Mathematics and Intuition - An Uncomfortable Alliance
Peter G. Casazza [d1]
Introduction
Mathematicians have an "intuition" for Mathematics which helps themto quickly recognize when a result fits into the reasonableframework of the theory. But, when results are put to the test of"the real world", even our intuition often goes astray. The problemseems to be that the very simplicity of the notation can oftendisguise the underlying principles. Thus, we end up with anelementary formulation of a result, which mathematically is quitetrivial, but which in the setting of the real world, is quiteprofound. The purpose of this note is to look at some veryelementary forms of this phenomenon. To give the reader a chance tobecome involved and test their intuition, the answers to some ofthe questions appear at the end of this article.
How large is LARGE?
We read almost every day about how the earth is becomingoverpopulated. Suppose we put this to the real world test. I form acube which is 1000 yards on a side. This is not a small cube.Its length is ten football fields laid end to end. However, wecould imagine laying these ten football fields end to end throughthe middle of the campus, and doing the same for the width, and theheight. That is, this cube would fit nicely into the center of ourcampus. Would you believe, that this cube will hold every man,woman andchild on earth and still have a lot of empty space left over? Ifthe entire population of the earth could be put in a box in thecenter of our campus, is the earth really overpopulated? I leavethat for you to think about.
Problem 1:Take a piece of paper and fold it in half, fold it in half again,fold it in half again, and do this 51 times. The average piece ofpaper is .003 inches thick. How high is the stack of paper?
When teaching Calculus class, we often have to draw the graph ofthe function 
on the blackboard. We all know that the graph "risesrapidly". But, the next time you see someone draw this graph on theboard, put it to the following real world test.
Problem 2:How high is the graph of 
, when x = 42 centimeters?
The United States government now spends one and a half trilliondollars a year. This is 1,500-billion dollars. (Be careful here. The word "billion" comes from Italian where it stood for a"million-million". In the United States it was "mis-translated" andended up representing a "thousand-million".) To grasp themagnitude of such a number, we would have to understand how large abillion is first. Actually, a billion itself is already anincredibly large number. For example, if I invested a billiondollars in a bad company, and this company lost $10,000 per day,seven days a week, three hundred and sixty five days a year, then Imight worry if I will have any money left at retirement. But, I amstill alright, since even at this rate, it will take the companyover 200 years to lose my investment. On the other hand, if Itook the cap of my pen and magnified it a billion times, then itwould be 95,000 miles long and 8,000 miles wide, and the entireearth would fit inside it (The diameter of the earth is about7,900 miles.) In fact, a billion is so large, that it was notuntil 10:40 a.m. on April 20, 1902, that exactly a billionminutes had ellapsed since the birth of Christ. I hope that I havenow convinced you that a billion is truely a large number. However,it cannot be too large since there are more than one billion waysto give change for a $5 bill.
According to American History, in 1626 Peter Minuet boughtManhatten Island from the Native Americans for 26 worth of goods.Manhatten island is now one of the most expensive pieces of realestate in the world. Probably, we should pay the Native Americanssomething more comparable to the value of the property. On theother hand, consider the following:
Problem 3:Suppose the Native Americans had put $26 in the bank in 1626at 6% interest. How much money would they have today?
With family and relatives visiting one Thanksgiving, there were14 of us in the house for dinner. We set up one long table in theliving room with 14 chairs around it. My mother told everyone togo in and sit down for dinner, and as usually happens, everyonewent in and stood by the table having trouble deciding where to sit. My mother became upset that dinner was getting cold while everyonewas "milling around" the table. So, I explained to her that it isreally a complicated proceedure for 14 people to sit around atable. There are, in fact, 14! ways that 14 people can sitaround a table (If we count "rotations" as being different ways -and 13! otherwise.) This is a lot. To put this in propercontext, answer the following problem.
Problem 4:If the 14 people sat down at the table, then stood up and satdown at the table again in a different way, then stood up andchanged positions again, and again,.... If we give them one minuteto change seats each time, how long will it take them to sit aroundthe table in all possible ways?
Suppose I have a company make me a piece of string 22,000 mileslong. This is approximately the circumference of the earth at theequator. I am going to take my string and wrap it around the earthgoing over the mountains, the oceans, the deserts until I end upwhere I started. Unfortunately, the company made a mistake. Thestring is 3 yards to long. Not wanting to waste the string, I tiethe ends together and put my fingers under it and slowly spread itaway from the earth until it stands equally distant from the earthat all points. After I have evenly distributed the extra 3 yardsalong the 22,000 mile circumference of the earth, how far from theearth does the string stand at each point? It is probably fartherthan you think, since I can walk under it standing up. Now, let mehave the company make me a piece of string 1 inch long which Iwrap around a marble. Unfortunately, the company again makes amistake and sends me a string which is 3 yards too long. Asbefore, I tie the ends together, put my fingers under the string andspread the string evenly away from the marble at all points. So,after I have evenly distributed the extra 3 yards along the 1inch circumference of the marble, how far from the marble does thestring stand at each point? Surprise! This string stands exactlythe same distance from the marble as the other string stands fromthe earth.
Problem 5:I am riding up a ski lift which is going 5 miles per hour and I am very impatient. I decide that when I get to the top, I am goingto ski down so fast that my average speed for the trip (up and down)is 10 miles per hour. How fast do I need to ski down themountain?
What is the largest number I can write using three integers between0 and 9?A little thought might lead one to:
Unfortunately, there is no such number as this. That, of course,is due to the fact that
Therefore, (1) does not make sense. A little thought will leadyou to the larger of the two. This number,
is truely a large number. How large is it you ask? Well, this is9 multiplied by itself 387, 420, 489 times. This number is largerthan the number of snowflakes it took to form the ice age (whichhas been estimated at a billion to the billionth power). In fact,this number is larger than the number of snowflakes that have everfallen. In fact, it is 4 million times larger than "Eddington'sestimate" for the number of electrons in the universe(Do you remember Sir Arthur Eddington [1882-1944] the EnglishMathematician, Astronomer, and Physicist?). In fact, even thoughbacteria are microscopic, this many bacteria would fill up the entire Milky Way Galaxy.
Problem 6:Suppose I decided to type the number in (2) on my typewriter.How much paper should I buy?
Probability
Probability theory is probably the least understood area by thegeneral population (except for certain gamblers).As a simple example, consider the History Professor friend of minewho was scared of flying and asked me one day: "What is theprobability that there will be a bomb on an airplane?" I respondedthat I really didn't know, but that it was certainly less than onein a million. So he asked: "Well, what is the probability thatthere are two bombs on an airplane?" I responded that (as long asthese were independent events) it would be the square of theprobability of having one bomb, which is 1 in a trillion - a truelyastronomical number. So, from that day forward he always carried abomb with him when he flew since it reduced the risk of having abomb on the plane from 1 in a million to 1 in a trillion.
Problem 7:Each morning when I get up, there are 4 loose socks in my drawer -2 red ones and 2 blue ones. In my "groggy" state, I choose twosocks at random and put them on and go to school. How often do I goto school with a matched pair of socks?
Problem 8:In a popular TV game show (The Price is Right) a contestant isasked to pick one of three doors. Behind two of the doors isnothing, and behind one of them is a "fabulous" prize. After thecontestant picks a door, the host opens one of the other doors toshow that it contains nothing, and then asks if the contestant wantsto keep the door they picked, or switch to the other unopened door. Should the contestant switch?
A very interesting problem is the "Birthday Problem". There are150 people in a lecture room and I offer to bet you even moneythat two of them have the same birthday. Is this a fair bet? Notreally, since the odds are against you 4,100,000,000,000,000 to1. So what happened? Given n people, each one can have theirbirthday on any one of the 365 days of the year (We will excludeleap years). So there are 
possibilities for theirbirthdays to occur. For n people to have birthdays on differentdays, there are 365 possible days for the first person's birthday,364 for the second, 363 for the third, etc. Thus, theprobability that no two people have the same birthday in a group ofn people is:
Of course 
is the probability that at least two people have the same birthday. For example, if n = 20, 
,and if n=22, then 
, and finally, if n=23, then 
. Hence, if there are 23 people in a room, there isalready a 50-50 chance that at least two of them have the samebirthday. Let's try this out in real life. Clinton is the42nd president of the United States. If we put n=42 into ourformula, we get that 
. i.e. There should be greater than a 90% chance that at least two Presidents of theUnited States were born on the same day of the year. Also, thereshould be greater than a 90% chance that at least two Presidentsof the U.S. died on the same day of the year. We are o.k. however,since Polk and Harding were both born on November 2, whileJefferson, Adams, and Monroe all died on July 4.
Mathematics and ``Real Life''
I made up a problem for my Algebra class one time which wasdesigned to stimulate a discussion of "problem solving". I toldthem that the rabbits ate up my whole garden last year, so this yearI decided to build a fence around it to keep them out. I went tothe barn and found one 44 foot section of fence and 48 one footsections. If I do not want to cut up any of the fence, what is thelargest garden I can enclose? My class decided that there were twopossibilities. They could add a one foot section to the end of the44 foot section, then make a rectangle which was 45 by 1 whichwould use up all the fence and give 45 square feet. Anotherpossibility, using all the fence, was to build a rectangle which was44 by 2 to get 88 square feet. They decided that 88 squarefeet was best. Being prepared for this, I brought in a can marked"gasoline", poured it on the 44 foot section of fence and burnedit to the ground. Then I took the remaining 48 one foot sectionsand built a square which was 12 by 12 for an area of 144square feet. Then I asked them how I enclosed nearly twice the areathey did with only about half the fence they used? After somediscussion, they realized several important facts.(1) Increasing perimeter does not always increase area. (2) Oneshould be careful not to add constraints to a problem which werenever there. i.e. I did not tell them to build a rectangle, or touse "all the fence". Eventually, they decided they could do betterthan I did by using the 48 one foot sections to fence against aportion of the 44 foot section. It is a standard problem inCalculus that you get maximum area for fixed perimeter when fencinga rectangle "against a river" by making your length twice yourwidth. In our case, this meant building a 12 by 24 footrectangle with the 44 foot secion of fence as the fourth side,giving a garden of 288 square feet. Of course, this was anAlgebra class, so why did I give them a "standard Calculus problem?" Actually, this is a "dumb" Calculus problem. This problem caneasily be solved using only Algebra, and should be done that way.Calculus should be done on problems which need Calculus. All we hadto do was let the length of the two sides be x, so the length ofthe long side of the rectangle is 48-2x, and the area isA(x)=x(48-2x). Now we observe that, using the standard Algebratrick of completing the square:
Thus, A(x) is clearly maximized when we are "subtracting" theleast amount (in particular, when we are subtracting 0). So themaximum occurs when x = 12. So this is the end of the story,right? Not really.
Problem 9:What is the maximum area I can enclose with my fence?
When we go to a pizza parlor, they generally cut the pizze intolittle pie slices through the center. Why do they do that? Oneobvious reason, is that it makes all the slices the same size so wecan evenly split them up amongst ourselves. But, if there were onlytwo of us, is this the only possibility?Well, there are many. One possibility is to pick a point otherthan the center and cut all slices at equal angles through thispoint. If we cut the pizza into 4 slices however, one of the"slices" could contain almost the whole pizze. So this doesn't workso good. So this is worthless. NOT SO FAST.
Problem 10:Suppose we cut the pizza "off center" like above, but by cutting itinto 8 pieces all at equal angles? Is it possible for both of usto get an equal amount of pizza?
Calculus and the ``Real World''
One of the first summation formulas we see - often in Algebra - is:
As an application of this formula, we could recall the story of theKing of India and the inventor of the game of Chess. As the storygoes, the King was so pleased by this invention, that he offered theinventor anything in his kingdom. The inventor said that all hewanted was 
grain of wheat for the first square of thechessboard, 
grains of wheat for the second square, 
grains of wheat for the third square, etc. The king was astoundedthat, having offered this person anything in his kingdom, theinventor only asked for a "few" grains of wheat. However, to theKing's surprise, he could not fulfill his promise. As we can seefrom (2), the King owed the inventor 
grains of wheat.This is a lot of wheat. How much is it, you ask? Well, it isenough to cover the entire surface of the earth 1 inch deep.
A related problem is called the "Towers of Hanoi". It is said thatin the great temple of Benares, at the "Creation", God put a brassplate with 3 diamond needles in it. He then placed 64 goldendisks (with holes through their centers) of different diameters overone of the needles so that the largest was on the bottom, thesecond largest is next, and so on. God then told the Priests of thetemple to move the 64 golden disks from their diamond needle toanother needle. But, they must follow the rules: (1) They canmove only one disk at a time; (2) They may never place a largerdisk on top of a smaller one. When the priests are done moving thedisks, the world is to end.I began to worry the other day about whether the Priests are aboutdone with their task. In this case, I should probably increase mylife insurance immediately. However, a little thought shows us thatit will take moves for the Priests to complete theirtask (Exactly the number of grains of wheat that the King of Indiaowes the inventor of the game of Chess).
Problem 11:If we give the Priests 1 minute to move any one of the heavy disksfrom one needle to another, how long will it take them to completetheir task?
Later in Calculus, we see that if 
in formula (2) wecan let 
, to get the formula:
Letting r = 1/2, and subtracting 1 from both sides we obtain:
I must confess that this could easily be my favorite formula in allof Mathematics. It can be used to solve a number of interestingproblems. The first is the famous "frog and the wall" problem. Iplace a frog one foot from a wall and tell him to jump half way tothe wall, then jump half way to the wall again, then jump half wayto the wall again, and keep going. Does the frog ever reach thewall? Of course he does. Formula (3) tells us why. That is, inhis first jump the frog traveled 1/2 of a foot, in his second jumphe traveled 
of a foot, in his third jump he traveled 
of a foot, and so on. So by formula (3), the totaldistance traveled by the frog was 1 foot and he was only 1 foot fromthe wall to start with. "But wait", you say. "He never actuallygets there since he will be jumping forever." My favorite formulacomes to the rescue again. I will just tell the frog to make hisfirst jump in 1/2 a second, his second jump is a second,and so on. So, my frog can complete all his jumps in only 1second, and arrives at the wall. You are still not convinced?Suppose I put the frog on a 1 foot high table and push him off. Doyou not doubt that he will hit the floor - and quickly? But, howdid he get to the floor? He first got half way to the floor, thenhalf way again, then half way again, .... This is what the otherfrog did, isn't it?"No" you say. "There is a major difference. The other frogstopped at each ofthese points while this one moved continuously through them to thefloor." But, I can take care of that. I will push another frog offthe table at the same time and tell him that when he gets half wayto the floor, he should stop for 1/2 a second, and when he getshalf way to the floor again he should stop for a second,and so on. His total "stopping time", again by my favorite formula,is 1 second. So the second frog will reach the floor only onesecond behind the first frog, but the second frog stopped at each ofthe points. I hope I have convinced you that the frog can reachthe wall. It is interesting that this problem was actually solved,in a different formulation, by Zeno in 550 BC.
Another interesting use of my favorite formula is in the"Supermachine paradoxes". I have a Supermachine which can doanything I tell it to do as long as I give it a positive amount oftime to carry out the task. So I tell it to turn a light bulb on in1/2 of a second, off in of a second, on again in of a second, and so forth. In one second, my supermachineturned the light bulb on and off an infinite number of times.
Problem 12:When my supermachine has completed its taks, is the lightbulb nowon or off?
Saying that the lightbulb is burned out is not an acceptable answersince this is a "mathematical" light bulb, so it never wears out.This question is a true logical paradox. And using yoursupermachine, you can make up any number of such paradoxes. Forexample, I could have my supermachine paint my house red in 1/2 ofa second, green in of a second, red again in ofa second and so on. After one second, my house has been paintedred an infinite number of times, and green an infinite number oftimes. So what color is my house? Or maybe I should have thesupermachine fly me from St. Louis to New York City in 1/2 of asecond, and back in of a second, and back to New York City... I think you see the point.
Problem 13:Make up your own "Supermachine paradoxes".
There is something that has always confused me. One of the firstthings we teach when we introduce Series in Calculus is that theharmonic series is divergent. That is:
If these numbers really do add up to infinity, I should be able toget them to add up to at least say, 193 on my computer without toomuch trouble.
Problem 14:How long should it take my computer to add enough terms of theharmonic series to get to 193?
Sometimes, it depends upon how you look at a problem that defiesour intuition. A typical example is the Banach-Tarski Paradox. Itis a "Theorem" really, which was proved by S. Banach and A. Tarskiin 1924 (Wagon 1985). The Banach-Tarski Theorem (A special case of it)says that given any two spheresA,B in R 
."
Suppose we take the graph of the function 
R given by 
, andrevolve it about the x-axis. I will get a"trumpet" but with an infinitely long neck. I can easily computethe volume of my trumpet by the standard Calculus formula for thevolume of a surface of revolution. This yields:
So my trumpet has a finite volume (after proper scaling) of 
gallons. On the other hand, using the formula for the surface areaof a surface of revolution, we obtain:
So my trumpet has an infinite surface area. Again, let us put thisto the real world test. The trumpet is sitting on my front lawnand is rusting away. I decide that I had better paint it to protectit from the elements. So I travel to my local paint store to getsome paint. I tell the manager that I need to paint my trumpet soit will stop rusting. He asks: "How much area do you need topaint?" I respond: "The trumpet has infinite surface area." Hesays: "You are in serious trouble. That would require an infiniteamount of paint, and I certainly do not have that much." "But", Isay, "it has a volume of gallons (Which we know is 3.14159....). So sell me four gallons of paint and I will fill it with paint. This should keep it from rusting." The owner is astounded andreplies: "Are you telling me that you can fill this object withpaint, but that this will not paint it?"
Problem 15:Can you explain the apparent "contradiction" above?
Getting Help from the ``Real World''
There are some problems which in a mathematical setting are quitedifficult, but when placed in a "real world" setting, actuallybecome clear. A simple example is the "up and down the mountain"problem. It goes like this. At 7:00 a.m. one morning I hike up atall mountain. It is a slow and tiring trip. I stop often torest, but finally make it to the top of the mountain at 5:00 p.m.The next morning at 7:00 a.m. I retrace my path back down themountain. This direction is easy and I move quickly down themountain arriving at the bottom at noon. Can you show that there isa point on the trail that I reached at exactly the same time when Iwent up the mountain as when I came down the mountain? To answerthis, we just "superimpose" the two days on top of each other. Soat 7:00 a.m., one of us is starting up the mountain and the otheris coming down the mountain (but a day later). Certainly we mustmeet ourselves someplace along the trail. Q.E.D.
Problem 16:I have two glasses. One is half filled with wine and one is halffilled with water. I take 1 tablespoon of pure wine and mix it intothe water. Then I take 1 tablespoon of this mixture and put itinto the wine. When I am done, do I have more wine in the water ormore water in the wine?
Problem 17:I have a convex polytope without uniform density. For each face ofthe polytope, consider the line from the center of mass which isorthogonal to the palne containing that face. The line, of course,may not actually go through the face. Prove that at least one ofthese lines passes through a face.
You can find out more about these problems an many others byconsulting the references at the end of this article.
Answers to the Problems
Answer 1:Your stack of paper is now 106,619,310 miles high and has alreadypassed the sun in our Solar System
Answer 2:The function , at x = 42 centimeters is already more than5.3 light years high and has passed Alpha Centurian (The neareststar to us in the Milky Way Galaxy) which is only 4.3 light yearsfrom earth.
Answer 3:A lot. The "interest" on their account this year would be over onetrillion dollars.
Answer 4:Would you believe that it will take more than 160,000 years?
Answer 5:Bad news! You cannot do it no matter how fast you ski. You see,you cannot double your average speed for a trip once you have gonehalf way. Since average velocity is distance over time, if youdouble your distance and double your velocity, time must remainunchanged - which is impossible.
Answer 6:You will need a sheet of paper more than 1,000 miles long to typethis number.
Answer 7:Unfortunately for you, 2 days out of 3 you will have on anunmatched pair of socks.
Answer 8:The contestant doubles their chances of winning by switching to theother door.
Answer 9:If you take a course called the Calculus of Variations, you willsee a proof that the largest area that can be enclosed for a fixedperimeter is a circle. The largest area we can get with our fenceis to fence a "semicircle" (actually, half of a 96-gon) made outof the one foot pieces of fence against the 44 foot section to getnearly 367 square feet. This is better than the 88 square feetthey started with, isn't it?
Answer 10:If we number the slices in order in the clockwise direction from 1to 8, then the even numbered slices and the odd numbered sliceshave exactly the same total area. For a proof of this and someinteresting variations, see (Carter and Wagon 1994).Also, try this on a "friend" sometime and see their reaction - andif you can convince them that you really did both get the same area.
Answer 11:It will take the Priests more than 58,454,206,609 centuriesto move all the disks from one needle to another. I guess I canleave my life insurance the way it is.
Answer 12:This is a true logical paradox. i.e. An unanswerable question.
Answer 13:Wherever your imagination takes you is a correct answer.
Answer 14:If my computer could do 1 million calculations per second, and itwas running for 60 seconds per minute, 60 minutes per hour, 24hours per day, 7 days a week, 365 days a year, and if it hadbeen running continuously for the last 5 million years, then itwould only be at 191 right now. So does the harmonic seriesreally "diverge" in the "real world"?
Answer 15:The manager made a common "real world" mistake. He assumed thatyou needed an infinite amount of paint to paint an infinite surfacearea. But, he should have fallen back on my favorite formula andthen he could have painted thiswhole infinite surface area with only one tablespoon of paint. Allhe needed to do was have my supermachine paint the first one footof the trumpet with 1/2 a tablespoon of paint (and do it in 1/2seconds), and the next one foot with a tablespoon of paint(and do this in seconds), and so on. Then he could havepainted the whole trumpet with one tablespoon of paint, andcompleted the task in one second.
Answer 16:They are exactly the same. And even if I repeat the proceedureover and over, they still remain always the same. As a"Mathematics" problem, this can be nasty. But as a real worldproblem, it is obvious. That is, each glass still contains the samevolume of liquid as it did before. So any molecule of wine in thewater is taking the place of a molecule of water which must be overin the wine - and vice versa. That does it.
Answer 17:Take the polytope and throw it on the floor. When it stopsrolling, the center of mass must be directly above the face layingon the floor.
References
- -
- Larry Carter and Stan Wagon.Proof without Words: Fair Allocation of a PizzaMathematics Magazine, 67, 1994, page 267
- -
- Peter G. Casazza.Paradox LostAlabama Journal of Math., Fall 1977, pages 63-73
- -
- Peter G. Casazza.Are There Any Uninteresting Natural Numbers?Alabama Journal of Math., Fall 1977, pages 27-38
- -
- Peter G. Casazza (Published under the "Pen Name" of Cathy Peters). Give Me a Tablespoon of Paint and I will Paint the Whole UniverseAlabama Journal of Math., Fall 1978, pages 33-39
- -
- Peter G. Casazza (Published under the "Pen Name" Patricia Caldwell). On Pepperoni Pizzas, Ham Sandwiches, and other Important Mathematical ProblemsAlabama Journal of Math., Spring 1979, pages 20-25
- -
- Peter G. Casazza (Published under the "Pen Name" Cora Green).Is Probability Probable?Alabama Journal of Math., Fall 1979, pages 9-17
- -
- Peter G. Casazza.Maximizing Areas without CalculusAlabama Journal of Math., Vol. 7, No. 2, 1984, pages 29-43
- -
- L.J. Upton.Problem 660Mathematics Magazine, Vol. 41, 1989, pages 120-122
- -
- Stan Wagon.The Banach-Tarski ParadoxEncyclopedia of Mathematics, Vol. 24, 1985, Cambridge University Press.
[d1] Department of Mathematics, The University of Missouri, Columbia, Missouri 65211, U.S.A., petecasazza.math.missouri.edu. This article is based on a tal