Irregular denominations

[kassyopeia]

Hi all,

 

Forgive the following somewhat lengthy introduction, but I suspect this is a somewhat unusual topic for this forum, so I'd rather include too much detail than too little. Here goes:

 

I'm a programmer and active at the XVBT? forum. We just had an interesting discussion about how to choose that combination of "monetary elements" (coins, bills, whatever) which gives a target amount using the minimum number of elements. I'll include an abridged part of the OP to clarify:

[A] C# console program that outputs the set of banknotes that:

1- sums in value to a given integer k

2- contains the smallest number of banknotes

The following types of banknotes are available: 500-note, 200-note, 100-note, 50-note, 20-note, 10-note, 5-note, 1-note

 

Example: k = 13

Here the correct set is: 1 of 10-note, 3 of 1-note

these 4 banknotes will sum to 13 and 4 is the smallest possible number of banknotes in this case.

 

<snip>

 

Yesterday, I gave this a try, and I ended up using the following algorithm:

Always subtract the greatest possible banknote from k, until k is 0.

 

<snip>

 

Now, is this algorithm always true regardless of the available types of banknotes? No it's not.

Imagine that the following banknotes are available: 500-note,

280-note

, 200-note, 100-note, 50-note, 20-note, 10-note, 5-note, 1-note

 

Imagine that k = 560

The algorithm will give us this set: 1 of 500-note, 1 of 50-note, 1 of 10-note

these 3 banknotes will sum to 560 but we could have used the following 2 banknotes: 2 of 280-note

 

So, subtracting the greatest possible banknote first can be misleading. It seems that it works if and only if no banknote is greater than the half of its preceding banknote when they are sorted in a descending order (correct me if I'm wrong). So, because 280 is greater than 250 (500/2) it doesn't work.

 

My question: is there an always working algorithm?

After some preliminary discussion of general approaches to packing and covering problems, we returned to a discussion of the specific problem, and not unsurprisingly it turns out that the situation is particularly simple in this case: Denominations are generally designed in such a way that smaller elements will integrally divide larger ones, which greatly reduces the complexity of the solution path.

 

For instance, in one dollar there are 100 cents, 20 nickels, 10 dimes, 4 quarters or 2 half-dollars. Generalizing, any "regular" decimal currency consists of 2s (5*2=10), 1s, 5s (2*5=10), and 25s (4*25=100). I found a single recent example, the Venezuelan céntimos, that continues the series with a 12.5 denomination (8*12.5=100).

One the other hand, a quick trawl of wikipedia found a handful of currencies that include "irregular" denominations of 3 and 4 (see reasonably complete lists in posts #14 and #15 in the thread). A notable example is the Cuban peso, which has both a 40¢ piece and a $3 note. It takes 25 40s to get to 1,000, and 3 doesn't integrally divide any power of ten at all.

 

So, my question is - are there any (current, recent or historical) examples of denominations that include even stranger denominations? This doesn't necessarily refer to denominations other than 2 and 5; in non-decimal systems the regular dividers are obviously different ones. For example, the British Sterling system in use in the 20th century was entirely regular - 4 threepences or 2 sixpences in a shilling, etc.

 

Thanks for your attention and TIA for any replies,

Cas

[zjemller]

Hi all,

 

Forgive the following somewhat lengthy introduction, but I suspect this is a somewhat unusual topic for this forum, so I'd rather include too much detail than too little. Here goes:

 

I'm a programmer and active at the XVBT? forum. We just had an interesting discussion about how to choose that combination of "monetary elements" (coins, bills, whatever) which gives a target amount using the minimum number of elements. I'll include an abridged part of the OP to clarify:

After some preliminary discussion of general approaches to packing and covering problems, we returned to a discussion of the specific problem, and not unsurprisingly it turns out that the situation is particularly simple in this case: Denominations are generally designed in such a way that smaller elements will integrally divide larger ones, which greatly reduces the complexity of the solution path.

 

For instance, in one dollar there are 100 cents, 20 nickels, 10 dimes, 4 quarters or 2 half-dollars. Generalizing, any "regular" decimal currency consists of 2s (5*2=10), 1s, 5s (2*5=10), and 25s (4*25=100). I found a single recent example, the Venezuelan céntimos, that continues the series with a 12.5 denomination (8*12.5=100).

One the other hand, a quick trawl of wikipedia found a handful of currencies that include "irregular" denominations of 3 and 4 (see reasonably complete lists in posts #14 and #15 in the thread). A notable example is the Cuban peso, which has both a 40¢ piece and a $3 note. It takes 25 40s to get to 1,000, and 3 doesn't integrally divide any power of ten at all.

 

So, my question is - are there any (current, recent or historical) examples of denominations that include even stranger denominations? This doesn't necessarily refer to denominations other than 2 and 5; in non-decimal systems the regular dividers are obviously different ones. For example, the British Sterling system in use in the 20th century was entirely regular - 4 threepences or 2 sixpences in a shilling, etc.

 

Thanks for your attention and TIA for any replies,

Cas

 

Bulgaria had a 2.1/2 stotinka coin and 3stotinka coin.

Also a comemorative coin of 1.95583Levs (1 euro) to comemorate Bulgaria's entery in the EU. USSR 15 kopeks

[bobh]

So, my question is - are there any (current, recent or historical) examples of denominations that include even stranger denominations? This doesn't necessarily refer to denominations other than 2 and 5; in non-decimal systems the regular dividers are obviously different ones. For example, the British Sterling system in use in the 20th century was entirely regular - 4 threepences or 2 sixpences in a shilling, etc.

 

Thanks for your attention and TIA for any replies,

Cas

The USA had 3 cent pieces from 1851-1889. I believe the reason for these was the fact that the U.S. postage stamp for regular mail cost 3 cents, so it was somewhat easier to buy stamps with these instead of the unwieldy large copper cents then circulating up until 1856.

 

Russia minted 7-1/2 and 15 rouble gold coins in 1897 (one-year issue). The smaller of the two had the same weight and alloy standard as the older 5 rouble coins. After that, they issued 5 rouble coins of smaller weight (4.3 grams). The reason for this was the transition from the older to the newer currency standard introduced in 1897 by Witte. There was also a donative coin (very rare) of an ounce of gold with denomination of 37-1/2 roubles issued by Nicholas II in 1902. There were just over 200 of them struck, and they were given to visiting dignitaries by the Tsar as a special gift (hence the term "donative").

[Lithuania]

Portuguese has just realeased a 2 1/2 euro coin, and it's circulating!

[tabbs]

Portuguese has just realeased a 2 1/2 euro coin, and it's circulating!

Oh, is it? Frankly, I don't think those actually circulate. Keep in mind that such collector coins are not even legal tender in the currency union except for the issuing member state. Guess those 2.50 pieces "circulate" just as the Austrian €5 or the German €10 coins "circulate" ... Which is also why we should leave "coins" such as the Spanish €12 out.

 

Personally I am glad about not having to deal with 0.25 or 2.50 pieces (be it coins or notes). Let's assume a cashier gives you some change back, and needs to have the amount of "47" (regardless of currency and unit). In Euroland for example he can simply take care about the "40" first (by taking four 10 or two 20 pieces, etc.), and then deal with the "7". In a quarter based system he would have to get a "25" first, which changes the "7" at the end into a "2". Need to pay 84? Same thing - "80 plus 4" vs. "75 plus 9".

 

The Dutch used to make 2 1/2 cent coins (until 1941) and 2 1/2 gulden coins (until 2001). In Germany, during the Reich, there were denominations such as 3 mark (which originally was the equivalent of the earlier Taler) until 1932 ... and they had a 4 pfennig coin that was supposed to counter inflation. After one year the piece was discontinued.

 

Christian

[Scottishmoney]

[Topher]

For modern denominations, I can think of, there is 15, used in both the Bahamas (cents )and the former USSR (kopeks).

 

What about accounting for the old LSD? especially when you factor in the groat, crown, half crown, guinea, etc. I suspect an entirely different algorithm would be needed there.

 

If you go back far enough in time (where money was actually WORTH something), then you'll find 3c, 4c, 12c, etc. All kinds of denominations that we would consider "odd" today.

[gxseries]

Good grief, I was expecting a few British people to talk about their pounds and shillings - that was a mess when it appeared. I don't even know how they did their maths

 

I pulled the video off youtube but I don't remember the url:

 

http://www.gxseries.com/numis/austype/aus_decimal.htm

[YeOldeCollector]

Good grief, I was expecting a few British people to talk about their pounds and shillings - that was a mess when it appeared. I don't even know how they did their maths

 

I pulled the video off youtube but I don't remember the url:

 

http://www.gxseries.com/numis/austype/aus_decimal.htm

 

Yep, gotta love groats, halfpence, shilling, sicpence, guinea and you could go back to Nobles, Ryals, Double Crowns etc...

[Delta]

Good grief, I was expecting a few British people to talk about their pounds and shillings - that was a mess when it appeared. I don't even know how they did their maths

 

Yep, gotta love groats, halfpence, shilling, sicpence, guinea and you could go back to Nobles, Ryals, Double Crowns etc...

 

.... inches feet yards furlongs miles acres pints gallons pounds ounces stones hogsheads ....

[kassyopeia]

Hi guys, thanks for the replies thus far, didn't expected that many.

 

Just to re-iterate one point:

This doesn't necessarily refer to denominations other than 2 and 5; in non-decimal systems the regular dividers are obviously different ones. For example, the British Sterling system in use in the 20th century was entirely regular - 4 threepences or 2 sixpences in a shilling, etc.

So, the distinction I'm making here is really independent of decimal vs. non-decimal currency bases. In a decimal system, the "regular" denominations are 2, 2.5 and 5 because those are the results of dividing 10 by 2, 4 and 5. 3.333... would also be regular, but doesn't represent a multiple of the base denomination (1 cent or equivalent).

 

Along the same lines, in a duodecimal system or subsystem like old pennies and shillings, the regular denominations would be 2, 3, 4 and 6. 5 would actually be irregular in this system, because it doesn't divide 12 integrally. From that point of view, a sexagesimal system can be considered a "better" currency basis than a decimal one, because its "divisor density" is ideal - it gives us e.g. all of {2, 3, 4, 5, 6}.

Hence, in the sense I'm after, many of the historical non-decimal currencies are just as regular as most of the modern decimal ones.

 

So far, the clear "winner" is the set of Russian coins mentioned by bobh, with zjemller's Euro-conversion coin considered out of competition.

[Scottishmoney]

The USA at one time or another has minted the following denominations of coins:

 

1/2 Cent

Cent

2 cents

3 cents nickel

3 cents silver

half dime

five cents

10 cents

20 cents

25 cents or quarter

50 cents or half dollar

1 dollar, silver, gold, cupro-nickel

2.5 dollars gold

3 dollars gold

5 dollars gold

10 dollars gold

20 dollars gold

25 dollars - AGE bullion

50 dollars, AGE bullion

 

Plus whatever nominal denominations are on platinum.

 

BTW the 2¢ and 3¢ coins were minted as late as 1873 and 1889 and saw some circulation into the very early part of the 20th century.

[gxseries]

The story doesn't end yet with Russian coins as they occupied various territories and they had unique "dual" currencies. Under some of the more bizarre coinages, you had something called for the following countries

 

Sadagura

 

1 para - 3 dengi, 2 para - 3 kopek

 

Poland

 

3 ruble - 20 zlot, 1.5ruble-10 zlot, 3/4 ruble - 5 zlot, 30 kopek - zlot, 15 kopek - 50 gros.

 

And then you have a pretty interesting coinage for the Baltic state such as:

 

96, 48, 24, 4 and 2 kopek. Now how on earth does that work??? They might be the pioneers in successfully removing 1 cent

[banivechi]

Romania issued in 1906 a gold coin 12 1/2 lei. Cyprus had funny numbers on coins: silver 3, 4½, 9 and 18 piastres. Romania issued also in 1941-43 in occupied Transdniestra bills of 600 and 1200 lei.

[thelawnet]

For British coins see

 

http://en.wikipedia.org/wiki/List_of_Briti...notes_and_coins

 

Many dozens of denominations

[kassyopeia]

Thanks again to all who contributed, the responses were very edifying!

[gxseries]

Found this neat set on ebay: http://cgi.ebay.com/ws/eBayISAPI.dll?ViewI...em=250268516622

[ccg]

Cyprus had funny numbers on coins: silver 3, 4½, 9 and 18 piastres.

 

If I remember correctly, they corresponded to the groat, sixpence, shilling, and florin (4d, 6d, 1/-, 2/-)

I wonder what in local history and monetary background would cause them to use a different unit rather than the LSD system used in many other colonies?