On The Octogram Of Seth LaPod

The latest wager that the Baron put to Sir R----- had them competing to first chalk a triangle between three of eight coins, with Sir R----- having the prize if neither of them managed to do so. I immediately recognised this as the game known as Clique and consequently that Sir R-----'s chances could be reckoned by applying the pigeonhole principle and the tactic of strategy stealing. Indeed, I said as much to the Baron but I got the distinct impression that he wasn't really listening.

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Further Still On An Ethereal Orrery

Recently, my fellow students and I constructed a mathematical orrery which modelled the motion of heavenly bodies employing Sir N-----'s laws of gravitation and motion, rather than clockwork, as its engine. Those laws state that bodies are attracted toward each other with a force proportional to the product of their masses divided by the square of the distance between them, that a body will remain at rest or in constant motion unless a force acts upon it, that if a force acts upon it then it will be accelerated in the direction of that force at a rate proportional to its strength divided by its mass and that, if so, it will reciprocate with an opposing force of equal strength.
Its operation was most satisfactory, which set us to wondering whether we might use its engine to investigate the motions of entirely hypothetical arrangements of heavenly bodies and I should now like to report upon our progress in doing so.

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On The Hydra Of Argos

When the Baron last met with Sir R-----, he proposed a wager which commenced with the placing of twenty black tokens and fifteen white tokens in a bag. At each turn Sir R----- was to draw a token from the bag and then put it and another of the same colour back inside until there were thirty tokens of the same colour in the bag, with the Baron winning a coin from Sir R----- if there were thirty black and Sir R----- winning ten coins from the Baron if there were thirty white.
Upon hearing these rules I recognised that they described the classic probability problem known as Pólya's Urn. I explained to the Baron that it admits a relatively simple expression that governs the likelihood that the bag contains given numbers of black and white tokens at each turn which could be used to figure the probability that he should have triumphed, but I fear that he didn't entirely grasp my point.

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Further On An Ethereal Orrery

Last time we met we spoke of my fellow students' and my interest in constructing a model of the motion of heavenly bodies using mathematical formulae in the place of brass. In particular we have sought to do so from first principals using Sir N-----'s law of universal gravitation, which states that the force attracting two bodies is proportional to the product of their masses divided by the square of the distance between them, and his laws of motion, which state that a body will remain at rest or in constant motion unless a force acts upon it, that it will be accelerated in the direction of that force at a rate proportional to its magnitude divided the body's mass and that a force acting upon it will be met with an equal force in the opposite direction.
Whilst Sir N----- showed that a pair of bodies traversed conic sections under gravity, being those curves that arise from the intersection of planes with cones, the general case of several bodies has proved utterly resistant to mathematical reckoning. We must therefore approximate the equations of motion and I shall now report on our first attempt at doing so.

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On Pennies From Heaven

Recall that the Baron and Sir R-----'s most recent wager first had the Baron place three coins upon the table, choosing either heads or tails for each in turn, after which Sir R----- would follow suit. They then set to tossing coins until a run of three matched the Baron's or Sir R-----'s coins from left to right, with the Baron having three coins from Sir R----- if his made a match and Sir R----- having two from the Baron if his did.

When the Baron described the manner of play to me I immediately pointed out to him that it was Penney-Ante, which I recognised because one of my fellow students had recently employed it to enjoy a night at the tavern entirely at the expense of the rest of us! He was able to do so because it's an example of an intransitive wager in which the second player can always contrive to make a choice that will best the first player's.

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On An Ethereal Orrery

My fellow students and I have lately been wondering whether we might be able to employ Professor B------'s Experimental Clockwork Mathematical Apparatus to fashion an ethereal orrery, making a model of the heavenly bodies with equations rather than brass.
In particular we have been curious as to whether we might construct such a model using nought but Sir N-----'s law of universal gravitation, which posits that those bodies are attracted to one another with a force that is proportional to the product of their masses divided by the square of the distance between them, and laws of motion, which posit that a body will remain at rest or move with constant velocity if no force acts upon it, that if a force acts upon it then it will be accelerated at a rate proportional to that force divided by its mass in the direction of that force and that it in return exerts a force of equal strength in the opposite direction.

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On Onwards And Downwards

When last they met, the Baron challenged Sir R----- to evade capture whilst moving rooks across and down a chessboard. Beginning with a single rook upon the first file and last rank, the Baron should have advanced it to the second file and thence downwards in rank in response to which Sir R----- should have progressed a rook from beneath the board by as many squares and if by doing so had taken the Baron's would have won the game. If not, Sir R----- could then have chosen either rook, barring one that sits upon the first rank, to move to the next file in the same manner with the Baron responding likewise. With the game continuing in this fashion and ending if either of them were to take a rook moved by the other or if every file had been played upon, the Baron should have had a coin from Sir R----- if he took a piece and Sir R----- one of the Baron's otherwise.

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Finally On Natural Analogarithms

Over the course of the year my fellow students and I have spent much of our spare time investigating the properties of the set of infinite dimensional vectors associated with the roots of rational numbers by way of the former's elements being the powers to which the latter's prime factors are raised, which we have dubbed -space.
We proceeded to define functions of such numbers by applying operations of linear algebra to their -space vectors; firstly with their magnitudes and secondly with their inner products. This time, I shall report upon our explorations of the last operation that we have taken into consideration; the products of matrices and vectors.

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On The Rich Get Richer

The Baron's latest wager set Sir R----- the task of surpassing his score before he reached eight points as they each cast an eight sided die, each adding one point to their score should the roll of their die be less than or equal to it. The cost to play for Sir R------ was one coin and he should have had a prize of five coins had he succeeded.

A key observation when figuring the fairness of this wager is that if both Sir R----- and the Baron cast greater than their present score then the state of play remains unchanged. We may therefore ignore such outcomes, provided that we adjust the probabilities of those that we have not to reflect the fact that we have done so.

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Further Still On Natural Analogarithms

For several months now my fellow students and I have been exploring -space, being the set of infinite dimensional vectors whose elements are the powers of the prime factors of the roots of rational numbers, which we chanced upon whilst attempting to define a rational valued logarithmic function for such numbers.
We have seen how we might define functions of roots of rationals employing the magnitude of their associated -space vectors and that the iterative computation of such functions may yield cyclical sequences, although we conspicuously failed to figure a tidy mathematical rule governing their lengths.
The magnitude is not the only operation of linear algebra that we might bring to bear upon such roots, however, and we have lately busied ourselves investigating another.

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