The Beauty of Euler’s phi function

The recent conversation I had about number theory has brought it back into my awareness. In particular, the concept of beauty in numbers. I’m definitely not any kind of graphical designer or fashion expert but I do appreciate what I think of as beautiful, and there are certain areas of mathematics that are just beautiful.

But who am I to say what is beautiful? What really is beautiful? Rather than trying to talk about these things in terms of the abstract concept of beauty, I wanted to try to nail down some of the things I like about it.

In our early years we learn about shapes. Sometime later we learn about things like “regular polygons”. These are polygons where all sides have the same length. We also learn about stars, but the stars we learn to draw most often is the 5 point star that we can draw without lifting the pencil.

A natural question becomes are there other stars we can draw without lifting a pen? A 4 point star? a six point star? a seven point star?

Before we go into the ? function, lets make sure we’re on the same ground. We need to talk about common divisors.

Suppose we have two numbers, lets call them m and n. A common factor of m and n is a number that divides into both of them. For example a common divisor of 4 and 6 is 2 since 4 = 2 * 2 and 6 = 2 * 3. Two numbers are called relatively prime if their only common factor is 1. Remember that 1 is a factor of every number.

Euler’s ? function (called the totient function) of a number n is defined as the count of numbers less than n that are relatively prime to n.

n12345678910111213
?(n)112242646410412

To understand what’s going on in that table above, lets look at a number like 10 and ask what are the numbers relatively prime to 10?

n123456789
Factor 101*102*51*102*52*52*51*102*51_9
Factor n1*11*21*32*21*52*31*72*41*10
GCF121252121

So from this example we see that the numbers relatively prime to 10 are 1, 3, 7, and 9, so ?(10) = 4.

A nice property of Euler’s phi function is that for any n > 3, if ?(n) is 3 or greater, then we can draw a star with that many (n) points without lifting the pencil.

To do this, we first need to talk about modular arithmetic. If we have two numbers, a and b and want to add them modulo some number, written
(a + b) mod n
We take the remainder of (a + b) when this number is divided by n.

For example, if we wanted to calculate (3 + 5) mod 7 we would first compute (3 + 5) to get 8 and then realize that 8 = 7 * 1 + 1. This gives a remainder of 1, so (3 + 5) mod 7 would be congruent to 1.

If we are considering drawing an n pointed star, we can start with a number that is not 1 and is relatively prime to n and continually add that number to itself. What will happen is that because this number is relatively prime to n, it will visit every other number before returning to the number 0.

What is more is that there may be more than one n pointed star that we can draw. The number of stars is (?(n) – 2) / 2. So for 10, it will be (4 – 2) / 2 = 1. This can be seen below.

I wanted to allow users to begin to see more of this beauty, so I wrote a script showcasing it.

Pascal’s Triangle

I had a recent conversation with a friend who asked me “what makes number theory interesting?”. I loved the question, mainly because it gave me an opportunity to talk about math in a positive manner. More importantly though, it was an opportunity to talk about one of my favorite courses in mathematics (along with discrete mathematics and set theory). As much as the current day seems to focus on joining Number Theory with Cryptography, when I answered this question I wanted to make sure I didn’t go that route. Numbers are beautiful in their own right, and one of the things about Number Theory that was so interesting was simply the ability to look at all the different questions and patterns and properties of numbers discovered.

To answer this question, I started listing numbers to see if she noticed a pattern, but I did it with a “picture”.

.
..

….
…..
……

and I asked two questions

  • How many dots will go on the next line
    and
  • After each line how many dots have been drawn in total?

Lets answer these questions:

Dots# on this line# in total
.11
..23
36
….410
…..515

There were a lot of directions I could have taken this conversation next, but I decided to stay in the realm of triangles and discuss Pascal’s triangle. This is a triangle that begins with a 1 on the first row and each number on the rows beneath is the sum of the two cells above it, assuming that cells not present have a value of zero.

So the first five rows of this triangle are

1
11
121
1331
14641

This is an interesting and beautiful triangle because of just the number of patterns you can see in it.

  • Obviously there are ones on the outside cells of the triangle.
  • One layer in, we get what are called the Natural or Counting numbers (1, 2, 3, 4, 5, …) .
  • One layer in, we can start to see the list of numbers that I was showing my friend (1, 3, 6, 10, 15, …).

There are several other properties of this triangle and I wanted to allow users to begin to see them, so I wrote a script highlighting some of these patterns.

Set Partition Problems

This is my first post of 2019 and my first post in a while. There was one posted a few months ago, but not really geared towards the algorithms and learning focus of the site. I have been doing a lot of coding in my spare time, but honestly life has just gotten in the way. Its not a bad thing, but life is life and sometimes I have to prioritize the things. In particular, I have been having ideas and actually coding things up but the time it takes to clean up code, write a blog entry and finding a nice way to visualize these things has been something that I haven’t been able to really focus on as much as I’ve wanted to.

That said, I want to talk to you about the Set Partition Problem today. You can go to Wikipedia to get more information about this problem, but I will give you a brief introduction to it and then talk about two different approaches to it. The problem assumes that we are given as input a (multi)-set S. The reason we say it is a multi-set and not a simple set is because we can have the same element appear multiple times in the set. So if S1 = {2} and S2 = {2, 2}, then although as sets they are both equal to the set {2} = S1, as multi-sets allow for multiple instances of an element. The elements of S are assumed to be positive integers.

So given this multi-set S, we ask the question of can the elements of S be divided into two smaller multi-sets, C1 and C2 where

  • C1 [union] C2 = S
  • C1 [intersect] C2 = [empty set]
  • [Sigma]_[x in C1] = [Sigma]_[x in C2].C1 [union] C2 = S, C1 [intersect] C2 = [empty set], and [Sigma]_[x in C1] = [Sigma]_[x in C2]

The first two bullets above say that the sets C1 and C2 form a partition of S. The third bullet says that the sums of the elements in the two children multi-sets are equal.

This problem is known to be NP Complete. This means that it is one of the more difficult decision problems. Because of this finding an algorithm that solves this problem exactly will generally take a long running time. And finding an algorithm that runs quickly will more than likely be incorrect in some instances.

I will show you two approaches to this problem. One is based on Dynamic Programming (DP) and one is based on Greedy Algorithms. The DP version solves the problem exactly but has a slow running time. The greedy algorithm is fast but is not guaranteed to always give the correct answer.

Dynamic Programming is based on the principle of optimality. This says that in order to have a correct solution to the overall problem, we need optimal solutions to all subproblems of this problem. This is done by keeping track of a table which can be used for looking up these subproblems and the optimal solutions to these subproblems.

For this problem, we will build a table where the rows represent the final sums and columns represent subsets of the given multi-set (containing the first 0…n elements). The question we are repeatedly asking is “can we find a subset of the set in this column whose sum is exactly the given rowsum?” Here is an example of the DP algorithm on the multi-set {5, 6, 5, 6, 7}.

Greedy Algorithms are generally based on sorting the elements based on some principle and using that to try to answer the underlying question. The main problem with this approach is that it is very short sided because they do not look at the overall picture. Such algorithms are known for finding local optima that are not always globally optimal. The benefit to these problems though is that they are generally easier to code and easier to understand.

For the Set Partition Problem , the Greedy approach is to sort elements in descending order. Once this is done, the goal is to keep two subsets while iterating through the array, adding the element to the smaller of the two sets whenever possible.

For more examples, check out Set Partition Problems

Binary Puzzles

As you can probably tell, I’m a big fan of puzzles. On one hand you can say that a good puzzle is nothing but particular instance of a complex problem that we’re being asked to solve. What exactly makes a problem complex though?

To a large extent that depends on the person playing the puzzles. Different puzzles are based on different concepts and meant to highlight different concepts. Some puzzles really focus on dynamic programming like the Triangle Sum Puzzles or the Unidirectional TSP Puzzles.

Other puzzles are based on more complicated problems, in many cases instances of NP-complete problems. Unlike the puzzles mentioned above, there is generally no known optimal strategy for solving these puzzles quickly. Some basic examples of these are ones like Independent Set Puzzles, which just give a random (small) instance of the problem and ask users to solve it. Most approaches involve simply using logical deduction to reduce the number of possible choices until a “guess” must be made and then implementing some form of backtracking solution (which is not guessing since you can form a logical conclusion that if the guess you made were true, you reach either (a) a violation of the rules or (b) a completed puzzle).

One day a few months back i was driving home from work and traffic was so bad that i decided to stop at the store. While browsing the books, I noticed a puzzle collection. Among the puzzles I found in that book were the Range Puzzles I posted about earlier. However I also found binary puzzles.

Filled Binary puzzles are based on three simple rules
1. No the adjacent cells in any row or column can contain the same value (so no 000 or 111 in any row or column).
2. Every row must have the same number of zeros and ones.
3. Each row and column must be unique.

There is a paper from 2013 stating that Binary Puzzles are NP Complete. There is another paper that discusses strategies involved in Solving a Binary Puzzle

Once I finished the puzzles in that book the question quickly became (as it always does) where can I get more. I began writing a generator for these puzzles and finished it earlier this year. Now i want to share it with you. You can visit the examples section to play those games at Binary Puzzles.

Below I will go over a sample puzzle and how I go about solving it. First lets look at a 6 by 6 puzzle with some hints given:

 01   
0 10  
11 0  
 1  0 
   0  
1  1 0

We look at this table and can first look for locations where we have a “forced move”. An obvious choice for these moves wold be three adjacent cells in the same row or column where two have the same value. A second choice is that when we see that a row or column has the correct number of zeros or ones, the remaining cells in that row or column must have the opposite value.

So in the above puzzle, we can see that the value in cells (2, 2) and (2, 5) must also be a 0 because cells (2, 3) and (2, 4) are both 1. Now we see that column 2 has 5 of its 6 necessary values, and three 0’s. So the last value in this column (2, 6) must be a 1 in order for there to be an equal number of 0s and 1s.

For some easier puzzles these first two move types will get you far enough to completely fill in all the cells. For more advanced puzzles though, this may require a little more thorough analysis. 

As always, check it out and let me know what you think. 

Range Puzzles

Range Puzzles

I have always liked puzzles. I really enjoy discovering a new puzzle. Sometimes when I have discovered a new puzzle, I enjoy it so much that I can’t seem to do enough of them. In such cases, I quickly run out of these puzzles to do and find myself looking for either a new puzzle or a way to generate them on my own.

Such was the case when I discovered the “Range Puzzles”. The rules are simple:

Every cell is marked either blue or gray
No two gray cells can be next to one another
The grid must be a connected (i.e. there is always a path from every cell to every other cell using horizontal and vertical connections.
Some cells have a number inside. This indicates the number of cells that can be viewed (horizontally and vertically in both directions) by this cell, including the cell itself.

To try out these puzzles, visit Range Puzzles at LEARNINGlover.com