Since our objective is to propose a dozenal number system and weights and measures based on this numeration, we would also need a dozenal coinage and banknote system.

Has anybody yet investigated this requirement? What would constitute an optimal set of coins? What criteria would be used to define an optimal set?

While it is possible to determine a minimal set of coinage, this would not necessarily be an optimal set for day to day usage.

Has anybody any ideas regarding this requirement or has it already been ascertained?

I think we should require:

• All denominations are A-series preferred numbers (or B-series for decimal), and
• The ratio between any two consecutive denominations is at least 2 and at most 4.

There have been several suggestions for the UK.
See: dozenal coinage
where there are details.

I prefer the scale 1, 3, 6, *10 as in our shillings and pence (real money...)
d for penny (of course)

(coin): 1d, 3d, 6d, 1s, 3s (maybe a 6/- for commemorative purposes)
(note): R1 (1 "Royal", to quote Aitken) (=*10s)
then R1, R6, R*10 R*20 or R*30, R*60 etc.

If we combine the two criteria from my suggestion, the possible options are:

• 1, 2, 4
• 1, 2, 6
• 1, 3
• 1, 3, 6
• 1, 4
• 2, 6

That would have been the obvious choice in 1971, but today it would allow only two of your existing coins to be used: 50p (as 6 oz) and £1. The 1-2-6 series would also allow the £2 coin to remain in use.

On this side of the pond, I'd recommend the 1-3 series: The $1 and the 25¢ (3 "new dimes") coins could remain in use.

Since £1 now buys about the same as a shilling did when I was in my teens, I'd replace the £1 with a shilling and the 50p with a sixpence. We would need a penny (1d) and a threepence (3d).
The rest of the decimals (1p 2p 5p 10p and 20p) get scrapped. The £2 could be kept as a 2s coin.
All the banknotes would have to be changed.

When the decimals were announced the coins that had to go first were the half-crown and the halfpenny, followed on Decimal Day by the penny, threepence and sixpence; the sixpence was allowed for a while as two-and-a-half p, but soon died. So once the decimals were in use only the shilling and florin survived - just two of the existing coins.

The major cost would be in printing banknotes.

Then maybe we could introduce this system in Zimbabwe. They're already prepared to print new banknotes.

Just as long as they stop using small billions ...
What with the credit crunch etc I'm sick of our Govt referring to trillions when they should be saying billions.

My ha'p'orth, so to speak.

I guess the most obvious choice would be 12 pennies to a shilling and 12 shillings to a pound (these names could change, of course). In order to avoid inflation issues, the new pound could be the same as the present pound, only divided differently. Essentially, the new penny would be pretty much worthless and liable to be withdrawn in a few years time, and the new half-groat (2p) if we had it would be c.1;4p and the thruppenny bit, if we had one, would be almost exactly 2 present pence. Works for me. But other proposals I like are a bit more radical, for example an incorporation of restoration of the gold standard.

I don't really see the point in having more than two divisions of the currency. What would be a much better idea (and I must admit, this isn't my idea but from another forum), would be to use just one unit and then use prefixes of that, taking the metric system as an example. Personally I'd keep the base the same (€, $ or £) and then use centi- for cents or pence, and then have kilo- etc. optional but used instead of billion and trillion to avoid confusion.

And in these "present economic circumstances", returning to the gold standard seems like a good idea!

So why bother with it? Have the smallest coin be a twelfth of a pound (which I'll call an "ounce" unless someone has a better idea). And of course, a 2oz or 3oz coin to fill in the gap between the 1oz and the 6oz (50p).

If you insist on having prices with two dozenal places, then introduce a new dozen-pound unit, like Shaun's suggestion.

Oh, how I wish I had bought some gold last year!

And it wasn't that long ago we adopted the US practice and it seems that the large billion, trillion etc. are still the norm in France and Germany, at least up to 1987 Bosch thought so.

As for the UK, this page from an undated little book of various measures by Philips (it must be between 1914 and 1930 as the first date is quoted in noting when gold coins were no longer in circulation, and the list of planets doesn't include Pluto) shows we were using the large versions of these number names.

I don't think there is a problem with the devaluation of a currency and the coins in use in that currency. Whatever subdivisions of the currency unit you decide to implement, these can continue in use as long as you like. If the currency loses value against its competitors, what is the problem then in just revaluing the base unit (e.g. the Pound, Royal, Lion) upward, allowing all the coins in circulation to be retained. For example, if the Pound loses value against other currencies by say 40 pergross (i.e. 1/3 leaving the pound at 2/3 its previous value), then revalue the pound by the reciprocal 3/2. Thus the coins within the pound are revalued by the same fraction and can still be significant aliquots of the base currency.

All this of course ignores any economic arguments one way or the other.

This thread seems to have veered off the original intention, defining the criteria for evaluating the options for aliquots coinage of the base unit.

Dan, you made the following suggestions.

* All denominations are A-series preferred numbers (or B-series for decimal), and
* The ratio between any two consecutive denominations is at least 2 and at most 4.

You do not however provide an argument as to why these should be criteria. You then suggest the following alternatives that meet your two criteria as follows.

* 1, 2, 4
* 1, 2, 6
* 1, 3
* 1, 3, 6
* 1, 4
* 2, 6

Then Shaun, you point to the DSGB site and three articles on Dozenal Money. I like the approach by David James. and I also like the addition by Andrew Sieber. However, I do think David limited his choices of possible combinations and so have extended the range of combinations and compared these against one another.

The criteria set out by Andrew while good, again fail to compare all combinations. However, I do think his criteria have merit and should be retained, although I am not sure they should all have the same weight in determining the overall winner.

So we have a a few suggestions and a few evaluation criteria which I shall list below, along with a couple of my own which although they may seem trivial and self evident, should be explicitly stated.

1. The base unit of currency should have a numeric foundation of 10*.
2. The base unit of currency could be divided to one or two dozenal subdivisions of 10*.
3. There should be no negative elements of currency.
4. There must be an identity element of coinage, either .1* or .01* of the base unit of currency depending on the level of subdivision respectively.
5. The values of coins should be evaluated to meet criteria of usefulness, manageability and cost of production, and the weighting of these criteria established to determine the best fit combination of values. (Note. The optimal set of values in terms of minimizing the number coins to represent all values up to the currency basis may not necessarily be the best overall choice.)
6. The number of transition by adding one coin and the number requiring replacing coins to move from one value to the next in the series 0 to 10* may be a valid criterion.
7. The minimum number of coins in total to represent all values from 0 to 10* may be a criterion.
8. The size of coins and the proportions between them may be a valid criterion in terms of differentiation of coins, particularly for the blind.
9. Do coins that are a factor of 10* have an advantage over those that are relatively prime to 10*?
:A . The value of coins to be used should be independent of the current value of the currency against other currencies.
:B . The Should the values of notes follow the pattern of the coins or are the criteria different?
10. The costs of coin production and circulation should be evaluated in inverse proportion to the number of different coins.

Any suggestions to solidify or quantify any of these points are welcomed. I might add that following the example of David James, I have also created a table of the various options of coin sets and the total number coins needed to represent all values from 0 to 10*. I did not include values of 5 or 7 in this table except for the set 1, 2, 3, 4, 5, 6. Of course, the set with coins of all values from 1 to 6 requires the smallest total number of coins to represent all values from 0 to 10* at the expense of producing 6 different coins. The ideal result will obviously lie some between this and a single coin of value 1.

(In this post, substitute "pence" for "cents", and "pound" or "euro" for "dollar" as appropriate to your country of residence.)

I was going to, but didn't for some reason. My reasoning was:

All denominations are A-series preferred numbers (or B-series for decimal), and

A B-series preferred number for decimal simply means a factor of 100 (times a power of 10). Thus, denominations are limited to 1, 2, (2.5), 4, 5, 10, 20, 25, 40, 50¢ or $1, $2, $2.50, $4, $5, etc. As far as I know, all major currencies use a subset of these denominations.

The intent, of course, is so you can stack up a whole number of same-denomination coins and get exactly a dollar, which is presumed to simplify counting your money. (But technically, this wouldn't apply to a 40¢, so I stated my criterion wrongly. Or you could just count them in multiples of $2.)

The same argument would apply to a dollar of *100 new cents, with the B-series being 1, (*1.4), (*1.6), 2, 3, 4, 6, 8, 9, *10, *14, *16, *20, *30, *40, *60, *80, *90 new cents. But this is a long list with denominations too close to each other, so it's gotta be pared down, and the simplest way to do that is to use the A-series: 1, 2, 3, 4, 6, *10, *20, *30, *40, or *60 new cents.

The ratio between any two consecutive denominations is at least 2

This (I think) is necessary for the "greedy" change-making algorithm to work. It also imposes a reasonable upper limit on the number of denominations, as it would be impossible to have more than 6 denominations less than a 100-cent dollar or 7 denominations less than a *100-cent dollar.

...and at most 4.

This is just a personal preference that comes from my experience with American coinage. We don't have an intermediate denomination between the penny (1¢) and nickel (5¢), and I always end up with a glut of pennies in my coin jar. Whereas having 4 quarters to a dollar doesn't bother me. Of course, that could be partly because an excess of quarters was never in problem when I was in college and spent my quarters on laundry.

For a more mathematical argument for why the maximum ratio between coins should be at most 4, suppose that we used only 1oz and 6oz coins. Then

• 0 oz = 0 coins
• 1oz = 1 coin
• 2oz = 2 coins (1oz + 1oz)
• 3oz = 3 coins (1oz + 1oz + 1oz)
• 4oz = 4 coins (1oz + 1oz + 1oz + 1oz)
• 5oz = 5 coins (1oz + 1oz + 1oz + 1oz + 1oz)
• 6oz = 1 coin
• 7oz = 2 coins (6oz + 1oz)
• 8oz = 3 coins (6oz + 1oz + 1oz)
• 9oz = 4 coins (6oz + 1oz + 1oz + 1oz)
• 10oz = 5 coins (6oz + 1oz + 1oz + 1oz + 1oz)
• 11oz = 6 coins (6oz + 1oz + 1oz + 1oz + 1oz + 1oz)

Average coins per transaction: 3

(This assumes that the last digit of a price is uniform randomly distributed, which is fairly accurate if you're making multiple purchases from a shop where there's sales tax.)

OTOH, if we use a 4oz coin instead of a 6oz coin, then:

• 0oz = 0 coins
• 1oz = 1 coin
• 2oz = 2 coins (1oz + 1oz)
• 3oz = 3 coins (1oz + 1oz + 1oz)
• 4oz = 1 coin
• 5oz = 2 coins (4oz + 1oz)
• 6oz = 3 coins (4oz + 1oz + 1oz)
• 7oz = 4 coins (4oz + 1oz + 1oz + 1oz)
• 8oz = 2 coins (4oz + 4oz)
• 9oz = 3 coins (4oz + 4oz + 1oz)
• 10oz = 4 coins (4oz + 4oz + 1oz + 1oz)
• 11oz = 5 coins (4oz + 4oz + 1oz + 1oz + 1oz)

Average coins per transaction: 2½

Thus, a 4×3 system of coinage is more efficient than a 6×2 system of coinage. In fact, either 4×3 or 3×4 is the optimal system if you allow only one coin between the 1oz and the ¤1, thus making 4 a reasonable upper limit on the ratio between denominations.

By that standard, that book could also be post-2006 as well. But if so, that's awfully low-quality paper.

Someone on Halfbakery once suggested a negative penny to make it easier for people to pay for 99¢ items. However, it was pointed out that people would be likely to cheat the system by discarding their negative coins instead of depositing them. So I agree with this criterion.

I didn't mention this criterion, but I should have (thus removing {2, 6} from my list). It would seem weird otherwise: Makes the unit of currency too abstract. I can imagine children learning about money seeing a "half dollar" coin and a $2 bill and instinctively asking "what does a dollar look like?"

Of course, there are plenty of examples of countries without a ¤1 denomination (like the Vietnamese dong), and several more without a ¤0.01 denomination (like the Australian dollar). But this is always the effect of inflation, or a temporary situation before introducing a new currency. Has there ever been a case of a currency designed without a ¤1 denomination?

This is true. I once read on Slashdot that the "optimal" currency would include an 18¢ coin, but that would make counting money difficult in a decimal system.

The type of edge the coin has would help differentiate them too. And color helps for the non-blind. (This is why the U.S. redesigned the dollar coin in 2000, though people still don't use it much.)

However, this issue is independent of the choice of denominations, unless we go back to an intrinsic value system.

If designing a currency from scratch, the criteria can be the same.

However, for existing currencies that use the denominations 25¢, 0.50¢, $1, and $2, it may be best for backwards compatibility sake to use the denominations 3oz, 6oz, $1, and $2 even though it breaks the pattern.

We could, in fact, choose the denominations of each currency by taking the factorization of each existing decimal denomination and replacing all the 5's by 6's. Thus, the U.S. would have coins for 1, 6, *10, *30, (*60), and (*100) new cents. And 1, (2-), 6-, *10-, *20-, *60, and *100-dollar bills.

Except that it has a table of Coins in Use that includes sterling coinage (Sovereign, Half-sovereign, Florin, Shilling, Sixpence, Threepence, Crown, Double Florin, Half-crown, Penny, Half-penny, and Farthing) with the note about gold coins not being in general circulation, and the Crown and Double Florin that are no longer minted.

There is also a note at the start of "THE METRIC SYSTEM" that states "The system [metric] is also legal, though not compulsory, in England and the United States.". I think that alone kills the argument for post 2006. :D

Not much has changed in the U.S.

Well, yes it has, though not what you might expect.

Caltrans, (California Transport Department) went down the road of metrication, spent I don't know how much money and several years preparing documents and training programs to convert all their projects to metric, only to immediately start the change back almost at the time of the completion of the metrication project. They are now completely back to using US Customary measures. I believe that most other states transport departments have done the same.

(All figures in this post are dozenal, even without the * symbol.)

David James' article states:

This efficiency, however, comes at the price of having the greedy algorithm for change-making fail. Using this algorithm makes combinations like {1, 4, 6} less efficient: 8d has to be represented as 6d+1d+1d (3 coins) instead of 4d+4d (2 coins). It does, however, make change-making simpler.

Under this constraint, the optimal sets of coins are:

• For 1 denomination, {1} averages 5.6 coins/transaction.
• For 2 denominations, {1, 3}, {1, 4} and {1, 5} all average 2.6 coins/transaction.
• For 3 denominations, {1, 2, 5}, {1, 3, 5}, {1, 3, 7}, and {1, 3, 8} all average 1.:A coins/tranaction.
• For 4 denominations, {1, 2, 3, 7}, {1, 2, 3, 8}, {1, 2, 4, 7}, {1, 2, 4, 9}, {1, 2, 5, 8}, {1, 2, 5, 9}, and {1, 2, 6, 9} all average 1.7 coins per transaction.
• For 5 denominations, {1, 2, 3, 4, 7}, {1, 2, 3, 4, 8}, {1, 2, 3, 4, 9}, {1, 2, 3, 5, 8}, {1, 2, 3, 5, 9}, {1, 2, 3, 6, 8}, {1, 2, 3, 6, 9}, {1, 2, 3, 6, :A}, {1, 2, 3, 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 3, 7, :A}, {1, 2, 3, 7, :B}, {1, 2, 4, 6, 9}, {1, 2, 4, 7, 9}, {1, 2, 4, 7, :A}, {1, 2, 5, 6, 9}, {1, 2, 5, 7, 9}, {1, 2, 5, 7, :A}, {1, 2, 5, 8, 9}, {1, 2, 5, 8, :A}, and {1, 2, 5, 8, :B} all average 1.5 coins/transaction.
• For 6 denominations, 74 different combinations all average 1.4 coins/transaction.
• For 7 denominations, :B2 different combinations all average 1.3 coins/transaction.
• For 8 denominations, 88 different combinations all average 1.2coins/transaction.
• For 9 denominations, 38 different combinations all average 1.1 coins/transaction.
• For :A denominations, all :A combinations average 1 coin per transaction.
• For :B denominations, {1, 2, 3, 4, 5, 6, 7, 8, 9, :A, :B} averages 0.:B coins/transaction. (It's less than 1 because the amount of 0 is included.)

Predictably, there is a trade-off between the number of denominations and the number of coins per transaction. How do we choose? In practice, this will be determined by things like the number of slots a cash register can have or the number of coins that can people can easily distinguish. But, for the sake of mathematical purity, define the inefficiency of a coinage system as the number of denominations multiplied by the average number of coins per transaction:

• 1 × 5.6 = 5.6
• 2 × 2.6 = 5
• 3 × 1.:A = 5.6
• 4 × 1.7 = 6.4
• 5 × 1.5 = 7.1
• 6 × 1.4 = 8.6
• 7 × 1.3 = 8.9
• 8 × 1.2= 9.4
• 9 × 1.1 = 9.9
• :A × 1 = :A
• :B × 0.:B = :A.1

Based on this criterion, the optimal number of coins to have is 2, which would be either {1, 3}, {1, 4}, or {1, 5}. Of these, I would most prefer {1, 3} for compatibility with existing "quarter" coins, and least prefer {1, 5} because the number 5 is dozenal-unfriendly.

The 3-coin systems fare just as poorly as the 1-coin system, which seems wrong to me. But I think we can all agree that having 4 or more denominations below the "new shilling" would be excessive.

Of couse, this all assumes that either the major monetary unit is divided into 10 parts. I'll consider the case of 100 parts next.

(Decimal again because there are more numbers to convert and I'm getting too lazy.)

In the previous post, I brute-forced all 2^10=1024 possible combinations. (The reason for this number is that a 0d coin is impossible and a 1d coin is mandatory, so we're left with ten independent binary decisions on whether to have an N-pence coin for N=2, 3, 4, ..., 11.) However, I cannot use a similar approach for a gross because 2^142 (about 5.58e42 or *4.67@33) is too huge of a number. So I'll have to restrict the solution set.

Coin denominations will be restricted to the FS(2) ("B-series") preferred numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 96, 108}. This gives us 15 denominations (excluding 1) and thus only 2^15=32,768 possible combinations. This should not take too long on a computer.

The optimal sets of coins are:

• For 1 denomination: {1} averages 71.5 coins per transaction.
• For 2 denominations: {1, 12} averages 11 coins per transaction.
• For 3 denominations: {1, 4, 24}, {1, 6, 24}, and {1, 6, 36} each average 6.5 coins per transaction.
• For 4 denominations: {1, 4, 18, 48} averages 4.875 coins per transaction.
• For 5 denominations: {1, 3, 8, 18, 48} averages 4.041666666666667 coins per transaction.
• For 6 denominations: {1, 3, 8, 18, 36, 96} and {1, 3, 8, 18, 48, 108} each average 3.625 coins per transaction.
• For 7 denominations: {1, 2, 4, 9, 16, 36, 96} averages 3.3125 coins per transaction.
• For 8 denominations: {1, 2, 3, 8, 12, 16, 36, 96} averages 3.0972222222222223 coins per transaction.
• For 9 denominations: {1, 2, 3, 8, 12, 16, 36, 72, 96} averages 2.9305555555555554 coins per transaction.
• For 10 denominations: {1, 2, 3, 4, 8, 18, 24, 48, 72, 108}, {1, 2, 3, 4, 9, 16, 24, 48, 72, 108}, {1, 2, 3, 4, 9, 18, 24, 48, 72, 108}, {1, 2, 3, 6, 9, 16, 24, 48, 72, 108}, and {1, 2, 4, 6, 9, 16, 24, 48, 72, 108} each average 2.7986111111111112 coins per transaction.
• For 11 denominations: {1, 2, 3, 4, 8, 12, 18, 24, 48, 72, 108} and {1, 2, 3, 4, 9, 12, 18, 24, 48, 72, 108} each average 2.6597222222222223 coins per transaction.
• For 12 denominations, {1, 2, 3, 4, 8, 12, 18, 24, 48, 72, 96, 108} and {1, 2, 3, 4, 9, 12, 18, 24, 48, 72, 96, 108} each average 2.5763888888888888 coins per transaction.
• For 13 denominations, {1, 2, 3, 4, 8, 12, 16, 18, 24, 48, 72, 96, 108} and {1, 2, 3, 4, 9, 12, 16, 18, 24, 48, 72, 96, 108} each average 2.5069444444444446 coins per transaction.
• For 14 denominations, {1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 48, 72, 96, 108}, {1, 2, 3, 4, 6, 9, 12, 16, 18, 24, 48, 72, 96, 108}, and {1, 2, 3, 4, 8, 9, 12, 16, 18, 24, 48, 72, 96, 108} each average 2.4583333333333335 coins per transaction.
• For 15 denominations, {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 48, 72, 96, 108} averages 2.4097222222222223 coins per transaction.
• For 16 denominations, {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 96, 108} averages 2.3611111111111112 coins per transaction.

Using the same scoring as before:

• 1 × 71.5 = 71.5
• 2 × 11 = 22
• 3 × 6.5 = 19.5
• 4 × 4.875 = 19.5
• 5 × 4.041666666666667 = 20.208333333333336
• 6 × 3.625 = 21.75
• 7 × 3.3125 = 23.1875
• 8 × 3.0972222222222223 = 24.777777777777779
• 9 × 2.9305555555555554 = 26.375
• 10 × 2.7986111111111112 = 27.986111111111111
• 11 × 2.6597222222222223 = 29.256944444444446
• 12 × 2.5763888888888888 = 30.916666666666664
• 13 × 2.5069444444444446 = 32.590277777777779
• 14 × 2.4583333333333335 = 34.416666666666671
• 15 × 2.4097222222222223 = 36.145833333333336
• 16 × 2.3611111111111112 = 37.777777777777779

Thus, the choice is between {1, 4, *20}, {1, 6, *20}, {1, 6, *30}, and {1, 4, *16, *40}. The 3-coin solutions all share the factorization *100=4×6×6 but differ in the order they have the factors in. The 4-coin solution has 2 non-integer ratios between denominations, which might be inconvenient.

And if you want to see how all of the coin combinations fared:

"Has there ever been a case of a currency designed without a ¤1 denomination?"
Cyprus, on decimalisation.
£ and mil system; smallest coin was 3 mil.

The Wikipedia article on Cypriot pound says that they indeed had a 1-mil coin, albeit introduced later than the 3-mil.

Yes, but the 1 mil came much later (1963), so the currency was designed without the 1 mil coin.

A rather awkward currency system, what with the impossibility of paying for anything that cost 1, 2, 4, or 7 mils. Which suggests another criterion for the ideal coinage:

The least-valued coin must be equal to the greatest common factor of all denominations.