Samuel Pepys, the famed diarist of 17th Century London, was born in modest circumstances and educated as a scholarship student at Cambridge. His education was exclusively in the classics. There were not many math/science students at that time. (Newton's lectures at Cambridge several decades later were very poorly attended, and it would be more than a century before Yale had its first math professor.) In 1662, when Pepys was "clerk of the acts" of the Navy Board, involved in purchasing stores for the Royal Navy, he undertook at the age of 29 to learn arithmetic. He found the multiplication tables particularly challenging, and he rose regularly at 4 am and stayed up late to study them. They were doubtless useful for someone in his line of work. The next year he enjoyed teaching his new-found arithmetic skills to his wife, but opted to skip division.

During the restoration of the Stuart monarchy, Pepys was one of the most influential men in England. As Secretary of the Affairs of the Admiralty he presided over the recovery of the British Navy and helped make it dominant over those of France and the Netherlands, which had dealt Britain an ignominious naval defeat in 1667. Despite his lack of science training as a student, he had natural curiosity and became quite interested in science.

Six years after the Principia was published, and five years after the Stuart King James II had been exiled, forcing Pepys into retirement, Pepys at the age of 60 carried on a brief correspondence with Newton, who at 50 had already made his reputation as the preeminent scientist of the age. This correspondence shows that in addition to curiosity Pepys had the following traits, which are important for a science student of any era:

(1) disinclination to accept answers based solely on the authority of an expert

(2) ability to recognize when he did not truly understand, and willingness to admit his lack of understanding

(3) gumption to persist in asking for help, with modesty and good nature.

Between November 22 and December 23, 1693, Pepys in London exchanged six letters with Newton in Cambridge on a problem about gambling odds. The ostensible reason for Pepys's interest was to encourage the thirst for truth of his young friend, Mr. Smith, though he later revealed in a letter to George Tollet that he himself was about to stake 10 pounds (equivalent to about $1500 today) on such a bet. [There is genuine scientific relevance - see the Chemical Problem below]

The deeper reason for their eagerness to correspond at this time (there is only one other letter from Pepys in The Correspondence of Isaac Newton ) may have been that on September 13, Newton had sent Pepys a weird, hostile letter causing Pepys to worry about Newton's sanity. So both parties were trying to restore friendly relations.

The first letter to Newton introduces Mr. Smith, who has a "general reputation...in this towne (inferiour to none, but superiour to most) for his maistery [of]...Arithmetick". At the close of the letter Pepys states his problem:

Newton obliges very quickly but notes an ambiguity:

I was very glad to hear of your good health by Mr Smith & to have any opportunity given me of shewing how ready I should be to serve you or your friends upon any occasion & wish that something of greater moment would give me a new opportunity of doing it so as to become more useful to you than in solving only a mathematical question. In reading ye Question it seemed to me at first to be ill stated...

So Newton restates the question:

and he provides the answer with a brief explanation:

If the Question be thus stated, it appears by an easy computation that the expectation of A is greater then that of B or C, that is, the task of A is the easiest. And the reason is because A has all the chances of sixes on his dyes for his expectation but B & C have not all the chances on theirs. For when B throws a single six or C but one or two sixes they miss of their expectations.

and an offer of further help:

But whether I have hit the true meaning of the Question I must submit to ye better judgments of yourself & others. If you desire ye computation I will send it you.

Pepys expresses his deep gratitude but freely admits his ignorance (remember, he came late to arithmetic and desires further help, including to see the real calculation rather than just the conclusion:

...You give it in favour of ye Expectations of A, & this (as you say) by an easy Computation. But yet I must not pretend to soe much Conversation wth Numbers, as presently to comprehend as I ought to doe, all ye force of that wch you are pleas'd to assigne for ye Reason of it, relating to their having or not having ye Benefit of all their Chances; and therefore were it not for ye trouble it must have cost you; I could have wish'd for a sight of ye very Computation.

After so much "chawing of ye Question" Pepys suggests reframing the problem in a somewhat less numerical format involving the execution of Peter:

I must confesse, were I now (after soe much chawing of ye Question) to begin my pursuit afresh after a Solution of it; I think I should avoid some of the Ambiguitys that commonly hang about our Discoursings of it, by changing ye characters of ye dice from Numbers to Letters, & supposing them instead of 1, 2, 3, &c to bee branded with ye 6 initial Letters of ye Alphabet A, B, C, D, E, F. And the Case should then bee this;

Peter a Criminal convict being doom'd to dye, Paul his Friend prevails for his having ye benefitt of One Throw only for his Life, upon Dice soe prepared; with ye Choice of any one of these Three Chances for it, viz.

Newton reiterates his conclusion:

In stating the case of the wager, you seem to have exactly the same notion of it with me; & to the question; which of the three chances should Peter chuse were he to have but one throw for his life? I answer, that if I were Peter, I would chuse the first.

After providing the details of his computation, involving Progressions which are not trivial to follow but which allow him to enumerate all the possibilities, Newton concludes:

The question might have been thus stated, & answered in fewer words: if Peter is to have but one throw for a stake of 1000 l. & has his choice of throwing either one six at least upon six dice, or two at least upon twelve, or three at least upon eighteen, which throw ought he to chuse; & of what value is his chance or expectation upon every throw, were he to sell it? Answer: upon six dice there are 46656 chances, whereof 31031 are for him; upon 12, there are 2176782336 chances, whereof 1346704211 are for him: therefore his chance or expectation is worth the 31031 / 46656th part of 1000 l. in the first case, & the 1346704211 / 2176782336th part of 1000 l. in the second; that is 665 l. 0 s. 2 d. [*] in the first case, and 618 l. 13 s. 4 d. in the second. In the third case, the value will be found still less.

[*] Though he gets the fractions and the pounds right, Newton has some trouble with shillings and pence! The first case should be worth 665 l. 2 s. 5 p. (as pointed out by H. W. Turnbull, the editor of Newton's Correspondence). Note that Newton does not supply the calculation for the more complex third case.

Pepys freely admits that he still needs help in understanding WHY there is a difference:

Nor must I conceal my being at ye same loss how to comprehend, even when flinging 12 Dyes at one throw out of a Single Box (the said dyes being tinged, halfe Green, half blew) my being less provided for turning up a Six with either of these different colour'd Parcels while flung together out of ye same Box, then were ye Six blew to bee thrown out of one Box, & the 6 Green from another; in which latter Case, I presume each of them severally would bee equally entituled to ye producing of a Six with A's Six white Ones, & by consequence of 2 when flung together. I am conscious enough that this is but fumbling, & that it ariseth only from my not knowing how to make ye full use of your Table of Progressions: but pray bee favourable to my unreadiness in keeping pace with you therein, & give mee one line of further helpe.

Newton introduces James, as B, to accompany Peter, as A, and provide a clearer explanation of his reason for the difference:

Were James to have twice as many throws as Peter & as often as he throws a six to win half as much as Peter doth by the like throws, & by consequence were James to win as much at every two such throws as Peter doth at every one such throw, & half as much at every such single throw their cases would be equal. But this is not the case of the wager. As the wager is stated Peter must win as often as he throws a six but James may often throw a six & yet win nothing because he can never win upon one six alone. If Peter flings a six (for instance) four times in eight throws he must certainly win four times, but James upon equal luck may throw a six eight times in sixteen throws & yet win nothing. For as the Question in the wager is stated he wins not upon every single throw with a six as Peter doth, but only upon every two throws wherein he throws at least two sixes. And therefore if he flings but one six in the two first throws & but one in the two next & but one in the two next & so on to sixteen throws, he wins nothing at all tho he throws a six twice as often as Peter doth, & by consequence have equal luck with Peter upon ye dyes.

That A (in our Question) has an easier Taske than B, & a yet more easy one than C; such (I take it) being ye Doctrine of this Paper, & full glad I am of my soe seasonably meeting with it, as being upon ye very Brink of a Wager (10 pounds deep) upon my former Belief.

Pepys jokes about how he might weasel out of this bet but still remain vulnerable to a further related wager in a London coffee house. He persists in wanting to understand:

But Apostacy (wee all know) is now no Novelty, & therefore like others I shall endeavour to make ye best of mine, & face my Antagonist downe that I always meant thus. But then I must begg Your Ayde, that I may not be outbrav'd (as I have sometimes seen it done at Garraways) by a Cross-Offer, & for want of knowing well why, not know which to stick to. But this will require another Cast of Your Kindnesse; for I cannot bear the Thought of being made Master of a Jewell I know not how to wear.

Pepys closes by proposing to meet Tollet for a help session on that same afternoon (a Wednesday) saying "Friday is too far off". He obviously was quick to seek help not only from the professor, but also from the "teaching assistants". We can hope that it finally clicked for him.

Since there is no further documentary evidence on whether Pepys ever truly succeeded in learning how to "wear this jewel", it is hard to know how he would have done in Chem 125. He certainly gave this problem his best shot in a way that has helped students in Chem 125 succeed - he never pretended to understand when he did not, and he persisted in seeking help wherever it might be available.

Clearly Newton understood how to solve Pepys's problem, and he got the right numerical answer, so shouldn't he get 100% if this were an exam question in Chem 125?

Arguably not quite, because questions on Chem 125 exams often ask the student to explain why the answer is as it is - not just to write "yes" or "6.182", or "more ortho than para".

Pepys kept asking "Why," so Newton finally explained in terms that went beyond mathematical manipulation. In his December 23 letter he elaborated on the reason to which he had alluded cryptically on November 26. His explanation was that whereas Peter, who is throwing only 6 dice for at least one 6, is assured of winning every time he throws a 6, James, who is throwing 6 dice twice to make at least two 6s, has no such assurance, because he might couple a throw with one 6 with another that had none.

This would make it appear that Newton might think that this explanation is general, so that throwing four-sided (tetrahedral) dice in a group of 6 to get at least one 4 should be a better bet that throwing 12 such dice to get two 4s. The same principle would seem to apply, but in fact the chance of at least one success in throwing 6 tetrahedra is 1-(3/4)^6 = 0.822 , while the chance of at least double success with 12 tetrahedra is 1-(3/4)^12-12*(3/4)^11*(1/4) = 0.842.

To show that he had a comprehensive view Newton should have mentioned that there could be cases when throwing twice might be an advantage, because of the possibility of rescuing a throw with no 6s by a throw with 2 or more sixes. The more likely it is for a 6 to come up, the more important this favorable second term should be. The second term dominates (that is, two throws is favored over one throw) when the number of faces on the dice is small, or the number of dice being thrown as a group is large, so that the chance of multiple hits on a throw is larger than the chance of zero hits. When the number in the group thrown for one success is more that about 1.3 times the number of faces on a single die, two throws for two successes is favored.

Thus is would not be right to call Newton's answer perfect. Perhaps he perfectly understood the sources of difference and was just trying to keep it simple for Pepys, but he gave no hint that he understood. He clearly showed he could do the math, and he identified the dominant source in the specific case about which he was asked. So 95% seems about right, because it leaves room to award more credit for a really great answer. [Points would not be subtracted for Newton's understandable difficulty with shillings and pence, which was, after all, a side issue.]

Certainly Newton would have been capable of a perfect score on this question, but he had better things to do with his time. Once he had seen the answer key, I doubt that he would have complained or been offended by his 95%, and there is no doubt he would have creamed the course as a whole.

Not surprisingly perfect scores are rare on Chem 125 exams, but the assignment of letter grades is correspondingly generous.

George Tollet and John Jackson, Pepys's nephew, actually did mention the (lesser) advantage of throwing multiple sixes on the second throw of six dice, as well as the disadvantage of throwing none. But they had been helped by studying Newton's answer.

The primary purpose of this web page is obviously to provide an example of attitudes that are important for success in Chemistry 125. The course will focus more on "Why?" and "How do you know?" than on the simple factual answer to questions. Willingness to recognize and admit the need for help, and to seek it, is "crucial".

The problem in probability that supplies the setting for this lesson happens to be relevant to organic chemistry, for example in the area of isotope-ratio mass spectrometry.