Recently in recreational mathematics Category

Further On A Calculus Of Differences

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 sn with the operators

  Δ sn = sn - sn-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.

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On A Calculus Of Differences

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

  sn = Sn - Sn-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.

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Finally On The Wealth Of Stations

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!

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Further Still On The Wealth Of Stations

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.

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Further On The Wealth Of Stations

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.

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On The Wealth Of Stations

As much as I enjoy the Baron's tall tales, I am no more convinced of their veracity than I am of his fervent belief that the nobility enjoy the benefits of their station because they are of better mettle than the likes of you and I. No, I am rather of the opinion that serendipity, above worthiness, maketh man!
To investigate the possibility that luck might well account for the elevation of the few above the many, my fellow students and I naturally turned to the mathematical arts and we resolved to create a series of games with which we might model the ebb and flow of prosperity.

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Finally On An Arithmetical Pursuit

We have thus far figured many of the mathematical properties of the game of arithmetical pursuit; that there is no target that cannot be hit given a fortuitous deal of the cards, that there are something of the order of 210,000,000,000,000 possible results of admissible formulae and that, if randomly chosen, they are approximately governed by a power law distribution, being one in which the probability of observing a value of x is more or less proportional to x-α for some α greater than one.
The last question that my fellow students and I should like answered is that of how likely it might be that we should be able to hit the target.

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Further Still On An Arithmetical Pursuit

We have spent some time now figuring the mathematical properties of the game of arithmetical pursuit in which the goal is to arithmetically manipulate six randomly chosen integers to land as close as possible to a randomly chosen target, using only addition, subtraction, multiplication and division and admitting no fractions.
We have thus far shown that any target can be hit with the right deal of the cards and that there are something along the lines of 210,000,000,000,000 admissible formulae that might arise during the game.
The next question that my fellow students and I should like answered is just how frequently any given number might be the result of those formulae.

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Further On An Arithmetical Pursuit

You will recall that my fellow students and I have been investigating the properties of arithmetical pursuit; a game that we often play at afternoon tea in which the goal is to take six randomly chosen integers and create an arithmetical formula comprised of additions, subtractions, multiplications and/or divisions to yield an integer as close as possible to a randomly chosen target between one and one thousand.

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On An Arithmetical Pursuit

Whilst taking our afternoon tea, my fellow students and I occasionally pass the time with some arithmetical pursuit, being a game in which the goal is to be the swiftest to manipulate six randomly chosen integers with addition, subtraction, multiplication and division to hit a randomly chosen target, or at least to land as closely as possible to it, admitting no more than one appearance of each number and no fractions.

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