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On Tug O’ War – student

student from thus spake a.k.

The Baron and Sir R-----'s latest wager comprised of first placing a draught piece upon the fifth lowest of a column of twelve squares and subsequently moving it up or down by one space depending upon the outcome of a coin toss until such time as it should escape, either by moving above the topmost or below the bottommost square. In the former outcome the Baron should have had a prize of three coins and in the latter Sir R----- should have had two.
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Finally On A Very Cellular Process – student

student from thus spake a.k.

Over the course of the year my fellow students and I have been utilising our free time to explore the behaviour of cellular automata, which are mechanistic processes that crudely approximate the lives and deaths of unicellular creatures such as amoebas. Specifically, they are comprised of unending lines of boxes, some of which contain cells that are destined to live, dive and reproduce according to the occupancy of their neighbours.
Most recently we have seen how we can categorise automata by the manner in which their populations evolve from a primordial state of each box having equal chances of containing or not containing a cell, be they uniform, constant, cyclical, migratory, random or strange. It is the latter of these, which contain arrangements of cells that interact with each other in complicated fashions, that has lately consumed our attention and I shall now report upon our findings.
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Further Still On A Very Cellular Process – student

student from thus spake a.k.

My fellow students and I have lately been spending our spare time experimenting with cellular automata, which are simple mathematical models of single celled creatures such as amoebas, governing their survival and reproduction from one generation to the next according to the population of their neighbourhoods. In particular, we have been considering an infinite line of boxes, some of which contain living cells, together with rules that specify whether or not a box will be populated in the next generation according to its, its left hand neighbour's and its right hand neighbour's contents in the current generation.
We have found that for many such automata we can figure the contents of the boxes in any generation that evolved from a single cell directly, in a few cases from the oddness or evenness of elements in the rows of Pascal's triangle and the related trinomial triangle, and in several others from the digits in terms of sequences of binary fractions.
We have since turned our attention to the evolution of generations from multiple cells rather then one; specifically, from an initial generation in which each box has an even chance of containing a cell or not.
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On May The Fours Be With You – student

student from thus spake a.k.

In their most recent wager Sir R-----'s goal was to guess the outcome of the Baron's roll of four four sided dice at a cost of four coins and a prize, if successful, of forty four. On the face of it this seems a rather meagre prize since there are two hundred and fifty six possible outcomes of the Baron's throw. Crucially, however, the fact that the order of the matching dice was not a matter of consequence meant that Sir R-----'s chances were significantly improved.
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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.
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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.
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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.
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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.
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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.