Chapter 43 The Unexpected Hanging

Chapter 43 mainly focused on situations where we invalidate our own reasoning. Michael Scriven, a professor of the logic of science, starts off the chapter by giving an example of how “The unexpected hanging” paradox works.  The puzzle starts off with a man who is condemned to be hung.  The man was sentenced on a Saturday and the judge stated that the hanging will take place at noon but on one of the seven days of next week, without the prisoner knowing.  It will become a surprise for the prisoner who is getting hung. The prisoner and his lawyer discuss about how the sentence cannot be carried out because the judge’s order seems self-refuting.  The lawyer explains that the judge cannot hang the prisoner on Saturday because it is already the last day of the week and that he will still be alive on Friday afternoon of next week. Since he will still be alive on Friday afternoon, Thursday cannot be the day of the hanging either because if he is alive on Wednesday afternoon, he will know already that Thursday is the day. 

The prisoner continues to convince himself and be confident that he will not be hung at all next week because he will know ahead of time.  But that is the problem, the paradox of “The unexpected hanging” is still a controversial topic because there are no correct deductions.  It is easy to guess what will happen next, but what assures you that it WILL actually happen?  You may guess correctly, but it does not always result being true. At the end, the hanging does occur. The prisoner assumed that his prediction will be fulfilled, but it was falsified.

“Future events can be known to be a true prediction by one person but not known to be true by another until after the event.” The judge knew already what day the hanging was going to occur, but the prisoner did not. It was ultimately up to the judge to decide what day of the week the hanging was going to be happen, without the prisoner knowing when his last day of life was going to be.  The judge was clear on keeping his word of the sentence happening the following week but the prisoners’ expectation was completely different. 

Even though Chapter 43 did not deal with a lot of Mathematics, it still had me thinking on various sections. It came to my surprise that there are many examples of how philosophers have not fully concluded on how to resolve these kinds of paradoxes.  There are no actual answers on how we expect things to happen. Guessing is one way of knowing what will happen next but it is not always assured right.  I learned that we cannot come to a conclusion of how unexpected things and situations will occur.  There are no correct deductions on guessing what day there will be a pop quiz or a fire drill. That will be a surprise that we are not expecting. I really enjoyed reading this chapter! Ps-Don't procrastinate!

Chapter 41 had two main topics, induction and probability. I had a tough time wrapping my head around most of the topics throughout the chapter, but a couple were fairly easy to understand. The very first topic that was talking about was the patterned carpet. The patterns on the carpet were compared to the real world, since the patterns on the carpet are always changing. Just like how the patterns of the real world are not constant. After reading the first couple of paragraphs I was pretty confused, but this quote helped me understand the topic a lot better, "the patterns of the real world, as distinct from this imaginary one, are constantly changing, like a carpet that is rolling up at one end while it is unraveling at the other end."

 Like in our class, there were a lot of relevance to playing cards. The mathematicians used playing cards to prove some hypothesis'. One of the interesting ones was the one about cards with blue and green backs. One hypothesis stated that if you have three face cards with green backs, and two face cards with blue backs, then you are obviously more likely to pull a face card if you pick one with a green back. Another hypothesis stated that if you have three red cards with green backs, and to red cards with blue backs, then you are more likely to pull a red card if you pick a card with a green back. Both of these hypothesis were proven true. To take it a step further, you would think that if you wanted to pull and red face card, you would pick a card with a green back. However with the information the book gives us, this is not the case. There ends up being only one red face card with a green back, while there are two red face cards with blue backs, thus disproving ones initial thought. The chapter goes on with a bunch of different probability tricks involving, a women picking her husband, a man picking out with pie he should eat, pulling poker chips out of a hat, and another using spinners. 

I thought this chapter had some interesting, but confusing topics. What I thought was most surprising about the chapter was the part about the spinners. You can refer to the chapter to wee what the spinners look life. If you match up the spinner head to head to see which one get spinner gets the high number, then spinner A beats spinner B and spinner C, and spinner B beats spinner C. After looking at this data, you would think that spinner C is the weakest out of the three in every circumstance. However if you were playing against two other people to see who would get the number and you had your choice of spinner, you should pick spinner C. It turns out after some testing that spinner C is the best choice, and spinner A is actually the worst. I had to re-read this a couple times because at first I didn't think it made any sense. I found myself doing that for most of the chapter, because it was a lot of information to take in. 

Chapter 3 Palindromes: Words and Numbers

In Chapter 3 in “The Colossal Book of Mathematics,” Martin Gardner seeks to enlighten its readers on the topic of palindromes. This section dives into many aspects of the material. It varies from palindromes found in numbers, for example, one starts with any positive integer, then reverses it and adds the two numbers together, continuing this process until a palindromic sum is obtained (68 + 86 = 154 + 451 = 605 + 506 = 1,111). Furthermore, it talks about palindromes, whether they be unintentional or intentional, found in sentences and names like Yreka Bakery in Yreka, California. Thus, this chapter aims to uncover patterns not easily seen by first glance.

“A Palindrome is usually defined as a word, sentence, or set of sentences that spell the same backward and forward. The term is also applied to integers that are unchanged when they are reversed” (Gardner, 23). This chapter gives a plethora of illustrations and sources of work that have gone into the research of palindromes. For instance, Charles W. Trigg found that by using the above steps I stated in my intro, “he found 249 integers smaller than 10,000 that failed to generate a palindrome after 100 steps” (Gardner, 23). However, aside from these 249 exceptions, this conjecture works with all other numbers less than 10,000 to create a palindrome in 24 steps or less. What also is an astonishing find is that, according to the book, there are an infinite amount of palindromic squares, most of which seem to have square roots that are also palindromes. Cubic palindromes, likewise, are extremely rich in palindromes. Aside from numeric palindromes, additionally, there are palindromes in language. Interestingly, there are “no common English words of more than seven letter [that] are palindromic” (Gardner, 26). Cases of English palindromes are reviver, deified, and rotator. A very easy way to make a palindrome as long as you want is by simply following this form: ““’______,’ sides reversed, is ‘______.’”” This was suggested by Leigh Mercer, a British palindromist. What you put in the first blank is any sequence of letters and then the reverse in the second blank.  Palindromes don’t necessarily need to be in letter units either. They can be in word units, for instance, ““You can cage a swallow, can’t you, but you can’t swallow a cage, can you?”” (Gardner, 27)

I thought the chapter was really fascinating. What I thought was interesting was how much time that must have been put in to discover these palindromes. For example, Harry J. Saal used the configuration of adding the sum of any positive integer and its reverse counterpart with the number 196. He carried this number to 237,310 steps, and still couldn’t find a palindromic sum. These mathematicians have to be really dedicated into proving their hypotheses, which I admire. Furthermore, there are ideas yet to be proved, like how it has yet to be verified that there is an infinite number of palindromic prime numbers e.g. 101, 131, 151. Doing a little research of my own, the highest prime number found with a base 10 is 10^320236 + 10^160118 + (137×10^160119 + 731×10^159275) × (10^843 − 1)/999 + 1. This just shows that there is a lot more to be proven in this topic  A surprising fact that I thought was cool was how palindromes have been intertwined into our culture as well. There are competitions held to see who can create the best palindrome and there are names of towns that are palindromes like Adaven, Nevada. One of the first palindromes I experienced as a kid was the name of the main protagonist Stanley Yelnats from the book “Holes.” In all, this chapter is fairly easy to understand, but at times the math can be a challenge to follow. Getting passed the technical details of palindromes, this chapter provides great insight on an engaging topic.

Chapter 26 Supertasks

Chapter twenty-six is an interesting chapter. It has many concepts and within these different concepts there are many rules which lead to make these concepts true. Throughout this chapter the concepts that were being taught were, “finite sets”, “subsets”, “null set”, “transfinite numbers”, “cardinal number”, “aleph-null”, and “Zeno’s Paradoxes”. These are many different concepts yet they are all relatable in a sense, and the way these concepts relate is this theory of “infinity”.

    Finite sets are generated by x elements. And the way subsets are generated are by the elements in the finite set. For example, “a set of three elements, ABC, has 2^3=8 subsets: ABC, AB, BC, AC, A, B, C, and the null set. This is in a way related to what we were learning in class with the different combinations we had on the board. The big difference though is the fact that there is a null set. A null set is defined to be nothing essentially being 0 in a sense. In that it will always be capable of being used in all sets. What I had also gathered from the subsets is that an element cannot be matched with itself because it cannot be included within its own subset. Definition of a transfinite number is that, “it is the number of subsets of n-must be a higher order of infinity than n.” The term “aleph-null” is typically used for the lowest transfinite number. “It is the cardinal number of the set of all integers, and for that reason is often called a countable infinity. Any set that can be matched one to one with the counting number, such as the set of integral fractions, is said to be a countable or aleph-null set.” The last term, “Zeno’s Paradoxes.” Both terms seek to show this continued pattern that is shown through the Zeno’s runner theorem. What takes place in this theorem is that a runner half the distance in half a minute, a fourth of the distance in a fourth of the time, then an eighth of the distance in an eighth of the time, and so on until he has reached a total of one minute in which then means he has reached the end point. Yet if this were to happen this means that the runner were to have to run at the same pace throughout the entire minute until he has reached the final point.

    This chapter was very interesting but also very confusing in which made it frustrating to read at moments. There were many concepts which made me have to re-read at moments because of there being multiple concepts it caused there to be some confusion when matching the definitions to their concepts. Plus when first reading this chapter it was first hard to understand what i was being taught because I had never heard about a “Null set”, but I kept re-reading and asked for an explanation which then led me to find out that this chapter is interesting and in some ways relates to what we are being taught in class.

Chapter 37: Harary's Generalized Ticktacktoe

This chapter gives a different approach to play one of the most well known games in existence: Ticktacktoe. This generalized way that the chapter details about was first created by Frank Harary. The concept of ticktacktoe was simple: two people play on a 3x3 grid taking turns placing their marks on the grid. The objective was to line up three of your symbols (X or O) either vertically, horizontally, or diagonally. Older variations date back to the ancient times where two players are given three counters and would move them around the grid until they matched their three counters in a row. But Harary devised a new approach to the game.

He wanted to find out two things: What is the smallest in which a player can force a win? and "In how few moves can the first player win?". It introduces the concept of cell polyominoes. An example of a polyomino is a domino, where it has two squares together. That is classified as two cells. This variation of the game can be played on any polyomino, whether it is one cell, three cells, four cells etc. The shape varies depending on how many cells there are. To make it simple to understand, think of it like Tetris where you can change the shape of the blocks. That applies to this game in that you can use a different board for each game to play by changing the shape.

In order to know which board you are using, Harary coined the term "animal" cells where each board is given a name. Now the first task is to determine the "animal's" board number which is the length of the side of the smallest square in which the first player can, by playing the best strategy, force a win. If there is a number, the animal is called a "winner" and if not, then it is called a "loser". If two people play a board that is a "loser" then the second player can always force a draw but can never win. Once you know the board number, then you find the fewest moves it takes the first player to win. On a one cell board its one move and on a two cell board its always two moves. For three cell boards and up, it gets tricky to analyze it. The chapter itself illustrates the boards with the board number and moves for each as well as the "losers" of the boards. As in regular Ticktacktoe, you play marking the cells with square matrixes with crosses and noughts (X's and O's). It gets more complicated in determining which ones are winners and which ones are losers and the chapter goes into further details on that but what I wanted to show you is the basic setup in how to play this generalized way of Ticktacktoe.

To me, this chapter at first glance before reading it, I thought it would be something simple to understand because it was ticktacktoe but after reading it, I did come across some parts that were a bit confusing. It was interesting in how he developed a different way to play the game, but trying to understand it took me a while and I still have questions about it. I wanted to know when playing on a board how can you tell that the animal is a "loser"? What surprised me is how this simple game can be transformed into a strategy to be able to win on different boards and to know when you are unable to. I'm sure everyone has played Ticktacktoe before but this new approach can be difficult to understand at first, but after rereading it a few times you can get a better understanding of it.

Chapter 8, The Wonders of a Planiverse.

This chapter was very interesting, and thought provoking. Although the topic appears to be mundane to begin with, the ideas towards the middle of this chapter become more and more intriguing. I feel that this chapter stands to show how relevant seemingly irrelevant thoughts can be. To prove this point it is necessary to recap certain parts of the chapter. When the chapter begins, it explains the basics of what is being discussed, which is a two dimensional reality. This is said to be impossible, for it would pose many issues and questions that we simply do not have the answers to. The book says that one issue with studying a two dimensional world is that we have no axioms, and we can only perform gedanken, thought, experiments. Even though the only path to understanding a planiverse that we have is through gedanken experiments, a lot can still be accomplished. Within this chapter, many properties of this planiverse are discussed, and how they relate to our steriverse. There are countless properties of the planiverse that are different than our steriverse. A lot of this has to do with nature and physics. For example, the earth would revolve around the sun, similar to what our earth does, but in the planiverse the path of the earth would be either a perfect circle, or an incomplete ellipse. Another example is that water and wind could not move around objects as they do in our reality. This would change many things, such as the geology of the planiverse, also rain would be way more dangerous there than it is here. The inhabitants of the planiverse is another topic of debate. These inhabitants will need to pass each other at some point, but in order to do so they will need to jump over each other to get by. This reminded me of the older 2D video games such as Mario. Another interesting topic discussed in this chapter was board games. The book said that the inhabitants of the planiverse would be able to enjoy boardgames similar to those played in the steriverse. Although they could play games such as checkers and chess, they were quite simple and usually had a preconceived outcome. But Linear Go could become quite interesting if the number of spaces was extended dramatically. Although these games are possible, they would need different rules and would require a lot of space. The amount of space is another issue that the planiverse would need to solve. Inhabitants of the planiverse would need to live underground to save space for necessary vegetation. This means that roads, houses (above ground), and cities must be kept at a minimum, if at all. The only solution would be to move everything underground. Even though the planiverse comes with many issues, there is a good side to it. Due to the constraints it puts on building machines and simple objects, it gives us a new way of viewing the use of space. An example from the chapter is the lock. We could create a two dimensional, with a very small width, that would save space, and the keys would not have to turn. Even though this is possible, it does not mean it is probable. The lock is a great idea, but is very expensive. So, the planiverse is great at giving us a different outlook on problems, but it does not give us realistic solutions.

Chapter 13: Hypercubes

Chapter thirteen of The Colossal Book of Mathematics is, in my opinion, one of the more intriguing topics of math. Hyper cubes and the different dimensions these cubes can exist in was the topic of discussion throughout this chapter. Hyper cubes can range from the simplest of objects to some of the most brain twisting structures your eyes have every laid sight on.

             Humans have a three dimensional view on the world and can literally on see to the third dimension of an object or space. Hyper cubes can extend in dimensions far beyond what the naked eye can perceive and what our brains can manipulate. Even when we are claiming to see something in a four-dimensional state, we are not. Because of our limited perception humans have contemplated for one hundred plus years whether or not these dimensions exist and if they do how can we stretch object to these dimensions in real life? In order to achieve this we must first start with a point and move it one unit in a straight line. Now we must take that line and move the line one unit perpendicular to the line. We now have a square with four points in a two-dimensional view. Now we must take the four points on our unit square and shift them perpendicular to all three axes. This very shift is what brings us into the visually boggling fourth dimension or 4-space. This shape is called a tesseract, which has four perpendicular edges meeting at every corner. If you try and draw this peculiar shape you may find yourself erasing quite a few times trying to correct the shape. For us to understand why this shape has so many different points, lines, squares, cubes and tesseracts we must first look at a simple formula that can help us calculate these exact numbers for each nSpace. The formula of (2x+1)^n is the base formula to calculate all the analogous of the cube in various dimensions. If you keep multiplying the formula by itself you will see a pattern start to develop.

            The most interesting section of chapter thirteen is unfolding and cutting certain edges of a hyper cube in order to form other shapes of see them shape in a different dimension. Salvador Dali’s Corpus Hypercubus is a great physical example of an unfolded tesseract in 4-space. The reason I like this example the most out of all the examples is symbolizes the limited vision of the human eye and mind.

            It came to my surprise that after reading the chapter and watching a couple of videos of how hyper cubes and multi-dimensional shapes work that this topic is as complicated as I first thought it would be. I was honestly frightened by the title “Hypercubes” but after reading and understanding that it is a shape simply shifted along axes in different dimensions my mind was able to understand it and try and figure out how to stretch images into these unexplored dimensions.

            I believe every who reads about hypercubes and tesseracts will be overwhelmed at first but after reading about them and seeing how these shapes are formed through the stretching and shifting process will definitely give you a better grasp on the topic than you had before.