studenttag:www.thusspakeak.com,2013-08-07:/student//122018-10-19T19:00:06ZMovable Type 5.2.13On The Rich Get Richertag:www.thusspakeak.com,2018:/student//12.2712018-10-19T19:00:00Z2018-10-19T19:00:06Z
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.
student
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 Analogarithmstag:www.thusspakeak.com,2018:/student//12.2692018-09-21T19:00:00Z2018-09-21T19:10:39Zstudent
For several months now my fellow students and I have been exploring ℓ-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 ℓ-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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On Blockadetag:www.thusspakeak.com,2018:/student//12.2612018-07-20T19:00:00Z2018-07-20T19:00:23Z
Recall that the Baron's game is comprised of taking turns to place dominoes on a six by six grid of squares with each domino covering a pair of squares. At no turn was a player allowed to place a domino such that it created an oddly-numbered region of empty squares and Sir R----- was to be victorious if, at the end of play, the lines running between the ranks and files of the board were each and every one straddled by at least one domino.
student
Recall that the Baron's game is comprised of taking turns to place dominoes on a six by six grid of squares with each domino covering a pair of squares. At no turn was a player allowed to place a domino such that it created an oddly-numbered region of empty squares and Sir R----- was to be victorious if, at the end of play, the lines running between the ranks and files of the board were each and every one straddled by at least one domino.
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Further On Natural Analogarithmstag:www.thusspakeak.com,2018:/student//12.2592018-06-15T19:00:00Z2018-06-19T21:53:14Zstudent
My fellow students and I have of late been thinking upon an equivalence between the roots of rational numbers and an infinite dimensional rational vector space, which we have named ℓ-space, that we discovered whilst defining analogues of logarithms that were expressed purely in terms of rationals.
We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report.
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On Quaker's Dozentag:www.thusspakeak.com,2018:/student//12.2552018-04-20T19:00:00Z2018-05-01T21:42:36Z
The Baron's latest wager set Sir R----- the task of rolling a higher score with two dice than the Baron should with one twelve sided die, giving him a prize of the difference between them should he have done so. Sir R-----'s first roll of the dice would cost him two coins and twelve cents and he could elect to roll them again as many times as he desired for a further cost of one coin and twelve cents each time, after which the Baron would roll his.
The simplest way to reckon the fairness of this wager is to re-frame its terms; to wit, that Sir R----- should pay the Baron one coin to play and thereafter one coin and twelve cents for each roll of his dice, including the first. The consequence of this is that before each roll of the dice Sir R----- could have expected to receive the same bounty, provided that he wrote off any losses that he had made beforehand.
student
The Baron's latest wager set Sir R----- the task of rolling a higher score with two dice than the Baron should with one twelve sided die, giving him a prize of the difference between them should he have done so. Sir R-----'s first roll of the dice would cost him two coins and twelve cents and he could elect to roll them again as many times as he desired for a further cost of one coin and twelve cents each time, after which the Baron would roll his.
The simplest way to reckon the fairness of this wager is to re-frame its terms; to wit, that Sir R----- should pay the Baron one coin to play and thereafter one coin and twelve cents for each roll of his dice, including the first. The consequence of this is that before each roll of the dice Sir R----- could have expected to receive the same bounty, provided that he wrote off any losses that he had made beforehand.
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On Natural Analogarithmstag:www.thusspakeak.com,2018:/student//12.2522018-03-16T19:00:00Z2018-03-16T20:02:14Zstudent
Last year my fellow students and I spent a goodly portion of our free time considering the similarities of the relationships between sequences and series and those between derivatives and integrals. During the course of our investigations we deduced a sequence form of the exponential function e^{x}, which stands alone in satisfying the equations
D f = f f(0) = 1
where D is the differential operator, producing the derivative of the function to which it is applied.
This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithm ln x, albeit in the sense of being expressed in terms of integers rather than being defined by equations involving sequences and series.
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On Lucky Sevenstag:www.thusspakeak.com,2018:/student//12.2472018-01-19T20:00:00Z2018-01-19T20:19:57Z
The Baron's most recent game consisted of a race to complete a trick of four sevens, with the Baron dealing cards from a pristine deck, running from Ace to King once in each suit, and Sir R----- dealing from a well shuffled deck. As soon as either player held such a trick the game concluded and a prize was taken, eleven coins for the Baron if he should have four sevens and nine for Sir R----- otherwise.
The key to reckoning the equity of the wager is to note that it is unchanged should the Baron and Sir R----- take turns dealing out the rest of their cards one by one after the prize has been taken.
student
The Baron's most recent game consisted of a race to complete a trick of four sevens, with the Baron dealing cards from a pristine deck, running from Ace to King once in each suit, and Sir R----- dealing from a well shuffled deck. As soon as either player held such a trick the game concluded and a prize was taken, eleven coins for the Baron if he should have four sevens and nine for Sir R----- otherwise.
The key to reckoning the equity of the wager is to note that it is unchanged should the Baron and Sir R----- take turns dealing out the rest of their cards one by one after the prize has been taken.
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Finally On A Calculus Of Differencestag:www.thusspakeak.com,2017:/student//12.2452017-12-15T20:00:00Z2017-12-15T20:00:19Zstudent
My fellow students and I have spent much of our spare time this past year investigating the similarities between the calculus of functions and that of sequences, which we have defined for a sequence s_{n} with the differential operator
Δ s_{n} = s_{n} - s_{n-1}
and the integral operator
n
Δ^{-1}s_{n} =
Σ
s_{i}
i = 1
where Σ is the summation sign, adopting the convention that terms with non-positive indices equate to zero.
We have thus far discovered how to differentiate and integrate monomial sequences, found product and quotient rules for differentiation, a rule of integration by parts and figured solutions to some familiar-looking differential equations, all of which bear a striking resemblance to their counterparts for functions. To conclude our investigation, we decided to try to find an analogue of Taylor's theorem for sequences.
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On Share And Share Aliketag:www.thusspakeak.com,2017:/student//12.2412017-10-20T19:00:00Z2017-10-20T19:01:22Z
When last they met, the Baron challenged Sir R----- to a wager in which, for a price of three coins and fifty cents, he would make a pile of two coins upon the table. Sir R----- was then to cast a four sided die and the Baron would add to that pile coins numbering that upon which it settled. The Baron would then make of it as many piles of equal numbers of no fewer than two coins as he could muster and take back all but one of them for his purse. After doing so some sixteen times, Sir R----- was to have as his prize the remaining pile of coins.
student
When last they met, the Baron challenged Sir R----- to a wager in which, for a price of three coins and fifty cents, he would make a pile of two coins upon the table. Sir R----- was then to cast a four sided die and the Baron would add to that pile coins numbering that upon which it settled. The Baron would then make of it as many piles of equal numbers of no fewer than two coins as he could muster and take back all but one of them for his purse. After doing so some sixteen times, Sir R----- was to have as his prize the remaining pile of coins.
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Further Still On A Calculus Of Differencestag:www.thusspakeak.com,2017:/student//12.2392017-09-15T19:00:00Z2017-09-15T19:01:15Zstudent
For some time now my fellow students and I have been whiling away our spare time considering the similarities of the relationships between sequences and series and those between the derivatives and integrals of functions. Having defined differential and integral operators for a sequence s_{n} with
Δ s_{n} = s_{n} - s_{n-1}
and
n
Δ^{-1}s_{n} =
Σ
s_{i}
i = 1
where Σ is the summation sign, we found analogues for the product rule, the quotient rule and the rule of integration by parts, as well as formulae for the derivatives and integrals of monomial sequences, being those whose terms are non-negative integer powers of their indices, and higher order, or repeated, derivatives and integrals in general.
We have since spent some time considering how we might solve equations relating sequences to their derivatives, known as differential equations when involving functions, and it is upon our findings that I shall now report.
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On Divisionstag:www.thusspakeak.com,2017:/student//12.2352017-07-21T19:00:00Z2017-07-21T20:19:50Z
The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.
Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.
student
The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.
Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.
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Further On A Calculus Of Differencestag:www.thusspakeak.com,2017:/student//12.2332017-06-16T19:00:00Z2017-06-16T19:02:42Zstudent
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 s_{n} with the operators
Δ s_{n} = s_{n} - s_{n-1}
and
n
Δ^{-1}s_{n} =
Σ
s_{i}
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 Turnabout Is Fair Playtag:www.thusspakeak.com,2017:/student//12.2272017-04-21T19:00:00Z2017-04-21T19:04:40Z
Last time they met, the Baron challenged Sir R----- to turn a square of twenty five coins, all but one of which the Baron had placed heads up, to tails by flipping vertically or horizontally adjacent pairs of heads.
As I explained to the Baron, although I'm not at all sure that he was following me, this is essentially the mutilated chess board puzzle and can be solved by exactly the same argument. Specifically, we need simply imagine that the game were played upon a five by five checker board...
student
Last time they met, the Baron challenged Sir R----- to turn a square of twenty five coins, all but one of which the Baron had placed heads up, to tails by flipping vertically or horizontally adjacent pairs of heads.
As I explained to the Baron, although I'm not at all sure that he was following me, this is essentially the mutilated chess board puzzle and can be solved by exactly the same argument. Specifically, we need simply imagine that the game were played upon a five by five checker board...
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On A Calculus Of Differencestag:www.thusspakeak.com,2017:/student//12.2252017-03-17T19:00:00Z2017-03-17T20:06:04Zstudent
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
s_{1}, s_{2}, s_{3}, ...
we can define a series S with terms
n
S_{n} = s_{1} + s_{2} + s_{3} + ... + s_{n} =
Σ
s_{i}
i = 1
where Σ is the summation sign, from which we can recover the terms of the sequence with
s_{n} = S_{n} - S_{n-1}
using the convention that S_{0} 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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On One Against Manytag:www.thusspakeak.com,2017:/student//12.2212017-01-20T20:27:08Z2017-01-20T20:34:56Z
Recall that the Baron proposed a pair of dice contests in which Sir R-----, were he to best the Baron's score, stood to win a bounty of thirteen coins.
Upon paying his stake Sir R----- was to cast his die but, if unhappy with its outcome, could pay a further coin to cast it again. Likewise, if he were not satisfied with the second cast, he could elect to cast a third time for a further two coins. He could continue in this fashion for as long as he pleased with the cost rising by one coin for each additional cast of his die. The Baron was to have but a single cast of his die, with Sir R----- to determine whether after or before his own play according to his stake; seven coins for the former and eight for the latter.
student
Recall that the Baron proposed a pair of dice contests in which Sir R-----, were he to best the Baron's score, stood to win a bounty of thirteen coins.
Upon paying his stake Sir R----- was to cast his die but, if unhappy with its outcome, could pay a further coin to cast it again. Likewise, if he were not satisfied with the second cast, he could elect to cast a third time for a further two coins. He could continue in this fashion for as long as he pleased with the cost rising by one coin for each additional cast of his die. The Baron was to have but a single cast of his die, with Sir R----- to determine whether after or before his own play according to his stake; seven coins for the former and eight for the latter.
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