 You have an envelope and in the envelope there are three different puzzles Would you take out the four pieces that go that will make this square and there's two puzzles that I would like you to attempt right now All right number one is Can you turn these four pieces from a square into a rectangle that is not a square can you arrange the four Pieces into a rectangle. That's not a square. I'm going to give you the answer to puzzle number one right now But puzzle number two Is a curiosity There's the there's the five by 13 rectangle that you're trying to make but wait a minute Well, it's not yellow that's true But I gave you a square That was eight by eight that has an area of 64 The rectangle has an area of 65 So where did you get the extra? Where'd you get the extra square? It's a puzzle. It's a puzzle Where did you get the extra square? now This is a puzzle that I've played with for many many many years and It's only recently that I learned about some connections that are Bringing me to talk about this puzzle with you 64 equals 65 There's no use trying one can't believe impossible things. I dare say you haven't had much practice when I was your age I always did it for half an hour a day Why sometimes I've believed as many as six impossible things before breakfast In a little catalog of what's At the University of Texas Austin in the forward teacher Turner out and I just saw this about two days ago When one works with Alice marvelous things happen You should have a eight and a half by eleven piece of paper as well, and then you have this puzzle on the back Mr. Dodson drew this when he was a very young man for his family that went into the family newsletter called mishmash and I will leave this as a challenge for you. Can you get out of the forest? Warren Weaver wrote I wish to show how nonsense kept showing up kept twinkling through his mathematics Here's another maze that he drew My goal and I mean me Stewart. I want to show you how how the mathematics keeps coming through the nonsense. I want to thank Martin Gardner because Martin Gardner for one thing I guess I didn't know he had you know was one of the founders of this of this society I Also want to thank Martin Gardner because last year at the Gathering for Gardner after giving a six-minute version of this talk I'm going through my slides. I'm sitting at the table, and there's a man peeking over my shoulder Who hadn't been at my talk and he says to me? Why do you have all those slides of Dodgson's manuscripts? Who are you? And it was Mark Bernstein Mark Bernstein You know noticed the purple ink on my on my laptop and knew that I had something to do here And we started talking and next thing I know I'm standing up here. I Also need to tell you a little bit about Jerry Slocum if you don't know Jerry Jerry is the founder of the International Puzzle Party When I started doing research on puzzles. I Wrote to him and he invited me to his backyard museum his backyard puzzle museum and at the end of the day He told me that I had earned the right to apply for an invitation to attend the International Puzzle Party and Again that has brought different worlds together for me And if you'll notice he's standing by a door and there's another door to hit the right There were two doors to get into the museum. Both of them were puzzles You have to you have to solve what's on the on the door just to get into the museum Jerry's job as a mechanical engineer was to design cockpits for fighter planes Talk about a puzzle. He told me that in World War one most of the casualties from from aircraft were due to pilot error and So designing a cockpit to minimize that was was his career in the ultimate puzzle to try to save lives I got to thank Chris Morgan for this too because at the International Puzzle Party Chris was giving a talk two years ago about his new book games puzzles and related pieces and He showed this slide from Lewis Carroll's notes And he showed the puzzle that you have in front of you And he said but it wasn't a carol original He didn't invent it himself. So it's not in the book and he went on to the other stuff that was in the book And I'm going wait a minute. I want to know more about that puzzle and so Chris Directed me to some places for me to go look and so thank you Chris One of the things I learned in fact, it's not a carol original Forgive me if I keep going back and forth between Dodson and Carol and sometimes I'm not sure who I'm referring to But before those notes were drawn this puzzle was available commercially a wooden version Not only can you arrange it to make an increase but even the puzzle that you have in your hand? You can also arrange it to decrease the area to 63 We'll get back to this puzzle later. I promise I Found it in 1877 it had showed up in the messenger of mathematics journal. I Found that in 1868 it showed up in a German journal. I Found that in 1858 Sam Lloyd Who is considered the premier puzzle master of the late 1800s? He claimed that he had introduced it in 1858, but Sam Lloyd also had a reputation That That we have found no evidence that he really did it But one thing Sam Lloyd did invent was the get off the earth puzzle So let me show you just what this means the get off the earth puzzle. So I'm going to move to Let's make sure let's move to the document camera so you can see this puzzle okay, and There are 13 men Around the around the globe and if you rotate it Let's see right now. There are right now. I've got a point in the right direction right now They're a 13 if I rotate it this way Now they're a 12 Well This is what makes this puzzle so interesting and I should say that there were millions of these produced They got used for all sorts of advertising purposes in particular one of the more interesting sources of advertising was that Was it got used by William McKinley in the 1896 election where they would put the Republican platform on the back and put this on the front Now what does this imply? Okay, this was at a time when the working man's party and in California their official slogan was the Chinese must go So the Republican Party wasn't coming out explicitly and saying it but by handing this out. Oh my goodness 1774 I find in a book called rational Recreations in which the principles of numbers and natural philosophy are closely and copiously elucidated by a series of easy entertaining interesting experiments I Love the title. I find that The first edition in which came out in 1774 shows a three by ten. I'm really looking on the left It shows a three by ten that when you cut it up into four pieces you get two rectangles That show a gain of eight units of area Well, this was a mistake Actually, if you cut it up what you really get is what shows up in the fourth edition Where you get a four by five and a two by six? You gain two units of area not eight now. What makes this very interesting is that Hooper 1774 to 1794 but at the same time Gio writes a French recreational math book that had the same two mistakes and And Gio's came out ahead of Hooper I think we're seeing some plagiarism happening and Hooper wasn't very it took Hooper four additions to get it fixed So don't took Gio only two additions to get it fixed 1545 Serlio Says a man has a table. It's three. It's it's three feet by ten feet But he wants a table that's four feet by seven feet So let's cut the table on the diagonal into two triangles just like this So we have an area of 30 Let's slide that as so So there's our four by seven table and the two triangles add up to an area of three well What we don't know about this one obviously 30 does not equal 31 But what we don't know is did did Serlio do this intentionally or not this wasn't in a recreational mathematics book Okay, we don't know But yet this is the first This is the first thing this is the earliest reference to this kind of a geometric vanish puzzle Where things where the area seemingly disappears or or appears Lloyd patented this idea 1896 Didn't do him much good Because we have so many puzzles like this and in fact in 1889 Fred Howard patented a version of the puzzle as well I Didn't put this one in your packet because it would have taken too many too much to cut up this many for everybody was Okay, but this is one of the puzzles that's on the back tables And you're welcome to help yourself to what's on the back table and there's scissors if you want to try cutting these up Before you before the end of the day if you choose in this case you have a three by eight and When you cut it up you can make two things you can cut it up and arrange it into a five by five square With or without a hole now without the hole, and that's two puzzles without the hole 24 equals 24 Excuse me with the hole 24 equals 24, but then when you arrange it without the hole, where did it come from? Where did it come from? political Messages have not gone away. Here's the get-off-the-earth puzzle in the mid 1950s in Esquire magazine And I know this is I tried to enlarge it as much as I could but you probably can't read it It says swivel the arrow to northeast count the red Chinese Now swivel to northwest and count again. What happened to the 13th man? Why can't the American government solve its recognition problems this neatly? I'll I will leave that be 2006 I'll read to you and then this is they are quoting Hillary Clinton here One international war criminal always has to look over his shoulder if you think he can't be found. Oh someone been lying That Joker George has conveniently forgotten him, but it doesn't take a crystal ball to see the future So you have George Bush juggling? you have Hillary Clinton with a crystal ball and you have Osama bin Laden and when you rotate it Hillary is holding Bin Laden's head in her hands and George Bush is now juggling for the dog and there's one person gone. We've gone from eight people down to seven Same kind of puzzle Martin Gardner. I think my first introduction to this puzzle was many many many years ago In the gotcha book and the vanishing leprechauns where 15 leprechauns can be rearranged into 14 leprechauns And I thought about giving you that one. It's on the back table But then I saw one on this one. I figured let's let's go with the Cheshire cat Why don't you take it out of the envelope? What color is the Cheshire cat purple? How appropriate? So the question is Which one is the Cheshire cat? So put the three pieces together This I think this makes more sense if you can see it right in front of you instead of up on the screen But I'll do it with you up here on the screen All you have to do is take the two pieces that are on the top and reverse them just reverse them now There's a bit of a clue as to what's going on here when you look at this Because with the leprechaun puzzle The hints that are given it says which one vanishes. Where does he go and When he comes back? Where has he been? So let me ask you that question Which one vanished? It seems pretty obvious. We'd say the one in the middle vanished But then I'm gonna ask the question. Where did he go? So I put a third puzzle in your envelope. Oh But I better warn you this was Duttony was a dude was a contemporary I wonder if if Dodson knew Duttony, you know Chris did they know each other? That's certainly It certainly would make sense that they would know you know, they were certainly contemporaries Now according to Martin Gardner, somebody went to jail for doing this So please don't do this with real money anymore. In fact the government has has chosen to counter the counterfeiters You have nine hundred dollar bills They're already cut have you found them? I think they're the I think what color is those those are green of course And so if you cut your money and rearrange them like this well here Let me do this you notice that I've got fractions next to each one of them They all add up to one When we slide it down now we have ten hundred dollar bills Have you ever have you ever had to tape a hundred dollar or a piece of currency back together? It happens. Have you ever had a piece of currency that was had a piece torn off and was missing? That happens What does it take to be legal? What does it take to be legal? And it's not more than 50% you don't have to have more than percent more than 50% I have used a subtitle on this talk in the past It's called why us currency has the serial number printed twice on each piece of currency All right, the serial numbers up there twice on each one of them check your bills when you do this rearrangement Yes, you gain an extra bill But those in the middle let's see you don't have to have 50% But you have to have both serial numbers and they better match Okay so can we explain The Cheshire cat and the leprechauns What are these fractions add up to now? so in other words Which hundred dollar bill disappeared? That's a misleading question, isn't it? They all Well, they're all different aren't they? They're all different. It's all been rearranged and it's all hidden very cleverly Here it's pretty clear that they're all a little bit shorter than they used to be in fact They're all one-tenth shorter than they used to be What I love about this I teach mathematics up at Humboldt State University I work with future elementary teachers what I love about this is I can use elementary mathematics to explain this puzzle and To me that's you know, that's why I pursue this. That's where that's why I can make an X why I can make this This is part of my research. This is why I do this is because I'm bringing this stuff to the classroom How can I make mathematics interesting for my students and I'm gonna get back to Dodson on that one When Jerry invited me to his house now, he's got 80,000 puzzles in this museum. He said bring me a new one I haven't seen before I'm going great. How am I gonna do that? Well in the middle of the night The Reverend Dodson was also an insomniac. I gave this But this email that you see on the left was sent to me at five o'clock in the morning Right after I had done this talk at Western Michigan University and the man who sent it to me said I've been thinking about this I've been laying awake at night and You see these six men here Which one disappeared They all did didn't they? And I think that this one really demonstrates what I love about this is it demonstrates the paradox and helps to see through what's going on so Chris pointed me towards the parish collection at Princeton University, and I had the the wonderful Opportunity to spend a day there. I need to go back. I have to spend more time I feel like I'm just beginning Well, you know what I'm doing and you know beginning of what you're seeing here is only the beginning of this I found thousands of pages of mathematical manuscripts in boxes some of it very unorganized actually Some of them just scribbles of computations all in purple ink Some of it was absolutely gorgeous. There's a proof that the of trisecting an angle or Should I say That you know his explanation and his is not done what what he did here was not strictly compass and straight edge for those of you that Know more about this, but he did this when he was 12 years old When I went to parish the parish collection at Princeton Stan Isaacs who I know many of you know And thank you Stan for pointing me and getting me to look for this Stan was looking for a particular billiard ball problem So every time I found one I took a took a photo of it for Stan It says find the path of a billiard ball in a cube Which goes on the same path forever and strikes all six sides gravity neglected At the top of the page it says I thought this partly out and wrote out the result on another occasion Sometimes he would write at the top of the page not thought out Here the goal is to have a billiard ball travel around a non rectangular table Here's a third one Where he says a billiard ball A billiard table is nine feet four inches long And it is Let's see it is kind of read it better. It is nine feet four inches long. It is 10 feet wide And the goal is to it starts in the center and you have to Let's see the goal is to hit all it has to hit 16 cushions It and then hit into a corner pocket and it has to travel a total of 75 feet 10 inches What you see on the right is Dodson's solution to this Some beautiful mathematics here using using reflection symmetry the understanding that that all every one of these tables is just a mirror reflection And but he's laid it all out in one straight line I'd be happy to make these slides available if you're interested and want to get a closer look at these I found a fourth pool table billiard ball problem in the pillow problem in the pillow tales book A triangular billiard table A point is given A point is given by tri-linear coordinates a ball strikes all three sides and returns to the starting point find the Find the point where the ball strikes the second side Again many many problems like this I found these notes. I love this one. Okay, it says on the bottom Can there be traced in one line each or if not in how many Going by the root of always Crossing to a new oval and every point of contact. Basically what he's asking Can you draw this without picking up your pencil? That's your goal. Let me read a letter Or no, this is not a letter But this is what you see on the right Again, the goal on the left is to draw it without picking up your pencil 40 years after let's see After he died let's see one when his nephew stewart collingwood was writing his biography He got a lot of feedbacks from many of dodson's young friends She writes we met for the first time in the forberry gardens He was I believe waiting for a train. I was playing with my brothers and sisters in the gardens I remember his taking me on his knee And showing me puzzles the puzzle this puzzle was by the way a great favorite of his The problem is to draw three interlaced squares without going over the same lines twice or taking the pen off the paper Which is so thoroughly characteristic of him in its quaint manner So two very the two puzzles are very much the same And it made me think about this problem Which in the 1700s the people of konigsberg prussia they have They have two islands Seven bridges that connect the two islands with the mainland And one of the things that they like to do would be go to go for a walk And what they wanted to do was cross all seven bridges Without crossing any of them twice This was the beginning of graph theory And this was the beginning and it was Leonard oiler who was able to prove that it's actually impossible to cross all seven bridges Once without having to retrace your steps And then I learned that when when dodson traveled through europe he went to konigsberg Makes me wonder if he had any interest in this puzzle because he gave children very much a similar puzzle And he went to konigsberg. I have not found any evidence. He never wrote about About this particular puzzle and I haven't found anything out there. Please let me know if you know anything about this one of the You know and and he even took it to three dimensions because here I found in these notes Where he's trying to wrap a string one string all the way around a cube Very much a similar kind of a puzzle here Can you take one piece and go all the way around the cube wrapping it as so And then tucked in between the manuscripts. I found this little two little cards two letters that Mark said you have to share these Stuart and he said you can't not share these Let me read to you Dear mrs. Jarrans. I'm reading the one on the right dear mrs. Jarrans And he wrote this less than a year about just about a year before he died I would have much liked to come to you last night and was quite sorry to be Debarred by the form of your letter which was practically an invitation underscore At present I can tell all friends and truthfully that I never accept invitations and I can't surrender My present secure solitude even for the pleasure of an evening with you Please only send me information in the future and if you add I shall have joy To see you please spell it with a big j Sincerely yours cl dodson There's been a lot of question, you know about his relationships with adult women There has been this letter. I know I'm not the first one to find this because I know that Edward Wakeling included it In the diaries. So there there is some information about it, but Very curious And the fact that what we do know is that as he got towards the end of his life He was shunning any Adult social you know socializing with other adults because he wanted to do more mathematics That was what he wanted to do and he didn't want to give up an evening Although children were another story Where does the day begin? This was an interesting puzzle of his And I'm going to just I'm going to read to you It says if a man could travel around the world so fast that the sun would be always directly above his head And if he were to start traveling at midday on tuesday Then in 24 hours he would return to his original point of departure and would find that the day was now called wednesday At what point in his journey would the day change its name? Well when he wrote this when he first put it out there was no international date line And by the time he Before he died I put it was 1880 something when the international date line was established. So this was a real issue and I don't know if he had much influence in the creation of the international date line But certainly i'm sure that this question was out there It had to have caused some of the interest in the conf you know the confusion here imagine that you know this is certainly something and and That he thought about it and it puzzled him and it bothered him as well Charles Dodson would refuse mail that was addressed to lewis carol. He would deny that he was lewis carol And You don't see much overlap, but here is a page of his notes He writes 17th in decimals Is point zero five eight eight two three five two nine four one one seven six two four seven and it goes on and on and on And then he goes on it's to write a whole lot about this particular circulating decimal But look what says up at the top of the page Look wild no hymns. Oh my Where did that come from? Where did that come from and i'm very curious about this one because as I said carol and dodson were almost two different people But on this page it seems like there's something bringing them together here And what is it about this particular page? You know about this particular one that made him do this He collected all kinds of mechanical devices As a boy he would make puppets he collected music boxes Um, certainly mechanical puzzles. He always had a puzzle in his pocket Okay, that you know that there's the handkerchief mouse that he always had Um, he always had a wire puzzle by the way this little guy is needed to get some air here He needs to get some air and even take a handkerchief and fold it up and do a little pocket mouse okay And Isa bowman No, let's see You know, he had this let's see one of the young girls writes next I went out looking for my twin daughters I find them seated with cld seated between them and they're listening to him open mouth in the greatest state of enjoyment With his knee covered with my new toys Always toys In his diary he writes I met my blue china friend and had some talk with her and tried her with my usual card of introduction The wire puzzle On another day he writes on the railway going back. I made friends with a nice girl of about 15 who was on her way to school in eastborn I gave her my card and promised her a wire puzzle Here's a letter. He writes my dear alice And There were some references that this might be princess alice The queen's daughter Please excuse me if this is written very badly, but how can one attend to one's writing? You know when a great harry green thing is crawling all over the letter And he did draw a picture of a spider on this letter I shouldn't mind it so much if the thing would just keep still What I didn't like most is that it will crawl about all the time. Isn't it provoking of it? Please give this note to your mother and this puzzle to charlie The two wire men are england and ireland and the puzzle is to make them join and unjoin their hands They give us sort of rule for doing it, but even with the rule It's rather hard. Tell them it goes quite easily if you do it right away So he mustn't push hard to get the wire loops over each other And if he does it'll get into such a mess. You'll never get it right again Give charlie my love and take two or three crumbs of it for yourself Yours affectionately now We don't know that this What that puzzle looked like one of the things is he gave away a lot of wire puzzles, but we don't have those puzzles Chris morgan did some research and this was in james dalgettis collection And it certainly seems to fit the description. It certainly seems to fit the description that this might be it He writes he writes this afternoon I shared a bench on the marine parade with a gentleman and his wife and a nice little girl about 10 To whom I showed some puzzles a pleasant child, though not very bright On a sunday afternoon he writes i met mrs. Blakemore and dolly at about one on the parade and dolly not only spoke to me without crying But actually asked me to come in with them stayed about an hour showing her puzzles etc And it really looks as if they were good friends at last after five weeks estrangement Went over to brighton and talked to the girls school 20 girls for whom I did puzzles For more than an hour and a half Isa bowman wrote about how he would keep things in his apartment And so so oh my goodness Bob the bat well, this is actually this is the closest I could find to bob the bat and He would fly about that children loved it He kept it in his left-hand drawer What would he do when uncle wound him up? He would really fly Those of you that are engineers, you know, here's a man who's an insomniac He's always thinking about things and anyone who has tried as I often have done the process of getting out of bed at 2 a.m In a winter night lighting a candle and recording some happy thought which would probably be otherwise forgotten Will agree with me it entails much discomfort Now all I have to do Is to draw from under the pillow a small memorandum book containing my nictograph Write a few lines or a few pages and without even taking it outside the bed clothes Replace the book and go to sleep again and here's his nictograph that he invented And this is the code. This is what it would what it would look like when he would write So he was designing, you know, so clever so clever for those long nights And I'm also now I'm here I'm going to start watching the clock and go through a few of these and I need to say a little bit James Newman wrote that he was a mediocre mathematician without brightening the hour of a single student Or producing anything of lasting value to his subject Warren Reaver wrote in all of dodson's mathematical writing It's evident. He was not an important mathematician Jenny wolf wrote he was too focused on rules and procedures. He was unattractive to his students A student wrote he was singularly dry and his perfunctory manner in which he imparted instruction to us never betraying The slightest personal interest in matters that were of deep concern to us As I explore his diaries as I read from other people, I'm finding that this wasn't true Or maybe what I'm finding is that he was misplaced. He should have been an elementary school teacher And not a university lecturer Because when he after he retired he started to go to schools to work with children and they loved him They absolutely loved him. They would look forward to him coming Little girls, let's see You know, his fancy was sometimes suffered to peep out little girls would learn the rudiments of calculation at his knee They found the path that they had imagined imagined. So thorny set about with rosin's by reason of the delightful fun Which which he would turn task a task into a joy When the fun was over the little girl would find she'd learned the lesson just the same When I went up to oxford I learned from mr. Dodge and to look upon my mathematics is the most delightful of all my studies His lectures were never dry. So we can find both arguments even at oxford Do you ever play at games or is your idea of life breakfast lessons dinner lessons tea lessons bed lessons breakfast lessons It is a very neat plan of life and almost as interesting as being a sewing machine or a coffee grinder But it seems like that's the way he taught at least at oxford. At least that's what most of what we see But then here he is teaching games again Where one player names three or four letters in a word and the other has to guess the word many many word games like that and he one time he went To a lot he writes in his diary that he gave his first logic lecture at st. Hughes hall to 13 girls Including eval and hatch and eval and wrote 40 years later That my old friend mr. Dodson offered to come and give us a lecture on logic With great eagerness my fellow students prepared to meet the famous mathematical tutor Who was the author of alice in wonderland and assembled in the library armed with notebooks and pencils to their surprise The lecturer appeared with a large black handbag From which he proceeded to draw a number of white envelopes to be distributed among the audience Why are you laughing? Each envelope proved to contain a card marked with two square diagrams and nine counters some pink and some gray Notebooks and pencils were not required. We were to play a game. He turned logic into a game Stephanie you made a reference a couple hours ago about a venn diagram happening up here And i'm just beginning to discover that as his if he had lived longer He developed a something and similar to venn diagrams, which we will call carol diagrams And what i have been reading and learning and i need to go learn more is that if he had lived longer It's possible that he would be remembered as a logician and one who did contribute to the field of logic Because he did do quite a lot with it And it is said that these carol diagrams actually work much better With larger sets and again i'm not going to get into the details I'm not even comfortable with it myself. I'm just getting into it, but You know, but yes We had a venn diagram up here, but we also had a carol diagram up here as well In the preface to symbolic logic he writes It's used mostly in enabling one to asserts one mental one's mental powers It gives the clearness to see your way through a puzzle the habit of arranging your ideas in an orderly and get Addible form and the power to detect detect fallacies and other people's arguments In the preface to curiosa mathematica he wrote it may well be doubted whether in all the range of science There is any field so fascinating to the explorer so rich and hidden treasures so fruitful In delightful surprises as that of pure mathematics The charm lives chiefly i think in the absolute certainty of its results for that is what beyond almost all mental treasures The human intellect craves let us only be sure of something And i think that you know it shows not just a love for the topic But again i find that he has a way with words that i can't seem to get across to my students This love of mathematics just for its own sake, but he puts it in words here that makes me think You know he he's he could do it. He could do it and but he was in the wrong place and perhaps at the wrong time He loved nothing more than to invite children over to his apartment for tea and logic An excerpt an excerpt from his diary He loved number tricks And again many many many number tricks about who can get to a hundred first He loved numbers my dear guttured. I send you seven kisses to last a week Your loving friend my dear guttured gertrude. I send you 10 million kisses and remain your loving friend see al-dadsin My dear guttured gertrude. I send you four and three-quarter kisses My dear gertrude This will not do you know sending one more kiss every time by post the parcel gets so heavy It is quite expensive when the postman brought in the last letter. He looked quite grave Mind you don't get any more such letters. He said at least not from that particular little girl I promised him we would send each other very few more letters only 2470 or so Oh, he said a little number like that doesn't signify what I meant is you mustn't send many So you see gertrude. We must keep count now and when we get to 2470 we must not write anymore Your loving friend In this little letter here, he talks about morning dress affairs as opposed to evening dress affairs His dinner parties are morning dress affairs and he also writes at the bottom. I will come for you at six and a quarter fractions used to tell time an interesting way to do it or if If each if in picking a coral if each party declined to go more than three eighths of the way And if in making friends each was ready to go five eighths of the way Why we would have more reconciliations of quarrels I love it elementary mathematics To to reconcile The consumption of mid-jera has been during the past year zero After careful calculation I estimate that if this rate of consumption be steadily maintained our present stock will last us an infinite number of years And although there may be something monotonous and dreary in the prospect of such vast cycles We may yet cheer ourselves with the thought of how economically it can be done Dividing by zero. What a great way to just to to demonstrate it. I think I will Go through i'm going to watch because I certainly want to share this one My darling isa it's all very well for you and nelly and emcee to unite in millions of hugs and kisses But consider the time it would occupy your poor old very busy uncle Try hugging and kissing emcee for a minute by the watch and I don't think you'll manage it more than 20 times a minute Millions must be 2 million at least and then he goes 2 million divided by 20 2 million hugs and kisses divided by 20. That's 100 000 minutes divided by 60. That's 1666 hours divided by 12 hours in a day That's 138 days divided by six days a week. That's 23 weeks I wouldn't go on and any rites. I wouldn't go on hugging and kissing more than 12 hours a day And I wouldn't like to spend Sundays that way So you see it would take 23 weeks of hard work. Really. I can't spare the time I struggled to get my students to comprehend what a number like 2 million means Here Dodson is doing it back when supposedly the elementary curriculum was very dull And he's finding ways to bring it alive. He's finding ways to make them interested and make it connected He came up with rules for addition and subtraction and rules for for division He came he explored cyclic numbers Which all I want to say about repeating decimals that go on and on and on in a repeating pattern and all I'm going to say about it is that In his eight or nine wise words about letter writing. He wrote Don't repeat yourself When once you have said your say fully and clearly on a certain point and have failed to convince your friend Drop the subject to repeat your arguments all over again will simply lead to his doing the same And so you will go on like a circulating decimal Did you ever know a circulating decimal to come to an end? Again, he brings math into some really unrelated places Here's a notebook with his with his lecture notes for algebraic geometry Here's the pages I was looking for obviously I found a lot more I'm going to spare you the details that are on this page, but I will say this If you rearrange those four pieces You're going to notice a long skinny gap. Guess what the area of that gap is right in the center and what he did Is he got into exploring where we gain? He came up with some methods. By the way, I've gotten my my my cue here that I need to wrap things up. So he Came up with some generalizations about this puzzle that I wasn't aware of Notice in this diagram, he's emphasizing he's exaggerating this gap And what's interesting here is he's explaining it in a way that I hadn't explained it before I hadn't I hadn't used his explanation He's also using notation that I hadn't seen before as well He found he he found the area of that gap using a formula that I wasn't aware of either You know about doublets So I'm going to go on with I'm going to going to keep going with that And I guess what I will do here To finish this up He knew about the 15 puzzle In fact, he spent an evening with the 15 puzzle Which is sort of the the this puzzle Took america by storm And the only thing that's come close to that has been the rivix cube This puzzle is on the back Notice when you rearrange it one rabbit just you know, but it's easy to explain this one because where do rabbits go? Down the rabbit hole and if you ever burn a hole in a rug come read this because Cut if the you know, you can cut it up in a certain way so that you can rearrange the pieces And it will still be the same size and the hole is gone And for those of you that like to write here's an eight line poem. I won't read it to you But you can turn it into a seven line poem if you like And it's on the back table by the way it is on the back table He wrote It occurred to me that if we take letters and put them on a chess board and move them around they might form words Is this the beginning of where scrabble came from? Do I have time to finish with this? Very quickly very quickly I will finish With the martin gardener dollar because you see In from the country of money That if we move martin out of the way, whoops, you can't see it. Let me escape and go here and full screen So here is Here's a martin gardener dollar bill. It's a very special dollar bill Because if we cut it up as so and we remove martin and then we Take the pieces and turn them upside down and put them back together Where'd martin go? Well Let's see we turned martin over Thank you very much