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.
With me, it seems like the idea that to obtain a palindromic sum, you must first take a number and then reverse it and you will most likely end up with said sum after x amount of steps is pretty straight forward (but that could just be a case of hindsight bias). It really only seems logical that to add a number and its reverse over many steps will eventually lead to a palindromic sum but I guess the fact that there are 249 exceptions between 1 and 10,000 after 100 steps suggests otherwise. I do agree with you though that it is very interesting that mathematicians put all this time into finding these palindromes; however, it seemed like there was no clear reason as to what the significance of doing so was.
ReplyDeleteThe concept of a palindrome is something that does intrigue me, but finding out all the sums of a palindrome does seem like there is no significance in doing so. Furthermore, in examples of names I remembered that name Stanley Yelnats when reading the book "Holes" but i never really thought what the term was for that until now.
ReplyDeleteI've never really thought of palindromes in a math sense before. This chapter has taught me a lot about palindromes and what you can do with them.
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