Further On A Very Cellular Process – student

student from thus spake a.k.

You will no doubt recall my telling you of my fellow students' and my latest pastime of employing Professor B------'s Experimental Clockwork Mathematical Apparatus to explore the behaviours of cellular automata, which may be thought of as simplistic mathematical simulacra of animalcules such as amoebas.
Specifically, if we put together an infinite line of imaginary boxes, some of which are empty and some of which contain living cells, then we can define a set of rules to determine whether or not a box will contain a cell in the next generation depending upon its own, its left and its right neighbours contents in the current one.

On Fruitful Opals – student

student from thus spake a.k.

Recall that the Baron’s game consisted of guessing under which of a pair of cups was to be found a token for a stake of four cents and a prize, if correct, of one. Upon success, Sir R----- could have elected to play again with three cups for the same stake and double the prize. Success at this and subsequent rounds gave him the opportunity to play another round for the same stake again with one more cup than the previous round and a prize equal to that of the previous round multiplied by its number of cups.

On A Very Cellular Process – student

student from thus spake a.k.

Recently my fellow students and I have been spending our free time using Professor B------'s remarkable calculating engine to experiment with cellular automata, being mathematical contrivances that might be thought of as crude models of the lives of those most humble of creatures; amoebas. In their simplest form they are unending lines of boxes, some of which contain a living cell that at each generation will live, die or reproduce according to the contents of its neighbouring boxes. For example, we might say that each cell divides and its two offspring migrate to the left and right, dying if they encounter another cell's progeny.

On Two By Two – student

student from thus spake a.k.

The Baron's most recent wager with Sir R----- set him the challenge of being the last to remove a horizontally, vertically or diagonally adjacent pair of draughts from a five by five square of them, with the Baron first taking a single draught and Sir R----- and he thereafter taking turns to remove such pairs.

When I heard these rules I was reminded of the game of Cram and could see that, just like it, the key to figuring the outcome is to recognise that the Baron could always have kept the remaining draughts in a state of symmetry, thereby ensuring that however Sir R----- had chosen he shall subsequently have been free to make a symmetrically opposing choice.

Finally On An Ethereal Orrery – student

student from thus spake a.k.

Over the course of the year, my fellow students and I have been experimenting with an ethereal orrery which models the motion of heavenly bodies using nought but Sir N-----'s laws of gravitation and motion. Whilst the consequences of those laws are not generally subject to solution by mathematical reckoning, we were able to approximate them with a scheme that admitted errors of the order of the sixth power of the steps in time by which we advanced the positions of those bodies.
We have thus far employed it to model the solar system itself, uniformly distributed bodies of matter and the accretion of bodies that are close to Earth's orbit about the Sun. Whilst we were most satisfied by its behaviour, I should now like to report upon an altogether more surprising consequence of its engine's action.

On The Octogram Of Seth LaPod – student

student from thus spake a.k.

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.

Further Still On An Ethereal Orrery – student

student from thus spake a.k.

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.

On The Hydra Of Argos – student

student from thus spake a.k.

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.

Further On An Ethereal Orrery – student

student from thus spake a.k.

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.

On Pennies From Heaven – student

student from thus spake a.k.

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.