## Ideal Currency Denominations

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

Apeiron
Posts: 120
Joined: Tue Feb 12, 2008 5:34 pm UTC

### Ideal Currency Denominations

Is there any sort of ideal set of currency denominations?

If you were making currency denominations, how would you do it and why?

This is in relation to my general interest in money/currency and to setting details for D&D.

Thanks!

gmalivuk
GNU Terry Pratchett
Posts: 26767
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: Ideal Currency Denominations

The 1-2-5 pattern seems to work well for lots of countries. (Repeated, of course: 1, 2, 5, 10, 20, 50, 100, etc.)
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

DavCrav
Posts: 251
Joined: Tue Aug 12, 2008 3:04 pm UTC
Location: Oxford, UK

### Re: Ideal Currency Denominations

gmalivuk wrote:The 1-2-5 pattern seems to work well for lots of countries. (Repeated, of course: 1, 2, 5, 10, 20, 50, 100, etc.)

British currency FTW!

BlackSails
Posts: 5315
Joined: Thu Dec 20, 2007 5:48 am UTC

### Re: Ideal Currency Denominations

I read an article once claiming that because of how things are priced, and how taxes are, you could reduce the average number of coins involved in each purchase with a 17 cent coin.

gmalivuk
GNU Terry Pratchett
Posts: 26767
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: Ideal Currency Denominations

DavCrav wrote:
gmalivuk wrote:The 1-2-5 pattern seems to work well for lots of countries. (Repeated, of course: 1, 2, 5, 10, 20, 50, 100, etc.)

British currency FTW!

Yeah, when the Brits finally decided to decimalize their currency, they saw how well the 1-2-5 system was already working for dozens of countries, and so copied them.

BlackSails wrote:I read an article once claiming that because of how things are priced, and how taxes are, you could reduce the average number of coins involved in each purchase with a 17 cent coin.
How taxes are where, exactly? Because even if a 17-cent coin worked for some particular location with some particular sales tax, which I doubt, it wouldn't help anywhere with a different sales tax.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

BlackSails
Posts: 5315
Joined: Thu Dec 20, 2007 5:48 am UTC

### Re: Ideal Currency Denominations

gmalivuk wrote:
BlackSails wrote:I read an article once claiming that because of how things are priced, and how taxes are, you could reduce the average number of coins involved in each purchase with a 17 cent coin.
How taxes are where, exactly? Because even if a 17-cent coin worked for some particular location with some particular sales tax, which I doubt, it wouldn't help anywhere with a different sales tax.

It might have been a particular state, or just the most common sales tax in the US. I read this article a really long time ago, in discover I think.

kernelpanic
Posts: 891
Joined: Tue Oct 28, 2008 1:26 am UTC
Location: 1.6180339x10^18 attoparsecs from Earth

### Re: Ideal Currency Denominations

BlackSails wrote:I read an article once claiming that because of how things are priced, and how taxes are, you could reduce the average number of coins involved in each purchase with a 17 cent coin.

And I've always insisted that the US should introduce their $9.99[+standard sales tax] note. That way you wouldn't get really weird amounts of change. And now that we're at it, why the hell are dimes smaller than nickels? I'm not disorganized. My room has a high entropy. Bhelliom wrote:Don't forget that the cat probably knows EXACTLY what it is doing is is most likely just screwing with you. You know, for CAT SCIENCE! BlackSails Posts: 5315 Joined: Thu Dec 20, 2007 5:48 am UTC ### Re: Ideal Currency Denominations Dimes used to be made of silver, no? majikthise Posts: 155 Joined: Sun Jun 10, 2007 2:28 am UTC Location: Bristol, UK ### Re: Ideal Currency Denominations Is it common all over the US for prices to be [nice, round number] + [weird tax percent], or only in certain states/types of store? Is this a common method in many other countries? Is this a wok that you've shoved down my throat, or are you just pleased to see me? gmalivuk GNU Terry Pratchett Posts: 26767 Joined: Wed Feb 28, 2007 6:02 pm UTC Location: Here and There Contact: ### Re: Ideal Currency Denominations kernelpanic wrote:And now that we're at it, why the hell are dimes smaller than nickels? Because nickel is worth less than silver, which is what dimes used to be made from. Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true. --- If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome (he/him/his) squareroot1 Posts: 172 Joined: Fri Nov 06, 2009 8:27 pm UTC ### Re: Ideal Currency Denominations BlackSails wrote:I read an article once claiming that because of how things are priced, and how taxes are, you could reduce the average number of coins involved in each purchase with a 17 cent coin. Well, we already have 1/100, 1/20, 1/10, 1/4, and 1/2 (sometimes) of a dollar coins, so why not include one for approximately 1/6. Although, considering how well the (most recent) dollar coin worked out, I can understand the hesitance. AllSaintsDay Posts: 35 Joined: Thu Aug 28, 2008 1:01 am UTC ### Re: Ideal Currency Denominations majikthise wrote:Is it common all over the US for prices to be [nice, round number] + [weird tax percent], or only in certain states/types of store? Is this a common method in many other countries? It's definitely common over most of the US, but there are some exceptions. Concession stands at events or on the street typically have tax worked into the price for an even amount, and tax is also worked into gas prices, which aren't as often round numbers. Out here in Oregon, there isn't sales tax on typical goods. I think when I went to Italy, they worked the tax into every price, but I could be misremembering. PM 2Ring Posts: 3713 Joined: Mon Jan 26, 2009 3:19 pm UTC Location: Sydney, Australia ### Re: Ideal Currency Denominations Ternary. stephentyrone Posts: 778 Joined: Mon Aug 11, 2008 10:58 pm UTC Location: Palo Alto, CA ### Re: Ideal Currency Denominations PM 2Ring wrote:Ternary. Balanced ternary with negative denomination notes? GENERATION -16 + 31i: The first time you see this, copy it into your sig on any forum. Square it, and then add i to the generation. jestingrabbit Factoids are just Datas that haven't grown up yet Posts: 5967 Joined: Tue Nov 28, 2006 9:50 pm UTC Location: Sydney ### Re: Ideal Currency Denominations stephentyrone wrote: PM 2Ring wrote:Ternary. Balanced ternary with negative denomination notes? That's pretty funny. ameretrifle wrote:Magic space feudalism is therefore a viable idea. PM 2Ring Posts: 3713 Joined: Mon Jan 26, 2009 3:19 pm UTC Location: Sydney, Australia ### Re: Ideal Currency Denominations stephentyrone wrote: PM 2Ring wrote:Ternary. Balanced ternary with negative denomination notes? I don't think that'd work so well. But I assume merchants are permitted to give change. Vesuvius Posts: 62 Joined: Tue Mar 02, 2010 5:47 am UTC ### Re: Ideal Currency Denominations I had an idea some time ago of negative value notes/coins. Unfortunately, due to the amount of people who would 'accidentally misplace' such denominations, they'd be pretty useless. We don't use anything smaller than 5c in Australia. The 1/2 cent coins were removed years back. I'm pretty sure we decided they were unnecessary. As such, anything worth 12c is actually only worth 10, unless you buy two in which case it's worth 25. No-one cares about the displaced couple of cents. Of course, with the continued trend towards cards/microchips etc., the argument becomes pretty pointless. If you can just wave your hand to spend money, you may as well have items costing nine dollars and 20.935728 cents. DavCrav Posts: 251 Joined: Tue Aug 12, 2008 3:04 pm UTC Location: Oxford, UK ### Re: Ideal Currency Denominations gmalivuk wrote: DavCrav wrote: gmalivuk wrote:The 1-2-5 pattern seems to work well for lots of countries. (Repeated, of course: 1, 2, 5, 10, 20, 50, 100, etc.) British currency FTW! Yeah, when the Brits finally decided to decimalize their currency, they saw how well the 1-2-5 system was already working for dozens of countries, and so copied them. I think that, because ours was changed so recently, we got the most sensible system. 1p and 2p are bronze-looking, round, with 2p weighing twice as much as 1p. 5p and 10p are silver-looking, one twice the other, and round. 20p and 50p are straight-sided, silver-looking, and one is 2.5 times the weight of the other. £1 and £2 coins are thicker and round, again one twice the weight of the other, and our notes are of increasing size. Hence blind people know what's going on, and you can weigh similarly shaped coins together when in a bank. Compare particularly the first point with the US system, where all notes are the same size. Tirian Posts: 1891 Joined: Fri Feb 15, 2008 6:03 pm UTC ### Re: Ideal Currency Denominations majikthise wrote:Is it common all over the US for prices to be [nice, round number] + [weird tax percent], or only in certain states/types of store? Is this a common method in many other countries? Yeah, the big problem is that the sales tax level is very fragmented, totally different between states and IME often different from one county to the next. If you're a national hardware chain and you decide that hammers should be on sale for$10 this week, then you'd be taking a different profit on every hammer depending on where you sold it if the sales tax were automatically figured in, so that's generally not how it works (except, as people have mentioned, for gasoline).

Which is not to say that local businesses cannot address the situation themselves to the benefit of their productivity and customers' satisfaction. I spent two years working at a Domino's Pizza where we said "screw it", and included the sales tax into all the sales and rounded to the nearest quarter, interpreting coupon deals as accurately as we could but still with everything priced to the quarter. In those two years, I fielded ZERO complaints from people who felt that their national coupons weren't being properly honored.

jjono
Posts: 71
Joined: Mon Oct 27, 2008 11:35 am UTC

### Re: Ideal Currency Denominations

A related mathematical question is this:

Suppose you can choose exactly n coin denominations, each less than a dollar. Let c(x) be the minimum number of coins needed to give x cents in change using these denominations. Which denominations should you choose to so that $\sum_{x=1}^{99} c(x)$ is a minimum? (In a sense, you are trying to minimise the 'average awkwardness' of giving change for a dollar.)

Of course, this probably won't give denominations that are easy to work with in your head, which is what most real currencies try to do. But it still seems like an interesting question... Any takers?

Apeiron
Posts: 120
Joined: Tue Feb 12, 2008 5:34 pm UTC

### Re: Ideal Currency Denominations

Wow! What a lively discussion. Thanks for the input. i'll look in to the ternary thing in a bit.

i found that 1, 2 and 5 work very well, as does 1, 3, and 5. But 3 is prolly less useful because doing math with 2 is much easier.

In the US, i'd like to see the abolishment of the penny, then round all prices to a 5 or 0(this works well at overseas bases). And/or require that state sales taxes be shown on the price tag. i'd like to know what something is going to actually COST ME.

If there was a base2 currency it could use 1, 2, 4, 8 etc.

stephentyrone
Posts: 778
Joined: Mon Aug 11, 2008 10:58 pm UTC
Location: Palo Alto, CA

In the US (or at least northern california), I'm seeing fewer and fewer $10 and$50 bills in circulation (a lot of businesses don't even give $10 bills as change, and I haven't gotten a$50 as change for like 12 years); the $2 bill has already been almost non-existent for years. We seem to be drifting towards a 1/5/20/100 system, which is just peachy with me -- it's still a "dense" enough set of denominations, and fewer different types of bills to keep track of. GENERATION -16 + 31i: The first time you see this, copy it into your sig on any forum. Square it, and then add i to the generation. gmalivuk GNU Terry Pratchett Posts: 26767 Joined: Wed Feb 28, 2007 6:02 pm UTC Location: Here and There Contact: ### Re: Ideal Currency Denominations There don't seem to be any significant differences in the number of bills printed with those denominations, and I still get 10s as change whenever appropriate, so I think that might just be a regional thing. Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true. --- If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome (he/him/his) achan1058 Posts: 1783 Joined: Sun Nov 30, 2008 9:50 pm UTC ### Re: Ideal Currency Denominations stephentyrone wrote:In the US (or at least northern california), I'm seeing fewer and fewer$10 and $50 bills in circulation (a lot of businesses don't even give$10 bills as change, and I haven't gotten a $50 as change for like 12 years); the$2 bill has already been almost non-existent for years. We seem to be drifting towards a 1/5/20/100 system, which is just peachy with me -- it's still a "dense" enough set of denominations, and fewer different types of bills to keep track of.
In Canada, on the other hand, 2 dollar coins and 10 dollar bills are used often as well. 100, however, essentially faded from the system. Anyone making such a big purchase is likely to use a debit/credit card, and there is also the issue with counterfeit bills.

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: Ideal Currency Denominations

jjono wrote:Which denominations should you choose to so that $\sum_{x=1}^{99} c(x)$ is a minimum? (In a sense, you are trying to minimise the 'average awkwardness' of giving change for a dollar.)

The way to minimise that sum is to have one denomination of size x for all 0<x<100, so I don't think that it does what you want it to.

I think the 1:2:5 thing works well enough here in oz. I'm unsure how to create any kind of objective measurement of best denominations. Perhaps something like "a set of denominations is ideal if the smallest set of coins that can make any amount of change between 0 and 100 is minimal". Of course, we don't use anything that isn't a multiple of 5 in oz, and we don't use notes smaller than a $5, so I guess you'd have to change that methodology for here. ameretrifle wrote:Magic space feudalism is therefore a viable idea. NathanielJ Posts: 882 Joined: Sun Jan 13, 2008 9:04 pm UTC ### Re: Ideal Currency Denominations jestingrabbit wrote:The way to minimise that sum is to have one denomination of size x for all 0<x<100, so I don't think that it does what you want it to. He made the added restriction beforehand that you are only allowed to choose n different denominations. If n=99 then of course the problem is trivial. If instead of trying to minimize that sum, we try to minimize the maximum number of coins needed to make a certain amount of change, then you basically just get coins that correspond to different bases. For example, if n = 7 then you're best off choosing the values of 1, 2, 4, 8, 16, 32, and 64 cents, since then you can represent any value under$1 using no more than 6 coins, which is optimal. If n = 5 then you're best off choosing 1, 3, 9, 27, 81 since then you can represent any value using no more than 8 coins, which is again optimal.
Homepage: http://www.njohnston.ca
Conway's Game of Life: http://www.conwaylife.com

Robstickle
Posts: 326
Joined: Sat Nov 28, 2009 10:07 am UTC

### Re: Ideal Currency Denominations

Instead of £1, £2, £5 etc coins/notes we need £0.99, £1.99 etc coins/notes.

But then the fuckers would probably start charging £x.98 instead

Nitrodon
Posts: 497
Joined: Wed Dec 19, 2007 5:11 pm UTC

NathanielJ wrote:If instead of trying to minimize that sum, we try to minimize the maximum number of coins needed to make a certain amount of change, then you basically just get coins that correspond to different bases. For example, if n = 7 then you're best off choosing the values of 1, 2, 4, 8, 16, 32, and 64 cents, since then you can represent any value under $1 using no more than 6 coins, which is optimal. If n = 5 then you're best off choosing 1, 3, 9, 27, 81 since then you can represent any value using no more than 8 coins, which is again optimal. If n=5 and you choose 1, 6, 10, 35, and 42, you can represent any value using no more than 5 coins. This contradicts the optimality of both sets you listed. kernelpanic Posts: 891 Joined: Tue Oct 28, 2008 1:26 am UTC Location: 1.6180339x10^18 attoparsecs from Earth ### Re: Ideal Currency Denominations DavCrav wrote:I think that, because ours was changed so recently, we got the most sensible system. 1p and 2p are bronze-looking, round, with 2p weighing twice as much as 1p. 5p and 10p are silver-looking, one twice the other, and round. 20p and 50p are straight-sided, silver-looking, and one is 2.5 times the weight of the other. £1 and £2 coins are thicker and round, again one twice the weight of the other, and our notes are of increasing size. Hence blind people know what's going on, and you can weigh similarly shaped coins together when in a bank. In Mexico, 5 (very rare, extremely useless) and 10 (rare, very useless) cent coins are small and silver (in color), the 10c one larger. The 20 and 50 cents are either bronze-colored and with 12 sides, larger than the 10 cents. 1, 2, and 5 peso coins are bimetallic, silver outside and bronze inside, in increasing size.$10 coins are large, silver (they used to be .125 silver, now something like .000125) inside and bronze outside. Then there are the extremely uncommon $20 and$100, coins, like the $10 one but larger. There is also one of the largest coins in the world, the 1 kg .999 silver$100 coin (technically a bullion, but still can be exchanged at face value, so it's legal tender)
I'm not disorganized. My room has a high entropy.
Bhelliom wrote:Don't forget that the cat probably knows EXACTLY what it is doing is is most likely just screwing with you. You know, for CAT SCIENCE!

jjono
Posts: 71
Joined: Mon Oct 27, 2008 11:35 am UTC

### Re: Ideal Currency Denominations

I did a few computer runs of minimising the sum I put up earlier. The optimal denominations with n coins seem to hang around the powers of 100^(1/n), which I guess makes sense as with this distribution each denomination is doing "equal work". There are usually multiple optima though.

Minimising the maximum number of coins given as change seems like a harder problem, and I'm not really sure how you would go about it.

It occurred to me that another desirable property that the 1-2-5 system has is that you can always give change in the minimum number of coins by starting with the biggest coin less than or equal to the change amount, then the biggest coin less than or equal to the remainder, and so on. In other words, the greedy algorithm is maximally efficient in giving change.

I wonder what properties a set of denominations needs for this to work? Clearly it's enough that each denomination divides the next largest, but this can't be necessary as even the 1-2-5 system doesn't do this...

NathanielJ
Posts: 882
Joined: Sun Jan 13, 2008 9:04 pm UTC

### Re: Ideal Currency Denominations

Nitrodon wrote:If n=5 and you choose 1, 6, 10, 35, and 42, you can represent any value using no more than 5 coins. This contradicts the optimality of both sets you listed.

34 requires 6 coins doesn't it? You might have meant 25 instead of 35, but either way 99 requires at least 6 coins. Although you're right that things aren't as straightforward as I thought they were.
Homepage: http://www.njohnston.ca
Conway's Game of Life: http://www.conwaylife.com

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: Ideal Currency Denominations

1*10 + 4*6 = 34. Its an ingenious collection of denominations, that I assume can only have been computed not thought up.

@jjono: sorry for not seeing the subtlety of your initial idea. I think greedy is enough if 2*m <= n, for adjacent denominations, but I could easily be wrong about that.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Token
Posts: 1481
Joined: Fri Dec 01, 2006 5:07 pm UTC
Location: London

### Re: Ideal Currency Denominations

And, of course, 99 = 42 + 35 + 10 + 2*6. This is a brilliant example of how, when given a list of available coin denominations and a target total, the minimum number of coins necessary to hit the total is *not* best determined by the greedy algorithm.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

Tirian
Posts: 1891
Joined: Fri Feb 15, 2008 6:03 pm UTC

### Re: Ideal Currency Denominations

gmalivuk wrote:There don't seem to be any significant differences in the number of bills printed with those denominations, and I still get 10s as change whenever appropriate, so I think that might just be a regional thing.

It matters a *lot* what your local ATM's are dispensing. If your bank allows dispersal in $10 increments, then there will be some in circulation. The rest of the time, it depends. I've worked in retail food, and you don't bother getting$10 bills in "change" from the bank. So the number of Hamiltons in your register is a random walk that starts every shift at 0 and (IME) is more likely to drop than rise, so having none when they'd be good for making change for a twenty is a pretty frequent event.

gmalivuk
GNU Terry Pratchett
Posts: 26767
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: Ideal Currency Denominations

jjono wrote:It occurred to me that another desirable property that the 1-2-5 system has is that...the greedy algorithm is maximally efficient in giving change.

Also, those numbers are a hell of a lot easier to work with mentally than pretty much anything else proposed in this thread so far. Which is a very significant factor when you're actually designing a monetary system that you expect people to use.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: Ideal Currency Denominations

jestingrabbit wrote:I think greedy is enough if 2*m <= n, for adjacent denominations, but I could easily be wrong about that.

And of course I'm wrong: using {1,4,9} to make 12 is a counter example. I suspect its necessary, but not nearly sufficient. You want some sort of approximate uniformity in the ratios of adjacent denominations as well I think.

And yeah, being able to work with the numbers is critical. As is I think a certain amount of modularity (50 = 2*25 = 20*2 +10= 5*10 etc etc). You want to be able to break coins up, or else you'd have to mandate that every coin operated device would necessarily take every coin.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Nitrodon
Posts: 497
Joined: Wed Dec 19, 2007 5:11 pm UTC

### Re: Ideal Currency Denominations

It's not quite a necessary condition either, as shown by the counterexample {1,2,3}.

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: Ideal Currency Denominations

Yeah, maybe [imath]\lceil n/2 \rceil \geq m[/imath] gets it.

edit: This isn't even close.
Last edited by jestingrabbit on Fri Mar 19, 2010 5:51 am UTC, edited 1 time in total.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Qaanol
The Cheshirest Catamount
Posts: 3069
Joined: Sat May 09, 2009 11:55 pm UTC

### Re: Ideal Currency Denominations

There are several considerations for “ideal” here.

Minify the total number of coins that need to be counted for exact change.
Make it simple for a person with poor math skills to count out exact change.
Make it so the coins one is likely to receive as change are also likely to be used in similar quantities for purchases, so the net change in number of coins of each type in a pocket is expected to be around zero over many purchases.
Minify the total number of coins that a person “ought” to carry.

The first condition tends toward having many denominations of coin. The last tends toward have fewer denominations. The second toward having values that evenly divide the next-greater power of the base of the number system being used. Since we use base 10, that gives possibilities from the set {1¢, 2¢, 5¢, 10¢, 20¢, 25¢, 50¢, $1, 2$, …}.

Personally I’d lean toward 1¢, 5¢, 20¢, \$1, etc. (I suppose 25¢ could be just as good as 20¢.) That way each coin is needed in quantities no more than four at a time, and you don’t get things like the nickel only being used one at a time.
wee free kings

Rilian
Posts: 496
Joined: Mon Sep 10, 2007 1:33 pm UTC

### Re: Ideal Currency Denominations

gmalivuk wrote:
kernelpanic wrote:And now that we're at it, why the hell are dimes smaller than nickels?

Because nickel is worth less than silver, which is what dimes used to be made from.

"Actually, the first five-cent coin in U.S. history was made of silver and was smaller than today's dime.

That's because when coins were first produced by the U.S. Mint in 1793 the U.S. standard coin was the silver dollar, and additional coin denominations were made with a proportionate silver content to the dollar. This, in turn, established the size of each. For example a ten-cent coin, or dime, contained 1/10 the silver found in a dollar.

The five-cent coin (which contained 1/20 the silver found in a dollar) was eventually determined to be too small to handle, and the five-cent coin we know today as a "nickel" was created in 1866. The size of the coin was increased and its metallic content was changed from silver and copper to a combination of copper and nickel."
And I'm -2.

### Who is online

Users browsing this forum: No registered users and 8 guests