On Twenty-Niner

The Baron's most recent wager set Sir R----- the task of placing tokens upon spaces numbered from zero to nine according to the outcome of a twenty sided die upon which was inscribed two of each of those numbers. At a cost of one coin per roll of the die, Sir R-----'s goal was to place a token upon every space for which he should receive twenty nine coins and twenty nine cents from the Baron.

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On A Clockwork Contagion

During the recent epidemic, my fellow students and I had plenty of time upon our hands due to the closure of the taverns, theatres and gambling houses at which we would typically while away our evenings and the Dean's subsequent edict restricting us to halls. We naturally set to thinking upon the nature of the disease's transmission and, once the Dean relaxed our confinement, we returned to our college determined to employ Professor B------'s incredible mathematical machine to investigate the probabilistic nature of contagion.

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On Tug O' War

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

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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On We Three Kings

Recall that the Baron's most recent game involved advancing kings from the first and last ranks of a three by three chequerboard in a pawn-like manner until either he or Sir R----- reached the opposing rank or blocked all of the other's kings from moving, having the game in either eventuality.

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Further Still On A Very Cellular Process

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

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

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

Recall that the Baronâ€™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

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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Gallimaufry

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