tag:blogger.com,1999:blog-19678732.post4175106988372447355..comments2023-07-31T10:39:53.182-05:00Comments on The Blog of the American Chesterton Society: Dr. Thursday's Thursday PostNancy C. Brownhttp://www.blogger.com/profile/06169395014931291729noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-19678732.post-15236152548051817522007-10-24T06:46:00.000-05:002007-10-24T06:46:00.000-05:00Oh, my. I did not really intend this go delve that...Oh, my. I did not really intend this go delve that deep into number theory and its delights... it's like a snack, it's tasty, and it makes you want to eat more, but it only goes so far... though God knows I rely on number theory at work, as most of us do, even if we don't really know it!<BR/><BR/>Ahem. Just a quick correction. You don't mean factorial. You mean the <I>sum</I> from one to n of the integers. Factorial is the <I>product</I>. <BR/><BR/>Now, there are whole realms of puzzles (actually, real problems) which have "infinite" answers. They often come up as such little thought puzzles as you indicated. One branch of such things is called "Diophantine Equations" in which all the numbers must be whole (integers). Your house-numbering puzzle is indeed such an equation, for it is the solution for m and k positive integers where<BR/><BR/>(m-1)m/2 = km+k(k+1)/2<BR/><BR/>using the non-recursive formula for the beer-can pyramid I gave in the original posting. (Jack lives at m, there are m+k houses.)<BR/><BR/>But it is rather clever of you to point out that this is INDEED a case of "squaring the triangle"... you ought to send that in. (Where, I do not know, but you ought to. Hee hee)<BR/><BR/>Yes, Zeno - and his paradox. This is of course the place where the famous rebuttal comes in:<BR/><BR/>Zeno posits: "Since (his argument about the infinite number of halfway points omitted), <I>motion is NOT possible</I>. Shall we discuss it?"<BR/>Scholastic rebuts: "<I>Solvitur ambulando</I>."<BR/><BR/>Hee hee. That is, he replies "Let it be solved <I>walking</I>." You can read the Latin as "while walking" or "by walking"... It is of course the case we shall see tomorrow, when we cross the infinite sea without going mad.... since (1) it's not infinite and (2) you're holding tight to a poetic view of things and of course (3) you have a sense of humor.<BR/><BR/>It will be fun; now I have a day's work ahead of me, and then I must prepare the posting... please have with you a good-size piece of plain white paper, a ruler, a pencil (and eraser), and preferably a compass (the circle-drawing kind, not the north-pointing kind)... That is, if you wish to try it yourself - butI'm NOT crossing any infinite sea without my trusty compass! hee hee<BR/><BR/>--Dr. ThursdayAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-19678732.post-34605702416599211042007-10-23T23:59:00.000-05:002007-10-23T23:59:00.000-05:00By the way ... "Jack's Address" problem is a probl...By the way ... "Jack's Address" problem is a problem of "squaring the triangle". <BR/><BR/>If a number factorial has a whole square root, then that square root is the answer to the address problem. (8! = 36 / Jack's street has 8 houses, and his is #6, the square root of 36. Likewise 49! = 1225. Jack's street has 49 houses, and his is #35, the square root of 1225). But the pattern of when a factorial has a whole square root is indecipherable. The dolls that are inside the other dolls are odd and appear at unusual intervals, they are SURPRISING.<BR/><BR/>A factorial with a whole square root is, indeed, a triangle that can become a square.Kevin O'Brienhttps://www.blogger.com/profile/12239185608038738884noreply@blogger.comtag:blogger.com,1999:blog-19678732.post-81826100200580029812007-10-23T23:36:00.000-05:002007-10-23T23:36:00.000-05:00OK, Thursday, I'm not sure what you're driving at....OK, Thursday, I'm not sure what you're driving at. Are you being recursive or merely discursive in your discourse?<BR/><BR/>C. S. Lewis pointed out that one of the big differences between men and women is that women often have no antecedents for their pronouns. How often my wife is in the other room and says to me, "What should I do with this?" I am usually tempted to reply, "Put it there."<BR/><BR/>There was a word problem in an Old Farmer's Almanac that used to keep me up nights. It went like this ... The addresses of the houses to the left of Jack's house total the addresses of the houses to the right of his house. What is Jack's address and how many houses are on his street?<BR/><BR/>I assumed that the writers of this puzzle wanted us to assume that the houses on Jack's street were all numbered with consecutive positive integers and were not all even on one side and all odd on the other; I assumed this just to simplify the problem. However, once the first answer, "Jack's house is #6 and there are 8 houses on the block [1 through 5 totalling 15 and 7 plus 8 totalling 15)" rolled out, it became obvious that there were, indeed, infinite answers to this problem. I once figured out the next two answers. I don't recall them, but it's something like, "Jack's house is 35 and there are 49 houses on the street," and "Jack's house is 1156 and there are 1260 houses on the street", or something like that.<BR/><BR/>These are of course factorial problems, or triangular / pyramid problems. But I couldn't for the life of me figure the pattern that could be used to determine the infinite answers. What was the formula?<BR/><BR/>That's the amazing thing about formulas or pronouns, they apply adequately to so many different things, and they fit the pattern!<BR/><BR/>And of course, there's Zeno's Paradox (I believe it's called), the conundrum that you can never get anywhere because to get from point A to point B you have to first hit A.5, and to get to A.5, you have to first get to A.25, and so on. Therefore there has to be a doll with no seam. If the recursion were limitless, we'd never be able to bridge the gap between particles and waves, between the pattern and the formula that contains it, between the shackled reason of the madman and the unfettered reason of the poet.<BR/><BR/>Is THIS what you're driving at????Kevin O'Brienhttps://www.blogger.com/profile/12239185608038738884noreply@blogger.comtag:blogger.com,1999:blog-19678732.post-71772670820915603682007-10-21T09:43:00.000-05:002007-10-21T09:43:00.000-05:00Bottles of shampoo often carry instructions ending...Bottles of shampoo often carry instructions ending with the words 'rinse and repeat.' Followed literally, these instructions would oblige the user to shampoo his hair over and over again without ever stopping. (A ploy by the marketing division to boost sales?)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-19678732.post-91141683575337247482007-10-19T06:50:00.000-05:002007-10-19T06:50:00.000-05:00Because the idea is so important to what comes nex...Because the idea is so important to what comes next, I will give yet another example: "How to eat soup recursively" (I'll assume you have your own spoon.)<BR/><BR/>* * * *<BR/><BR/>To Eat a Bowl of Soup:<BR/><BR/>You are given a bowl containing some soup.<BR/><BR/>If there is just one spoonful of soup in the bowl, swallow that. You are done.<BR/><BR/>Otherwise, take out ONE spoonful of soup, and swallow that. Then Eat the Bowl of Soup [that remains]. <BR/><BR/>* * * *<BR/><BR/>You see, the instructions USE the instructions.<BR/><BR/>--Dr. T.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-19678732.post-52209680559393628972007-10-19T06:35:00.000-05:002007-10-19T06:35:00.000-05:00Did you merely post twice, or is that a recursive ...Did you merely post twice, or is that a recursive example of a comment? Hee hee. Good one!<BR/><BR/>--Dr. T.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-19678732.post-85440600072179863952007-10-19T06:33:00.000-05:002007-10-19T06:33:00.000-05:00Thanks, Richard - yes, that's a nice slangy way of...Thanks, Richard - yes, that's a nice slangy way of putting it. There are many other ways of coming at the idea - "nesting" or repetition is used in certain forms of art or music, though usually of a "simple" character... otherwise the pattern would get lost. And for the moment I am trying to explain an idea, so that we can look at something quite a bit more complex, yet having a pattern. Concentrate on the dolls - keeping the idea that while you are unpacking them, you really do not know HOW MANY MORE there really are. Another point which is not so easy to tell about is the idea that when you open a doll you would find <I>two</I> inside, both with seams, and so on... again the idea is NOT to look at the repetition, but the scheme by which the pattern repeats (e.g. when a doll has a seam, you open it, and deal with what's inside.) That' all I want for now. Next time we'll take a real example you can try at home.<BR/><BR/>Another example in words, which I was going to use (but did not) in my book on Subsidiarity is the children's ditty "The House that Jack Built", where each successive verse contains the preceding ones (we're then going from little to big). Again note that the verse is made by sticking on clauses ("THat beat the dog, THAT barked at the cat, THAT chased the rat, THAT ate the malt") to an ever-growing sentence THAT Jack built. Hee hee.<BR/><BR/>But as I think of it, "deja vu all over again" is perhaps a handy "literary" parallel one might use to model recursion... sort of a verbal (or numerical) "Groundhog Day" - though that is far closer to another, rather different kind of computing problem (What we call "Newton's method" is one - er - approximation, hee hee.)<BR/><BR/>Some of this does just sound like a fancy form of repetition - which is in some sense all that recursion is. Remember that I am purposely omitting the REAL (theoretical) stuff. I do give it away in the line "P(she)=she+P(she-1)" but we are not going into that at all. The stuff that comes next week is more like the repetition of baking cookies, but a kind of nightmarish form, where each time you take a tray out of the oven, instead of having (say) a dozen finished cookies, you have 12 trays of unbaked cookies to be put back... <BR/><BR/>If you or some reader cares to concoct a "recursive" story or other literary work, I will be very interested in reading it... if I have time. Hee hee.<BR/><BR/>thanks, Richard!<BR/><BR/>--Dr. Thursday<BR/><BR/>PS I forgot! Actually there IS a recursive book, which is one of my favourites. It is Michael Ende's <I>The Neverending Story</I>. I cannot explain further without spoiling it - but you ought to read it, it's great. (N.B. but NOT the movies which are only loosely based on the book.)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-19678732.post-839193760462200482007-10-18T21:37:00.000-05:002007-10-18T21:37:00.000-05:00Dear Dr. Thursday,If my question makes sense maybe...Dear Dr. Thursday,<BR/><BR/>If my question makes sense maybe i understood some of what you said.<BR/><BR/>Is this subject in anyway analogical to "deja vu all over again, but somewhat attenuated with each "re-occurance"?"<BR/> <BR/>richard, looking for a linguistic liniment to unstrain my brain... <BR/>;-)richardhttps://www.blogger.com/profile/10529849654880738731noreply@blogger.comtag:blogger.com,1999:blog-19678732.post-83292942848960285092007-10-18T21:36:00.000-05:002007-10-18T21:36:00.000-05:00Dear Dr. Thursday,If my question makes sense maybe...Dear Dr. Thursday,<BR/><BR/>If my question makes sense maybe i understood some of what you said.<BR/><BR/>Is this subject in anyway analogical to "deja vu all over again, but somewhat attenuated with each "re-occurance"?"<BR/> <BR/>richard, looking for a linguistic liniment to unstrain my brain... <BR/>;-)Anonymousnoreply@blogger.com