The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.

Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.

As I have previously reported, my fellow students and I have found our curiosity drawn to the calculus of sequences, in which we define analogues of the derivatives and integrals of functions for a sequence s_n with the operators

  Δ s_n = s_n - s_n-1

and

		n
	Δ-1 sn =	Σ	si
		i = 1

respectively, where Σ is the summation sign, for which we interpret all non-positively indexed elements as zero.

I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.

Last time they met, the Baron challenged Sir R----- to turn a square of twenty five coins, all but one of which the Baron had placed heads up, to tails by flipping vertically or horizontally adjacent pairs of heads.
As I explained to the Baron, although I'm not at all sure that he was following me, this is essentially the mutilated chess board puzzle and can be solved by exactly the same argument. Specifically, we need simply imagine that the game were played upon a five by five checker board...

The interest of my fellow students and I has been somewhat piqued of late by a curious similarity of the relationship between sequences and series to that between the derivatives and integrals of functions. Specifically, for a function f taking a non-negative argument x, we have

		x
	F(x) =	∫	f(x) dx
		0
	f(x) =	d	F(x)
		dx

and for a sequence s having terms

  s₁, s₂, s₃, ...

we can define a series S with terms

		n
	Sn = s1 + s2 + s3 + ... + sn =	Σ	si
		i = 1

where Σ is the summation sign, from which we can recover the terms of the sequence with

  s_n = S_n - S_n-1

using the convention that S₀ equals zero.
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.

Recall that the Baron proposed a pair of dice contests in which Sir R-----, were he to best the Baron's score, stood to win a bounty of thirteen coins.
Upon paying his stake Sir R----- was to cast his die but, if unhappy with its outcome, could pay a further coin to cast it again. Likewise, if he were not satisfied with the second cast, he could elect to cast a third time for a further two coins. He could continue in this fashion for as long as he pleased with the cost rising by one coin for each additional cast of his die. The Baron was to have but a single cast of his die, with Sir R----- to determine whether after or before his own play according to his stake; seven coins for the former and eight for the latter.

In our recent investigations we have found that games comprising of random returns upon funds, of random trades between players and of random outcomes of labour, trade and sustenance, with the latter subject to some bare minimum of expenditure, invariably rewarded a fortunate few at the expense of an unfortunate many, despite having rules that applied perfectly equitably to all.
For our final analysis, my fellow students and I have sought to develop a rule by which we might cuff the hands of providence!

In the Baron's most recent wager, he was to roll a twenty sided die marked with the digits zero to nine twice apiece and place it either upon a space representing tens or upon another representing ones according to his fancy, after which Sir R----- was to do the same. Then the Baron and Sir R----- were to roll a second die each and place them upon their empty spaces. If the number thus made by the Baron was smaller than that made by Sir R-----, then Sir R----- was to have a prize of twenty nine coins from the Baron, otherwise the Baron was to have one of thirty coins from Sir R-----.

My fellow students and I have spent some time investigating my suspicion that serendipity, beyond worth, might account for the relative fortune of the few over the many. To this end we have set to creating perfectly fair games mimicking the manner in which wealth accumulates amongst the populace, that we might discover whether their outcomes should elevate some small lucky band of players well above their fellows.
Thus far we have seen that games of both random returns and losses of players' funds and random trade between them most certainly do so, but their rules failed to take into account either the value of labour or the cost of sustenance, somewhat weakening any conclusions that we might have drawn from their study.
We have consequently spent some time creating further rules to rectify these deficiencies.

The Baron's latest game consisted of up to four turns throwing a pair of dice and cost nine coins to play. After each turn Sir R----- may have elected to stop playing and collect their sum as winnings.
On hearing these rules, it immediately occurred to me that he should only continue the game if he has thrown a sum less than that he might expect to win in future turns.
We can thus reckon the expected winnings before throwing the dice by considering the expected winnings after throwing them, conditional upon the cases of having done better and having done worse than expected in future turns.

I explained this insight to the Baron, but fear I may not have done so with sufficiently clarity since I was struck with the impression that he had not fully understood me. Hopefully I shall do better with this exposition!

Recall that my fellow students and I have resolved to investigate the role of chance in the spread of wealth amongst the populace by creating a series of games with which we might approximate its ebb and flow. Our first such game was of so simplistic a construction that it offered no meaningful insights into the matter, but at least shed some light upon the manner in which we might answer such questions as what are the chances that, in a perfectly fair game, a few players might fare significantly better than the many or what are they that a player having had a run of poor luck might conclude the game ahead of a fellow who had had better?
We have since spent some mental effort improving the rules of our game and it is upon these changes that I shall now report.