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Coin denominations (math problem/discussion)

Say you have a set of gaming coins that has 5 denominations but without a printed value. Consider a game like Modern Art where exchanging $100+ is pretty common by the end of the game, but you also need to be as granular as individual dollars. How would you choose the denominations of each coin? Besides the smallest coin, which must be $1. I have some ideas, but I’ll withhold to avoid biasing discussion.

Either doubling each time (1,2,4,8,16) or some coinage that means that any value needs the smallest number of coins. (1,2,5,10,20 or 1,5,10,25,50)

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I wrote some code to calculate minimum number of coins. Assuming three of the denominations are 1, 10, and 100, the denominations that would minimize the number of coins needed to “make change” from 0 to 99 (assuming each is equally likely) are 6 and 31. But I think people would find issue with using 1, 6, 10, 31, and 100 coins.

Edit: found a bug with my code.

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This would be my initial go to, as it’s the denominations we use in the UK. But for games, I think you’d only need to make a jump when there’d be a good reason to. So 1 and 2 / 10 and 20 are somewhat excessive. Changing 2 $1 pieces for a $2 is silly. And 3 coins to make $5 rather than 5 isn’t much of a convenience.

So I’d say 1 / 5 / 10 / 50 / 100 is as granular as you’d need to be.

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This is a common denomination pattern, 1, x5, x2, x5, x2…

I have a set of unmarked poker chips in 5 colors and I typically use 1, 5, 10, 20, 50… But I might do 1, 5, 10, 50, 100 depending on the game

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Question: current common US denominations are 0.01, 0.05, 0.10, 0.25, 1, 5, 10, 20, 50, 100. Now 0.01, 0.10, 1, 10, and 100 make sense because we use a base-10 numbering system. But why 0.05, 0.25, 5, 20, 50? Is there a good reason for the intermediate denominations to be some fractional part of the next larger denomination?

Using only (1,2), to make each value from 1 to 9 would take 25 coins. For (1,5), it’s also 25 coins. Which is why cash registers have to keep a lot of $1 and $0.01 around to make change. But if you instead had (1,3) or (1,4), that would be 21 coins.

Except that people are bad at spotting sets of 10 quickly. Typically, between 5 and 7 are the accepted maximum number of things you should group together in order to quickly visually parse without having to stop and count/index

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Since I had the code, I removed the 10 condition. To minimize the number of coins if 1 to 99 are equally likely are (1,5,18,25) or (1,5,18,29), averaging 3.89 coins. But most people would be hard-pressed to find the exact minimum number, so this code optimizes the wrong thing.

Edit: found a bug in my code.

Are you trying to argue for or against base-10 denominations? I would think that humans being unable to count more than 5 to 7 at a glance would argue for base-10, since humans would chunk each digit, reducing a group of coins into a single item to store in short-term memory.

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I was actually hinting that a strict 1, 5, 10, 50, 100 system would mean you’d never be pressed to spot more than 4 things in an optimized group

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Sure, but same can be said for 1, 3, 10, 30, 100.

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Another vote for 1, 5, 10, 50, 100.

It’s all about ease of counting, as Pillbox points out, and grouping into multiples of 10.

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I almost agree. Three 3-value coins and a single 1-value coin can be traded for a 10-value… But that’s not quite as easy as two 5-value coins trading for a 10. I truly believe that if people used the 1/3/10/30/100 pattern for a lifetime, it would be just as natural… But I’ll stick to 1/5/10/50/100 or, sadly, more often 1/5/10/25/50 that is commonly used as poker chip denominations

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Oh, I think I see what you’re saying now. When you said “optimized group,” I thought you ruling out examples where there would be more than one 5 (because you’d use a 10 instead). But breaking a 10 is a common operation in a poker game, and it’s easier to grab a couple 5s than three 3s and one 1. Ditto for the reverse operation. That is a good point, thanks.

Also consider other “make change” scenarios… An extreme example, perhaps, might be 1 from 100, returning 99:

  • 1/10/100: get back 9x 1s, 9x 9s - hard to spot check… Awkward silence while you count the change and silently insult the banker
  • 1/3/10/30/100: get back 3x 30s, 3x 3s - pretty easy to spot check, pretty easy to grab for the banker
  • 1/5/10/50/100: get back 1x 50, 4x 10s, 1x 5, 4x 1s - pretty hairy, actually… but easily parsed due to no single denomination exceeding 4

Other situations would favor 1/5/10/50/100 over 1/3/10/30/100, obviously. I think a big part of it is what you are used to. I, for one, am an advocate for them dozenal system… If we all used base 12, wouldn’t things be so much better?

To perhaps muddy the discussion further, the value of what you pay for should be some kind of an indicator for suitable divisions of coin. If there is little variation in the single units, then 1,5,10 etc is fine. But if there are a lot of things where the units end in 3, 8 or 9, you’ll probably find yourself wanting a 2 coin, or maybe a 3 coin.

Usually once value of items in games go over a certain value, then units often get rounded off (e.g. once past 50, increments will tend to be in 10s etc.). Therefore perhaps it is worth also considering whether coinage needs an even distribution or scales to where the game play ‘action’ happens. So if there’s there’s a lot of play at low values, and then jumps to more rounded high values, maybe 1, 3 (or 2), 5, 10, 100 could work better?

So one game that has a lot of action with low values is Sheriff of Nottingham. It’s denominations are 1, 5, 20, and 50. The 20 and 50 only really get used to do end-of-game scoring, and bribes and penalties (at least with my group) are handled with 1 and 5. So instead of taking 1, 5, 10 and swapping the 5 for a 3, it swapped the 10 for a 20.

Interestingly, my copy of Modern Art does the same thing. I have an old copy whose denominations are 1, 5, 20, 50, and 100. Thought there’s a good reason for Modern Art to have 20, since the sale price of paintings at the end of the year is always a multiple of 20. And most auctions during the game can be carried out using 1, 5, and 20. The 1:5 and 1:4 ratios here corroborate @pillbox’s groups of at most 5 to 7.

The only game I can think of that has a 3 coin is 7 Wonders, but that’s because 3 coins = 1 VP. It only has 1 and 3 denominations since only small numbers trade hands.

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I think we should all go back to the old British model …

"NOTE FOR YOUNG PEOPLE AND AMERICANS: One shilling = Five Pee. It helps to understand the antique finances of the Witchfinder Army if you know the original British monetary system:

Two farthings = One Ha’penny. Two ha’pennies = One Penny. Three pennies = A Thrupenny Bit. Two Thrupences = A Sixpence. Two Sixpences = One Shilling, or Bob. Two Bob = A Florin. One Florin and one Sixpence = Half a Crown. Four Half Crowns = Ten Bob Note. Two Ten Bob Notes = One Pound (or 240 pennies). One Pound and One Shilling = One Guinea.

The British resisted decimalized currency for a long time because they thought it was too complicated."

(From Good Omens)

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1, 2, 5, 10, 25

I can’t really cope with the coins in Scythe that have value three. Who thinks in threes?

On the other hand, I grew up playing poker with sixpences and pennies that my father held on to after decimalisation, so I can work with those. The coins in Brass confuse me a bit because they have the colour of sixpences and pennies, but they are actually worth five and one not six and one.

Familiarity is far more significant than maths here.