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Tuesday, 23 July 2019

The Bank Note Problem (Second Attempt)


For First Attempt, scroll down to 17 July 2019

In Ruritania, there are four denominations of banknotes. The Ruritanian National Bank issues them in unequal quantities, according to perceived need, and on a rotating basis changes the designs to make forgery more difficult and less remunerative. One design is changed every three years, and once changed the old notes for that denomination cease to be valid. So each design has a life of twelve years.

In the past, the designs comprised abstract and complicated backgrounds (so-called guilloché or burelage ) combined with unique fonts in mostly calligraphic styles, all designed to defeat attempts at forgery. But at the urging of a modernising Ruritanian government, some decades ago now, the bank changed its policy and all new designs have represented dead people who are remembered for their achievements. The modernising government wanted to see different kinds of people and different kinds of achievement represented, but did not bind the bank to any particular formula.

This posed the bank a problem. How many different kinds of people are there? How many kinds of achievement? Without answers to those prior questions it was very hard to know how to proceed. The Bank did not at first identify this problem and started out without any clear answers, rather hoping that “obvious” cases would present themselves, as indeed they did ; Mr Shakespeare, Ruritania’s most famous and acclaimed dead playwright, had his image uncontroversially placed on a twenty pounds sterling banknote in 1970. It was true but irrelevant that there have always been doubts about whether images of Mr Shakespeare look at all like the man they purport to represent.

The lack of clear principles of choice immediately encouraged subjects of Ruritania to come forward with humble petitions addressed to the Governor of the Bank proposing that such and such a person, or group of persons, or achievement, or group of achievements, should be represented on the next banknote scheduled for replacement. In the common parlance of the United States, these humble petitions - however worded in terms of justice, fairness and representation not to mention Greatness - were necessarily instances of log rolling.

That posed the Bank a problem. Should it respond to the biggest logs rolled its way or should it seek to establish some principles of fair representation? The Governor decided that Principles should be sought. A committee was formed to find them.

After the usual lengthy deliberations, the committee proposed that two categories of person should be recognised (Male and Female) and four categories of achievement (Arts, Science, Politics, War). The committee pointed out that these numbers had been arrived at having regard to the reality of four bank note combinations. All eight possible combinations of the categories (which they summarised as MA, MS, MP, MW; FA, FS, FP, FW) could be represented on four notes in just two complete banknote cycles. There would be no awkward remainders to deal with.

As for the actual personages to be represented, the committee concluded that (a) that they all be dead - there was no disagreement about that - and (b) that it was up to the Governor to decide between several possible selection procedures enumerated as follows:

·         a committee of experts and / or the Great & the Good to pick the person to be featured in any of the eight categories;
·         a simple lottery the tickets for which would bear names selected by some method or other;
·         a weighted lottery with the number of tickets for each name equal to the number of signatures on humble petitions submitted in favour of that name - this was seen as a concession to log rolling;
The committee also felt that a further accommodation of the public was possible:
·         A list of names selected by the Bank’s own committee could be submitted to public vote according to one of the recognised procedures (first past the post, and so on).  
This last procedure had built into it a guarantee against any possibility of an overwhelming public vote in favour of Banky McBankface.

The committee did identify one unresolved problem. Since banknotes of the four denominations are issued in unequal quantities, it might be thought that the representational value of the image on them should be weighted according to the number of banknotes on which that image would appear. The committee noted that this would remove an element of simplicity from its proposals and would require assistance from someone able to do advanced mathematics and not just simple sums.

The committee then took cover. 

When it learned of the committee’s recommendations, the government of Ruritania was appalled. There were far too few categories of person and it was not sure that “War” was any longer a category of achievement. What about “Entertainment” or “Sport” - perhaps these could be combined into “Culture”?  If “War” was then added to “Politics”, that would preserve the four categories of achievement. A neat counter-proposal.

But as for categories of person, the government felt it had a duty of special care for the Ruritanian Minorities of which thirty nine were currently recognised. How did the Bank propose to ensure that those minorities featured appropriately on the four denominations with only its Male and Female categories available?

The Bank replied humbly that it thought that it could cope with the increased complexity demanded by the government but would need a few extra mathematicians, a new computer, and an answer from the government to two remaining questions: Are the thirty nine minorities to be represented equally within the Person categories or in weighted form according to the number of persons identified as being members of those minorities? And if the latter, should the number be those actually living or the number who have ever lived within the borders of Ruritania? The second question was given its point by the twin facts that all Persons had to be dead in order to qualify and that today’s Ruritanian Minorities were not distributed in the same proportions or same aggregate numbers as the Minorities of yesteryear.

The government appointed a small committee of mathematicians to come up with its reply to these two supplementary questions and it is hoped that a Nobel Prize (possibly for Mathematics but preferably for Peace) will result.

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