discoveries for you all!