There are a few famous math problems that have captured the imagination of the general public. And one in particular has been included in short stories, magazines, books, mathematics textbooks, and even movies. This problem has probably stimulated more work by amateur mathematicians, and more failed attempted solutions, than any other math problem in history. In the following set of Thinking Problems, this famous "Coconuts, Sailors, and a Monkey" Problem is included. But before we get to this one, we first examine two "warm-up" problems designed to introduce some of the algebra required for the full problem. And after the famous one, we extend the problem to its general form.
(Just to let you know, about a dozen years ago I gave this problem to the students, and their parents, attending a Math Club that I was running at the time. One of the mothers of one of the students solved this problem and won $20 from me, so this problem is not impossible to solve by non-mathematicians. It is just very difficult. You will notice that there are actually a number of subproblems included below. My progeny know this as "leading them down the garden path", meaning that if you solve the subproblems first, you will find that the solution to the full problem will be easier. On the other hand, you could just jump in and solve the full problem first.)
Below is the e-mail that I sent to a group of parents for a Math Club that I was organizing at the time. Yes, the e-mail is rather long, but it provides a few interesting details and introduction to this type of problem.
Preliminaries:
Hi all:Isn't it just marvelous what one can find at one's local libraries? I had the occasion of perusing the stacks recently, and uncovered a real chestnut that seems to have been forgotten or neglected. It involves coconuts and what are known as Diophantine equations.Lately, I have been thinking about proposing continued fractions as a possible choice for a 5 session trial Math Club topic, so obviously coconuts fit right in with my thoughts ... what is that you say, you are wondering what coconuts have to do with continued fractions? Well, never fear, I won't get to that connection below, you'll have to wait for the Math Club talks. Rather, I'd like to discuss four "Coconuts'' problems with you.Before we begin, let's go back to the ancient Greeks. And in particular, let's talk about perhaps the most famous of all Diophantine problems: Archimedes's Cattle Problem. Simply put, a Diophantine problem is one whose solution consists of the counting numbers (integers, both positive and negative). And Archimedes came up with a doozie of a problem counting the number of cattle on some plain in Sicily (Trinacia to the ancient Greeks). There are several alternative stories of the origin of this problem. Personally, I rather favor the legend proffered by Gotthold Ephraim Lessing in 1773 [see Heath, T. L., The Works of Archimedes] whereby Archimedes became upset with his fellow mathematician Apollonius of Perga who had criticized one of his works. Supposedly to get back at Apollonius, Archimedes devised this devious problem and sent it to him. Archimedes asks Apollonius to compute the solution if he thought he was a clever enough mathematician. It begins, "If thou art diligent and wise, O Stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Trinacian isle of Sicily, divided into four herds of different colors, one milk white, another a glossy black, the third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while ...". Several biographers and historians, including both Greek and Roman poets, have referred to Archimedes attempting to solve "his" problem, but no solution has come down through the ages to us. [Amthor, A., Zeitschrift fur Math. u. Physik, 1880, 25, 153.] Now, I'm not even going to fill in the details of Archimedes's problem here (I'll leave that for another Thinking Problem) for you, but I would like to quickly discuss its solution.Most people today believe that the great Archimedes (the first of the three greatest mathematicians of all time) never found a solution to "his" problem. This was especially cemented when in the late 1800s a mathematician was able to show that the solution to the Cattle Problem were numbers with more than 206,000 digits apiece. I think it is quite reasonable to assume that Archimedes never found this solution. The actual solution, by the way, resisted all attempts until the late 1890s when a civil engineer by the name of A. H. Bell [Bell, A. H., "Cattle Problem, By Archimedes 251 B.C.", American Mathematical Monthly, 1895, 2, 140.] and a couple of his friends formed a club of amateur mathematicians in Hillsboro, Illinois and took the calculation of the digits of these 206,000 digit solution integers as their pet project. The club spent four years and computed several hundred digits by hand. At that time, they estimated that it would take 1000 men more than 1000 years to complete their task by hand. The final solution was eventually achieved in 1965 when H. C. Williams, R. A. German, and C. R. Zarnke of the University of Waterloo submitted the problem to computer computation.While Archimedes obviously was working on Diophantine equations in the third century B.C.E. (BC), these integral solution equations received their common name most likely in the third century C.E. (AD). In particular, they are named after Diophantus [You will also occasionally see the spelling "Diophantos."] of Alexandria, a renowned Greek mathematician of his time who worked on such integral problems. But what time was that? No one really knows for sure! There is no record of exactly when Diophantus lived. But, based upon Diophantus's own writings, it is known that he must have lived after 150 BCE since he writes about "polygonal numbers" which were defined by Hypsicles around 150 BCE. In addition, Theon of Alexandria quotes one of Diophantus's definitions in 350 CE, so we know he must have lived before this date. Some of the best information stems from a letter written in the eleventh century by Michael Psellus. This letter has been interpreted by a specialist in the history of Greek mathematics, Thomas Heath, to indicate that Diophantus lived in the third century CE, although even this interpretation is controversial. In fact, a more recent author, W. R. Knorr, in 1993 reinterpreted this same Psellus letter to mean that Diophantus lived before the first century CE. This controversy reminds one of a quote by Napoleon Bonaparte, "What is history, but a fable agreed upon?"And the only, somewhat circuitous, reference to the life of Diophantus is found in a Greek anthology of mathematics problems, the Palatine Anthology, mostly compiled by Metrodorus around 500 CE. This compendium contains a poetic problem that is supposedly a repeat of Diophantus's epitaph, as quoted below, carved into the stone of his grave. [This epitaph, which is probably fictitious, is quoted in Newman, J. R. (ed.), The World of Mathematics, Vol. 1, Simon and Schuster, New York, 1956, p. 207.]"THIS tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life."Solving this age puzzle yields the times of Diophantus's boyhood, beard growth, marriage, son's birth, son's death, and his own death. You might find it entertaining to solve this fairly simple age problem yourself before reading the following footnote: [Converting this epitaph to an equation produces (1/6+1/12+1/7+1/2)L+9=L where L is the length of Diophantus's lifetime. Solving this equation yields that Diophantus's boyhood lasted 14 years, he grew a beard when he was 21, he married at the age of 33, his son was born when Diophantus was 38 years old, but Diophantus's son died at age 42, and Diophantus himself died at the age of 84.] So, with this scanty information, we suspect, but don't know for sure, that Diophantus lived in the 3rd century CE.The second most famous of all Diophantine problems is the third of the Coconut Problems that I give below. I think a version of this problem has been included in every book ever written on Number Theory or Diophantine Equations. And it is this problem that I recently found stated in an intriguing magazine article at the library. But, before telling you this story, I'd like to give you a couple preliminary "Coconut" problems as warm-ups. In particular, this first Coconut problem does not have to be solved in integers, you can use fractions, and it is simple enough that it can even be solved by trial-and-error, given a box of toothpicks, that is. Your solution of this problem (assuming that you carry out the algebraic solution) will prime you for the daunting task of solving the Third Coconut Problem.
The Coconut Puzzlers (reward applies to only the Third Coconut Problem):
- First Coconut Problem:
Five sailors and a monkey become shipwrecked on a desert island. They collect a pile of coconuts. The ship's Captain takes half of them plus a half a coconut for his stash. The First Mate absconds with half of what is left plus half a coconut. The Engineer then hoards half of what's left plus half a coconut. Next the Gunner takes half plus half a coconut. And finally, the lowly Deck Hand takes half of what remains plus half a coconut. Left over is exactly one coconut, which they toss to the Monkey. How many coconuts did the sailors initially collect? [Note: Remember that the solution is not restricted to be an integer; it can be any real number.]
If you are having trouble with the algebra, go get a box of toothpicks (or some other expendable objects that you can break in half) and do a trial-and-error solution. The trial-and-error solution will guide you in your algebraic proof.Here is a link to the solutions, both using toothpicks and using algebra, for the First Coconut Problem: First Coconut Problem Solution
- Second Coconut Problem:
Here is another warm-up problem. Again, the solution is not restricted to be integral, it can be a fraction or any real number.
Five sailors and a monkey are shipwrecked on a desert island. They collect coconuts during their first day (not necessarily a whole number of coconuts). That evening, one sailor awakens, decides to split the pile into five equal shares, takes and hides his share, gives one coconut to the monkey, and retires for the rest of the evening. Subsequently, a second sailor awakens, decides to divide the pile into five equal shares, hides her share, gives one coconut to the monkey, and retires. Each of the remaining three sailors separately perform the same late night divisions. In the morning, the five sailors distribute the remaining pile into five equal shares after giving the monkey one coconut. How many coconuts were in the initial pile?
- Third Coconut Problem:
Now, after those warm-up problems, let's tackle the third of these coconut problems. This is the famous one, the one that is included in all books on Number Theory. Interestingly, no one ever attributes this problem to an author, and I think the reason is because it is so prevalent and so old that no one knows its true origin. Of course, there are also multiple, slightly modified, versions of this problem in print. And because of this, I'd like to quote directly from a source I located during my recent library foray.
Ben Williams wrote a short story called "Coconuts" for the illustrious The Saturday Evening Post, [Did you know that The Saturday Evening Post was founded by none other than Benjamin Franklin in Philadelphia in 1728?] October 9, 1926. [Williams, Ben Ames, "Coconuts", The Saturday Evening Post, 1926, 199 (15), 10.] Yes, that's right, isn't it amazing where you will find math problems!
Williams's story concerned a competition between several building contractors for a particularly lucrative hospital contract. One contractor, Wadlin, being especially shrewd and knowing his competitor's love of mathematics, ensnared his chief competitor, Marr, in an exasperating problem. Here, in Williams's own words, the trap is set (the red-colored text is quoted from Williams's story in The Saturday Evening Post).
Marr nodded. "Can't get the last word from the subcontractors till tomorrow morning," he agreed. "But my part of it's all done." He glanced at Wadlin's notebook. "That doesn't look like specifications."
Wadlin said somewhat morosely, "Just a problem that interested me."
"Stuck, are you?"
"I haven't found the key yet," Wadlin confessed. "The method of attack."
"What is it?" Marr insisted. "I never saw a problem yet that would stick me."
"Let's have it," Marr insisted. "Let's have it."
The little man still hesitated, said at last reluctantly, "Why, I had it at noon from a man uptown. Haven't been able to get it yet." Marr made an impatient, peremptory gesture; and Wadlin said vaguely, "It's just a problem in indeterminates, I think."
"What is it, man?" Marr cried. "Let me in on it. I'll straighten you out in no time."
"I don't want to bother you," Wadlin argued. "It's not as simple as it looks. I've been at it six hours and more."
"What is it? What is it?" Marr demanded. "You act as though it were confidential."
Clever how Wadlin played on Marr's character flaws, in particular, Marr's sense of pride, to entrap the man. Wadlin very effectively lead Marr to believe that he was a better mathematician than Wadlin, and thus should be able to help Wadlin solve this problem.
Let's next return to Williams's version of the problem itself.
So at last Wadlin told him. "Well," he explained, "according to the way the thing was given to me, five men and a monkey were shipwrecked on a desert island, and they spent the first day gathering coconuts for food. Piled them all up together and went to sleep for the night.
"But when they were all asleep one man woke up, and he thought there might be a row about dividing the coconuts in the morning, so he decided to take his share. So he divided the coconuts into five piles. He had one coconut left over, and he gave that to the monkey, and he hid his pile and put the rest all back together."
He looked at Marr; the man was listening attentively.
"So by and by, the next man woke up and did the same thing," Waldin continued. "And he had one left over, and he gave it to the monkey. And all five of the men did the same thing, one after the other; each one taking a fifth of the coconuts in the pile when he woke up, and each one having one left over for the monkey. And in the morning they divided what coconuts were left, and they came out in five equal shares."
He added morosely, "Of course each one must have known there were coconuts missing; but each one was guilty as the others, so they didn't say anything."
Marr asked sharply, "But what's the question?"
"How many coconuts were there in the beginning?" Wadlin meekly explained.
And the hook was baited, the rod cast, and now for the final sinker.
Marr laughed. "Why, that's simple enough. You just ----"
But their victuals were served; the table was filled with viands; they could find no room for calculations. So when they were done Marr said, "Here, you come upstairs to the office and we'll work this out. It won't take long."
"I'd like to get it," Wadlin agreed, "before I go to sleep. There must be a formula, some way to work it."
"Won't take five minutes," Marr declared; and Wadlin said meekly:
"Well, I believe the record is fifty-eight minutes. But I've been at it hours."
Marr laughed. "Went at it wrong," he insisted. "Come along upstairs."
Like Marr, you too may be wondering why this coconuts problem is so difficult, you might even believe that you could solve it with relative ease. But, before you do so, I think another passage from Williams's story is in order. In particular, Williams describes a fairly typical encounter with this coconuts problem.
In Marr's office the big man bade Wadlin sit down; himself took pad and pencil. "Here, I'll show you," he explained. And, while Wadlin watched him with some attention, he wrote swiftly:
"x --- original pile; a, b, c, d, e --- shares of each man."
And he proceeded to form equations:
x = 5a + 1
x - (a + 1) = 5b + 1 .
And on in the same fashion, till abruptly he stopped and hesitated. "And that equals the remaining pile," he said. "How much was that?"
"We only know it was divisible by five," Wadlin replied gently.
"Call it y," Marr said carelessly; and Wadlin started to speak, then held his tongue, while Marr proceeded with the processes of substitution. Till presently Marr's pencil came to rest again, and he scratched his head.
"That leaves us two unknowns in one equation," he said reluctantly; and Wadlin nodded and commented in a meek tone:
"I ran up against that too."
"No known quantities in the darned thing," Marr protested.
"No," Wadlin agreed.
"Why, it can't be solved," Marr cried, but Wadlin dissented.
"It's been done," he assured the other man. "You see, you do know the monkey got five coconuts, and you do know there were five men. It's been worked out, but I don't know how to go at it."
For a moment there was silence; then Marr cried, "Wait a minute; I see it now." And began to write again.
Wadlin watched him with a contented little smile.
"Five must be a factor, somehow," he would say. "I think the number is some kind of power of five."
And when that line of search was exhausted he would propose that they seek to make a formula.
"There must be one," he urged; "one that would fit all such problems."
Their formula, when they got one, covered four lines of figures and letters, running clear across the sheet; when they solved it --- and that took an hour and ten minutes --- the result they got was 7; a manifest impossibility.
They checked the formula for errors and found four, and made a new one which would not solve; they strove and panted and perspired; and little Wadlin, with the utmost gravity, made long calculations which came to nothing; solved tremendous equations which gave an absurd result at the end. Sheer weariness drove them at last to temporary surrender; ...
Ben Williams didn't include the answer in his short story! And, in a subsequent issue of the magazine, the editor, George Horace Lorimer, admits to having a major problem. In the weeks after the story ran, The Saturday Evening Post received several thousand letters both asking for the answer or proposing a solution to this problem. Being inundated with letters, the editor sent Williams the following telegram: "For the love of Mike, how many coconuts? Hell popping around here." [While I was at the Boston Library, I skimmed through the subsequent year's worth of Saturday Evening Posts looking for the answer. Now I may have missed the "Coconuts" solution, as a couple issues were missing, but I didn't find the answer published subsequently in the magazine. I think that, just like Archimedes not having the solution to his "Cattle" Problem, Ben Williams may not have had the answer to his "Coconuts'" Problem and may not have known how to solve it either.]
(And to make a short story even shorter --- of course Marr and Wadlin spent the entire night, until Marr fell asleep, working on this coconuts problem. In doing so, Marr lost track of the correct time as his watch had wound down during the night because he had forgotten to wind it in his maniacal attack on the coconuts problem. In addition, Wadlin had awoken in the middle of the night and, being both extremely clever and immoral, reset Marr's mantel clock backwards. When Marr awoke in the morning he immediately checked his watch only to find that it had stopped since he had forgotten to wind it; he then looked up at the mantel clock and noted that it was still early so he immediately set upon the task of solving the coconut problem. With the mantel clock set to the incorrect time, Marr failed to enter his bid by the noon deadline that day, allowing Wadlin's team to win the hospital contract.)
As I mentioned above, this is really an ancient problem, with various versions of it dating much earlier. What Williams did was to slightly alter a much older problem. In particular, the most prevalent version of this problem concluded with the morning's five-way split leaving a last coconut, which was again given to the monkey. And this older problem has a long history of solutions and especially attempted solutions. In fact, there is an interesting legend about the solution of this original problem. But first of all, allow me to tell you how I would solve this problem. Really, there are two general methods, one is known as Euler's method, named after the great Leonhard Euler (1707-1783) who single handedly created the branch of mathematics now known as Graph Theory, and the other is continued fractions (okay, so I have told you how continued fractions are connected to coconuts, but I won't fill in the details here). Personally, I favor the continued fraction methodology because of its shear elegance, and I use the continued fraction method for solving this and other Diophantine indeterminate problems. In fact, Math for the Motivated includes a chapter discussing this extension of normal fractions and some of the marvelous consequences of continued fractions, not only for Diophantine problems but also for irrational numbers and even for explaining the line, "Flowers Can Count!" that is discussed in detail in the "Sequences and Series" Chapter 24 of Math for the Motivated. Now allow me to return to the legend.
Paul A. M. Dirac, a famous Nobel laureate physicist, was the first person to solve what was recognized as a major problem for early Quantum Mechanics [Quantum Mechanics is the theory supplanting Newtonian Mechanics for center stage in physics. It is the theory that the whole Universe follows ... okay, so it doesn't yet include gravity which is a major problem, so we still must somehow incorporate Einstein's General Relativity with Quantum Mechanics.], and that problem was that the Schrodinger Equation did not follow Einstein's Special Relativity. Thus, Quantum Mechanics explained the motions of molecules, atoms in molecules, electrons in atoms, protons and neutrons in the nuclei of atoms, and even the motions of quarks contained in the protons in those nuclei of the atoms of the Universe, but it only explained the slow moving particles. When the particles approached the speed of light, Schrodinger's Equation breaks down. P. A. M. Dirac, in the late 1920s, conceived a beautiful solution to this relativistic problem, now known as the Dirac Equation. [Another physicist, Max Born, who later won the 1954 Nobel Prize in Physics for his work on the statistical nature of wave functions, was quoted in 1928 as telling a group of visitors to Gottingen University, "Physics, as we know it, will be over in six months." His overconfidence was founded in Dirac's recent discovery of the equation linking Special Relativity with Quantum Mechanics. And Dirac later shared with Schrodinger the 1933 Nobel physics prize for their contributions to quantum theory.] And, falling out of his equation were negative energy solutions! [It is sort of like negative temperatures --- everyone knows about "absolute zero'' being the lowest possible temperature (0 degrees Kelvin, that is, and for equilibrium systems), so who would ever think about negative temperatures? No one, of course. The same closed-minded thinking grew up around the positive energy solutions to Schrodinger's equation. No sane physicist would ever think about negative energy solutions.] Most physicists at the time discounted the negative energy solutions as non-physical, calling them simply figments of the mathematics and thus unreal. The brilliance of Dirac, on the other hand, was to take the negative energy solutions to his equation seriously. Consequently, Dirac showed that a vacuum (what is known as the vacuum state, in the parlance of Quantum Mechanics) is not really empty as you might expect, rather it is perfectly full! In words, when all of the negative energy states are completely filled with particles, we call this the vacuum state. When a surplus particle then fills a positive energy state, then this corresponds to a particle of matter, such as an electron or proton. This is a remarkable idea, for it explains so many mysteries that had been noticed in the physics of subatomic particles. In particular, Dirac went on to predict the existence of antimatter, and he did so years before the first antimatter particle, the positron, was discovered experimentally. What Dirac said is that if one has enough energy, then one of the negative energy particles can be pushed out of its negative energy state. When this happens the freed particle reappears as a positive energy particle of matter, and the "hole" in the vacuum left behind by the missing particle corresponds to a newly created particle, but this particle has a positive charge --- it is thus an antiparticle composed of antimatter! And, when a particle meets an antiparticle (that is, the matter particle returns to fill the "hole" left in the vacuum), they annihilate each other releasing vast amounts of energy with the particle and antiparticle disappearing in the process (in reality, the particle fills the "hole" returning the vacuum to its totally full, but apparently empty, state and thus the particle and antiparticle appear to disappear)! Of course, all of these vacuum processes, what are known as vacuum fluctuations, have been observed empirically and lead to some rather remarkable effects like the Casimir effect. And in the 1970s Stephen Hawking employed precisely these vacuum fluctuations (particle/antiparticle generation) to show what is affectionately known as the "Hairy Black Hole Theorem". [This is in contrast to the "No-hair Black Hole Theorem" which is a classical General Relativity and Electrodynamics result stating that black holes only retain three properties: mass, electric charge, and angular momentum. The "Hair" on a Black Hole refers to the fact that Black Holes have additional properties besides mass, charge, and angular momentum. Black Holes also have an entropy, a temperature, and emit radiation at that temperature as if they were a normal hot body (what is known as a black body radiator).] This theorem tells us that black holes do emit something (unlike the physical attributes usually attributed to black holes which say that nothing can escape from a black hole, not even light) and thus they actually slowly "evaporate'' into nothingness. [This idea that black holes can emit radiation was an equally unpopular concept to Hawking's fellow physicists as was Dirac's ideas concerning the negative energy solutions to his equation. In fact, Hawking himself argued against such an occurrence initially, but was later convinced of its validity during a trip to Russia. Then, in February 1974 at Oxford when Hawking first proposed his "Hairy Black Holes" at a conference, the chairman John Taylor, a well-known mathematician, proclaimed after Hawking's talk, "I'm sorry Stephen, but this is absolute rubbish." Of course, Hawking's "Hairy Black Holes" are now widely accepted by physicists.] Don't worry, it will take roughly 10^{66} or more years [For comparison, the age of our Universe is estimated to be between 1x10^{10} and 2x10^{10} years old.] for even a small black hole to evaporate, so the black hole at the center of the Milky Way galaxy that holds our galaxy together is safe for our lifetimes. Ah, but I'm getting a little too far afield with my "connections", so let's return to the coconut problem. The importance of all of this talk of physics is to impress upon us that believing in negative solutions can be extremely important. This belief led Dirac to propose the existence of antimatter at a time when electrons and protons were believed to be the only subatomic particles.
A widely held legend has it that Dirac proposed a solution strategy for the Coconut Problem. And, his solution entailed what might be called "anti-coconuts". Now, given the above history of Dirac's prediction of antimatter based upon negative energy solutions to his relativistic quantum mechanical equation, it is very reasonable to make this assumption that Dirac also authored this new solution strategy. You see, this new strategy for the coconut problem also depends upon negative numbers. And I think this is enough of a hint as to the nature of Dirac's solution method, perhaps it is too much of a hint, I hope I haven't ruined it for you. In any case, there is an ingenious way to solve the coconut problem without knowing any theorems about continued fractions or Euler's method for solving Diophantine problems. You may find it intriguing to attempt to find Dirac's method. Now, I started by calling this little aside a legend, the reason for my using this term is because when Dirac was queried as to whether he had authored this strategy, he answered that no, it was not his but that he had heard the method from a mathematician at Oxford and liked it so much that he often reiterated this solution. So, once again, we don't know the true genesis of even this latter and later solution method.
I'll let you attack these problems now --- well, on second thought, let me give you the answer to the original version and you can work on Williams's modification. Remember that Williams's version ended without a coconut left to give to the Monkey while the original version again had one coconut left after the morning's split to give to the monkey. So, this is what you've been waiting for: the answer to the original problem is 15,621 coconuts. Don't fret, I haven't given away the kitchen sink. This answer will most likely be of little use to you for solving Williams's version, excepting that you might use it to check whatever solution method you devise since you can apply your method to the original problem to see if you obtain 15,621 coconuts. This will verify the correctness of your methodology so that you will be more confident in applying it to Williams's problem.
Good luck!
P.S. There are in fact an infinite number of solutions to this problem. The $20 reward is won only by the smallest solution.Note: Dirac's interpretation of negative energy states being completely `filled' in the vacuum, and thus when enough energy is supplied to `pop' one of these filling electrons out of the vacuum state leaving behind a `hole' that becomes the positron, is just one possible interpretation of these states. This is because in Dirac's Equation the energy and time appear together as a product, as in `Et'. Thus Dirac's negative energy states actually appear as `-Et' in one of his equation's solutions, and this is what Dirac interpreted to mean negative energies, i.e., Dirac said the minus sign went with the energy `E'. Richard Feynman (1918-1988), on the other hand, had a different interpretation for `-Et', he said that the minus sign went with `t' and not `E', thus Dirac's negative energy states were rather negative time, or backwards in time, states according to Feynman. Thus Feynman interpreted positrons to be electrons that were traveling backwards in time. [Some physicists disagree with both Dirac's and Feynman's interpretations for antimatter. Richard Muller, in his new book, Now: The Physics of Time, discounts both Dirac's `holes' in the infinite, filled, sea of negative-energy electrons as well as Feynman's electrons moving backward in time interpretations for positrons.] This led Feynman to his famous Feynman diagrams for performing Quantum Electrodynamics (QED) and Quantum Field Theory (QFT) calculations.
- Fourth Coconut Problem:
And now on to the last of the four coconut chestnuts. Now this problem can likewise be solved with a little elementary algebra in under a half-page of effort, but I would not recommend that you attempt this one until you are quite comfortable with the solution to the above famous coconut problem (the third one). This next one, as you will recognize, is just a generalization of the third problem. So it requires you to generalize your solution to the previous problem.
There are S sailors and a monkey ship wrecked on a desert island. The first day they spend gathering a pile of coconuts which are to be divided the next day. During the night, while everyone is sleeping, one sailor awakes, divides the pile into S equal shares, finds one coconut left over which she gives to the monkey. She then hides her share and gathers the remaining coconuts into a pile. And each of the remaining S-1 sailors repeats this performance. In the morning, the sailors find that the pile of coconuts is exactly divisible by S. What was the minimum number of coconuts in the original pile? And, what if there were M monkeys instead of one? (In other words, after each of the S late night partitions, there were M coconuts left over which were distributed to the M monkeys.)
Go at it ...
- Oh No! Not a Fifth Coconut Problem:
Okay, for those that are feeling a little overwhelmed by your attempts at generalizing the five sailor and one monkey problem to S sailors and M monkeys, below is an easier coconut problem to restore your confidence.
Say there were only two sailors, and after collecting the coconuts the first day they pile them up into a pile shaped like a pyramid with a triangular base (really just a regular tetrahedron). Late that night, one sailor has decided to split the original coconut pile into two piles, but not knowing how to count he decides to restack the coconuts into two new pyramids with triangular bases. (These two new pyramids are not necessarily equal in size at this point.) Both of these piles are also in the shape of a pyramid with a triangular base. What is the minimum number of coconuts that the two sailors initially collected?
Here are the solutions to the Coconut chestnuts:
This PDF contains the analysis and derivation of the solutions, including Dirac's solution (a very limited and specific strategy) and the Continued Fractions strategy that solves all linear Diophantine Indeterminate Equations. This PDF, while performing the necessary calculations, assumes that you already have a deep understanding of Continued Fractions. Continued Fractions actually are quite remarkable, and their properties are instrumental for describing and calculating irrational numbers, as well as for solving Diophantine equations. The above PDF solution file does not provide an introduction to these properties of continued fractions, however. If you are interested in the details of Continued Fractions, then read the following PDF file that contains the beginning sections of the my chapter on Continued Fractions:
The Continued Fractions Chapter relies heavily on Mathematical Induction in order to establish the theorems and properties for Continued Fractions, so if you need to review Mathematical Induction, a primer is included on this page: Mathematical Induction
And here is the Sum/Difference of Powers Theorem:
Copyright (c) 2010-2017 Craig G. Shaefer, all rights reserved.
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