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Offline Bimat

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Most Efficient Coin System
« on: July 09, 2012, 06:33:55 PM »
July 9, 2012, 12:00 PM

Numberplay: Denominations

By GARY ANTONICK

It’s often said that the most efficient coin system would consist of seven coins in denominations 1, 2, 4, 8, 16, 32 and 64 cents. This would allow you to carry the fewest coins necessary to make change for a dollar. We ran into this basic idea (until shattered by Mike) in our recent Mixed-Up Wires.

The problem with this system is that it involves numbers that aren’t very easy to add in your head. That’s troublesome if you’re a cashier. The system also a lot of different coins, which is troublesome if you’re a vending machine. So I wondered about something better. Fewer coins, for starters.

What about two coins? Would certainly make sorting a lot easier. And the math? Probably depends on the denominations. Which brings us to our puzzle this week.

If a currency system could only have two coins, what should the values of those coins be?

That’s it.

Source

(Check the link to see some interesting illustrations not related to the question above :))
It is our choices...that show what we truly are, far more than our abilities. -J. K. Rowling.

Offline Bimat

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Most Efficient Coin System
« Reply #1 on: July 09, 2012, 06:34:42 PM »
I'll repeat the question here:

If a currency system could only have two coins, what should the values of those coins be?

The question is in context of US system (dollar/cents) but we can generalize for any system, for example for India, you can give two Indian denominations like 50 paise and 1 Rupee or in case you are British, you may say 50p and £1.

Aditya
It is our choices...that show what we truly are, far more than our abilities. -J. K. Rowling.

Online Figleaf

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Re: Most Efficient Coin System
« Reply #2 on: July 10, 2012, 12:13:05 AM »
In any system, you must have a 1 or a multiple of 1. The problem is reduced to what the second coin would be. My solution:

  • Assume denominations can only be multiples of 1, 2 and 5 to be efficient for mental calculations.
  • Calculate a break-even value by a formula that equates (the present value of) the cost of producing coins and banknotes over a period of say 100 years. Note that the main variable in the formulas are the production cost, the useful life and the discount rate if you prefer to work with the present value (I don't)
  • Round up the solution found to the nearest multiple of 1, 2 or 5. Since nothing is specified on banknotes, this is your lowest value banknote
  • Calculate the simple average of the first denomination and the lowest banknote and round to the nearest multiple of 1, 2 or 5. That is the denomination of the second coin.

Example. Suppose that you find that the equilibrium value for India is 38.8 rupees (making this up). The lowest banknote value would be 50 rupees. The base denomination would be 1 rupee (10 paise makes no sense). The average is 25.5, so the second denomination should 20 rupees. If the equilibrium value would have been 55 rupees, the lowest value banknote would have been 100 rupees and the second denomination would be 50 rupees.

The calculations show the folly of admitting only two denominations. You would need enormous amounts of 1 rupee coins for small transactions and change.

Peter
An unidentified coin is a piece of metal. An identified coin is a piece of history.

Offline dheer

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Re: Most Efficient Coin System
« Reply #3 on: July 25, 2012, 06:54:20 AM »
I'll repeat the question here:

If a currency system could only have two coins, what should the values of those coins be?

The question is in context of US system (dollar/cents) but we can generalize for any system, for example for India, you can give two Indian denominations like 50 paise and 1 Rupee or in case you are British, you may say 50p and £1.

Aditya

It also depends on the buying capacity of the currency. If for British there are items that are available for 1p, then the lowest should be 1p ... for India 50p is the lowest that one can buy a candy for ...

So once we are stuck with this, then we would need to normalize it ... say if we have just 1p and 50p for UK, if someone wants to pay 20p .... he need to carry 20 coins thats to large number of coins in the pocket ... that is where the other number come into play ... there can never be a system with only 2 coins, apart from theory, it would never be practical ...
http://coinsofrepublicindia.blogspot.in
A guide on Republic India Coins & Currencies

Offline FosseWay

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Re: Most Efficient Coin System
« Reply #4 on: July 25, 2012, 08:47:46 AM »
The view that having seven denominations in active use is odd in some way (as expressed in the original quote) is very US-centric. It's the norm in most of Europe.

It strikes me that there are two important questions, which aren't linked so much to what all the denominations are, but rather to their contribution to the overall efficiency of a transaction. Firstly, you have to set the minimum value for which you feel a coin is needed at all. It's clearly inefficient to produce coins that cost more than their face value and/or are perceived as worthless by users and not used as intended.

That's really the only actual denomination you need to specify in general terms. What matters thereafter is that the denominations are spaced such that you don't need more than X coins of a given denomination to give change, while not confusing people and overburdening the system with a multitude of unusual denominations. From global experience, it would seem that X = 2. Whether you choose the 2/20 route or the 2.5/25 route makes no real difference.

A lot of currencies fall down on one or other of these efficiency criteria. Sterling and the euro have pointless denominations at the bottom of the scale but use the rest of the scale efficiently (neither leaves holes requiring loads of 10c or €1 coins, for example). The pre-euro German system and the UK before the advent of the 20p had usable denominations but a big hole that resulted in too many (max 4) 10-value coins being required for some transactions. Sweden has a sensible base denomination but lacks a 2. The US dollar falls down on both counts, by having a pointless cent but also no 2 and no 50 (in practical terms), requiring huge quantities of cents and quarters.

Are there any currencies that meet both criteria? Those eurozone countries that have officially ditched the 1c and 2c coins, perhaps. Or Australia and NZ. I'm not familiar enough in the latter case with the buying power of the lowest coin to be sure.

Of course, efficiency and utilitarianism in general are not necessarily the only criteria it's important to take on board when designing a currency system, never mind perpetuating an existing one. Cultural concerns and good old-fashioned inertia are also important, so that a system that looks good on paper may not actually be the best in a given context if the people using it don't like it.

Offline Bimat

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Most Efficient Coin System
« Reply #5 on: July 25, 2012, 09:47:09 AM »
Someone has posted this interesting solution (see the comments on the article, link given in first post):

___________________________________________________________________________________________________

My 14-year-old son found that using 9-cent coins and 10-cent coins, he could create any amount of change between $.01 and $1 with 10 coins or fewer:

"If a currency system could only have two coins, what should the values of those coins be?" He solved the problem with the rules being: 1) you have to be able to make change for every amount of money from $.01 to $1; 2) you can only use two denominations of coins; 3), you are allowed to trade coins with the person you are transacting with.

With these rules he tried to find a solution that would require the least number of coins required for every transaction. He found that with a 10-cent coin and a 9-cent coin you can create any amount of change under $1 with 10 coins or fewer:

There are three ways to make change using the 9- and 10-cent coins. One, by paying with 9-cent coins and receiving 10-cent coins in change, for example, giving 3 9-cent coins and receiving 2 10-cent coins to pay 7 cents total.

Two, by paying with 10-cent coins and receiving 9-cent coins in change, for example, giving 3 10-cent coins and receiving 1 9-cent coin to pay 21 cents total.

Three, by giving a combination of 9- and 10-cent coins and receiving nothing, a one-way exchange, for example, giving 8 9-cent coins and 1 10-cent coin to pay 82 cents.

Here is a link to the number table he made to show how each amount of change is made: http://tinyurl.com/c93l5vg

Is the best solution? Perhaps 8-cent and 10-cent would be more efficient?

___________________________________________________________________________________________________

Aditya
It is our choices...that show what we truly are, far more than our abilities. -J. K. Rowling.

Offline dheer

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Re: Most Efficient Coin System
« Reply #6 on: July 25, 2012, 10:01:59 AM »
Someone has posted this interesting solution (see the comments on the article, link given in first post):

___________________________________________________________________________________________________

My 14-year-old son found that using 9-cent coins and 10-cent coins, he could create any amount of change between $.01 and $1 with 10 coins or fewer:

"If a currency system could only have two coins, what should the values of those coins be?" He solved the problem with the rules being: 1) you have to be able to make change for every amount of money from $.01 to $1; 2) you can only use two denominations of coins; 3), you are allowed to trade coins with the person you are transacting with.

With these rules he tried to find a solution that would require the least number of coins required for every transaction. He found that with a 10-cent coin and a 9-cent coin you can create any amount of change under $1 with 10 coins or fewer:

There are three ways to make change using the 9- and 10-cent coins. One, by paying with 9-cent coins and receiving 10-cent coins in change, for example, giving 3 9-cent coins and receiving 2 10-cent coins to pay 7 cents total.

Two, by paying with 10-cent coins and receiving 9-cent coins in change, for example, giving 3 10-cent coins and receiving 1 9-cent coin to pay 21 cents total.

Three, by giving a combination of 9- and 10-cent coins and receiving nothing, a one-way exchange, for example, giving 8 9-cent coins and 1 10-cent coin to pay 82 cents.

Here is a link to the number table he made to show how each amount of change is made: http://tinyurl.com/c93l5vg

Is the best solution? Perhaps 8-cent and 10-cent would be more efficient?
Aditya
Congrats! You son looks like a true Maths prodigy
Again in theory, possibly YES. However if you are always buying candy worth 5 cents, you would have to give 5 ten cents coins and get back 5 coins of 9 cents. So the total coins in the transaction are 10. If you had a 5 cent coin in your pocket, you are well off having just that one coin.

So for an system to be efficient in practice, it would have to consider what are the various items a person buys and it is at what price points. There is no use of having a coin system that would cater to every possible denomination from 1 cent to 100 cents, the question is are there good one can buy worth odd cents? If not these should be removed out of equation.
For example today in India there is nothing one gets for less than 50 paise. So having a coin of 9 paise and 10 paise would make it very inefficient, for one has to always carry 5 times the number of coins that are actually required. Similarly the next price point in India is Rs 1. there is nothing in the market for 75 paise, 80 paise or any other paise between 50 paise and 100 paise. So having the 9 paise coin would be redundant.
http://coinsofrepublicindia.blogspot.in
A guide on Republic India Coins & Currencies

Offline Bimat

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Most Efficient Coin System
« Reply #7 on: July 25, 2012, 10:08:25 AM »
This guy has come up with a negative one cent. ;D Again, applies to US system only. :)

__

For US currency: my preference would be dime and dollar, and round off transactions to the nearest 10¢. Maximum number of coins would be needed for change amounts between $4.85 and $4.94 - 13 coins.

Or how about a 10 and negative-1 cent piece? So to make 49c in change, it's 5 dimes and 1 anti-penny. If we think change amounts would be biased towards terminal digits 5-9 (implying transaction amounts ending in 0-4), this reduces expected number of coins. It's like a "leave-a-penny" dish.

__

I found this reply interesting too; never noticed that those denominations follow this logarithmic pattern!

__

Right now I have only hand-waving arguments, so here they are FWIW.

I think the answer depends on the number system we use and the typical cost of the smallest non-electronic transaction we normally make. Any coin less than a quarter is pretty much useless in today's America, where coffee is around $1.25 /cup and a candy bar can go for $1. So the question is about the smallest denomination we feasibly want to live with, and for this we address the number system.

Since we have a decimal currency system the U.S., traditional currency multiples have been 1, 2, 5, 10, 20, ,50, 100, 500, 1000. What is interesting about this sequence is it is pretty much evenly spaced over a logarithmic scale:

number logarithm (base 10)

1 0
2 0.3
5 0.7
10 1
20 1.3
50 1.7
100 2
200 2.3
500 2.7
1000 3

The denominations of the Euro -- both for coin and banknote -- follow this pattern exactly. The underlying assumption seems to be that factors of 5 in this currency are easy to keep in one's head, since 2 fives make 10 and 10's are easy to keep track of . Assuming that 5 is as good number for a decimal system I would choose a 2 unit and a 5 unit currency note.

If we were to restrict ourselves to US coinage, I wold stick to dimes and quarters.

Other currencies (octal,hex, sexigesimal, etc.) need to be looked at also.

__

Aditya

It is our choices...that show what we truly are, far more than our abilities. -J. K. Rowling.

Offline FosseWay

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Re: Most Efficient Coin System
« Reply #8 on: July 25, 2012, 10:45:55 AM »
It's true that by using non-decimal systems you might get better results (12, or £sd, being the most obvious), but this leads to questions of what constitutes 'efficiency'. We're answering two different questions in this thread at the moment: "What is the most efficient system?" and "How few denominations do you theoretically need to be able to give change for any sum from $0.01 upwards?" The answer to the latter will pretty much invariably not be the answer to the former, regardless of how you define efficiency.

But the definition of efficiency must include the effects on individuals as well as on systems. For systems, the number of different denominations, the number of coins in use in each, and the total weight involved are all important questions, whereas the ease of performing the necessary arithmetic to get a transaction right isn't. (Any system will be computerised, and any computer can calculate any transaction-related arithmetic in any base, whether 10, 12 or what, more or less instantaneously.) On the other hand, for the private individual or the checkout assistant, the arithmetic thing is important, which makes base 10 an obvious choice even though 12 is more versatile. On the other hand, weight is probably less of an issue -- if you habitually carry a maximum of two of any given denomination around in your pocket or purse, it won't make much difference whether they're florin-sized 10ps or current ones, for example.

Ultimately, the most 'efficient' system from the point of view of the state and organisations that deal with large numbers of low-value transactions is to abolish coins (and notes) altogether and use electronic payments. But, aside from the implications of that for our hobby, that isn't necessarily the most efficient way of paying for something from the individual customer's point of view.

Offline EWC

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Re: Most Efficient Coin System
« Reply #9 on: December 29, 2014, 10:39:32 AM »
I see to recall that if everyone carried just 6 coins - value

1, 3, 9, 27, 81, 243

Then any two people would be able to make any integer payment between them - from 1 to 243 units

Please let me know if I mis-remember

Offline Bimat

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Most Efficient Coin System
« Reply #10 on: December 29, 2014, 10:52:01 AM »
I see to recall that if everyone carried just 6 coins - value

1, 3, 9, 27, 81, 243

Then any two people would be able to make any integer payment between them - from 1 to 243 units

Please let me know if I mis-remember

Interesting observation, however the question says there should be only two denominations. :)

Aditya
It is our choices...that show what we truly are, far more than our abilities. -J. K. Rowling.

Offline bgriff99

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Re: Most Efficient Coin System
« Reply #11 on: December 29, 2014, 11:00:16 AM »
Two denominations:   the cash coin and the tael silver, of course!