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The Almighty Buck Science

Making Change 1129

Roland Piquepaille writes "There are mostly four kinds of coins in circulation in the U.S: 1 cent, 5 cents, 10 cents, and 25 cents. But is it the most efficient way to give back change? This Science News article says that a computer scientist has found an answer. "For the current four-denomination system, [Jeffrey Shallit of the University of Waterloo] found that, on average, a change-maker must return 4.70 coins with every transaction. He discovered two sets of four denominations that minimize the transaction cost. The combination of 1 cent, 5 cents, 18 cents, and 25 cents requires only 3.89 coins in change per transaction, as does the combination of 1 cent, 5 cents, 18 cents, and 29 cents." He also found that change could be done more efficiently in Canada with the introduction of an 83-cent coin and in Europe with the addition of a 1.33- or 1.37-Euro coin. Check this column for more details and references." The paper (postscript) is online.
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Making Change

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  • by Swannie ( 221489 ) * on Friday May 16, 2003 @10:56AM (#5972431) Homepage
    I think the advantage to having a 10-cent piece is that it makes the math easy. Let's face it; can you imagine the average cashier at WalMart giving back 98 cents change with an 18-cent coin?

    • Re:I hate math... (Score:3, Insightful)

      by Anonymous Coward
      That's part of the process that doesn't seem to have been looked at well. While the model of 'giving change' may include a number of coins, it also includes the entire process including a cashier translating the change amount into coinage, then counting it back to the customer, and much of the time, the customer counting and checking it's the right amount.

      For the pure maths side of it it's pretty neat, all the same - just not completely useful when it comes to Real World Stuff
      • Re:I hate math... (Score:5, Insightful)

        by Khasmo ( 659750 ) on Friday May 16, 2003 @11:12AM (#5972652)
        Beyond that, it also means the cashier would have to figure out what the most efficient combination of change is . . for 0.36 don't give a quater two nickels and a penny, just give two 0.18 coins.

        The logic for determining change is really easy for a cashier. start with the largest coin and work your way down until it all adds up.
        • Re:I hate math... (Score:3, Informative)

          by Alsee ( 515537 )
          The logic for determining change is really easy for a cashier. start with the largest coin and work your way down until it all adds up.

          I can't think of an example where that doesn't work in a 1,5,10,25 system, but it is definitely not a valid rule in general. For example in a 1,40,41 system you can give 80 cents change with two coins, but your method would use fourty coins.

          • Re:I hate math... (Score:4, Insightful)

            by DarkFyre ( 23233 ) on Friday May 16, 2003 @11:39AM (#5972996)
            Yes, yes, congratulations on completing first year computer science (or math, for that matter).

            Don't you think that the current denomination systems are designed specifically so that the greedy change-making algorithm will work?

            The poster you were replying to seemed aware of that; They were merely saying that since the current denominational system has this property, it is easy to use. The problem with adding 32- and/or 18-cent coins is that the greedy approach may no longer make the most optimal change.
          • Re:I hate math... (Score:3, Informative)

            I can't think of an example where that doesn't work in a 1,5,10,25 system

            The reason you can't think of any examples in the 1,5,10,25 system is because 10 and 25 are both multiples of five. Therefore whatever you could make with a 25, you could also make with five 5s. So if you would ever have five 5s or two 5s, just use a 25 or a 10, respectively. In 1,40,41 system, 41 is not a multiple of 40 (or vice versa), so it makes finding the optimal number of coins a bit more difficult, since you have to find the
            • Re:I hate math... (Score:5, Insightful)

              by ukyoCE ( 106879 ) on Friday May 16, 2003 @12:00PM (#5973231) Journal
              Close, but I dont think the multiples-of-five is the reason why it works. As someone else said, it has to do with greedy algorithms, and our coin system was specifically designed so that you'd be able to start from the biggest coin and work your way down. We went over it briefly in one of my CS courses. I think it may have been that each coin must be at least twice the value of the previous coin. Something like that.

              And as many people have mentioned, the current system is probably the best because of the ease of addition/subtraction. An 18-cent coin would be a nightmare for most minimum-wage cashiers. The only problem with our current monetary system is that inflation has made pennies freakin worthless.
          • Re:I hate math... (Score:4, Interesting)

            by Captain Rotundo ( 165816 ) on Friday May 16, 2003 @12:35PM (#5973560) Homepage
            When I worked in retail I used to totally baffle my coworkers. They all depended on the register to tell them what to give back, then they would carefully count largest to smallest. (as they were trained) I never entered the amount the customer gave me in ther register, as it took too much time, and I would count back the change randomly (as I could/can easily figure out in my head what the best use of the coins is) I swear there were times that it seems they thought simple math was magic, and my manager thought I more than one occassion that I was doing it to steal. quite amusing.
      • Re:I hate math... (Score:3, Insightful)

        by jimlintott ( 317783 )
        When was the last time a cashier counted back your change. They usually give whatever the computer tells them. Counting back change is a lost art. There is no question that these unusual coin values would be more difficult. Ease in making change isn't simply number of coins given back.
        • Actually, ... (Score:3, Insightful)

          The computer tells the cashier to give back 98 cents, not "6 times the coin 18, ..." etc.

          So your argument is moot: The cashier does actually have to use math to give you back your change.
    • Pet peeve (Score:5, Insightful)

      by swillden ( 191260 ) <> on Friday May 16, 2003 @11:12AM (#5972649) Homepage Journal

      Math != Arithmetic

      I just hate it when people say... "Oh, you're a mathematician! Can you add up these numbers for me?"

      I suck at addition. Give me a theorem to prove!

    • by aborchers ( 471342 ) on Friday May 16, 2003 @11:20AM (#5972741) Homepage Journal
      ... so bear with me.

      The paper assumes all amounts in 0..99 are equally likely in making change. Is this a fair assumption? Wouldn't there be patterns in the amount of change returned based on varying "psychologically appealing" decimal retailer pricing schemes, local sales tax percentages, the average total price of a transaction, etc? Wouldn't the optimal selection of coins in making change therefore be influenced by the peaks and troughs of this change function?

      • I agree (Score:5, Informative)

        by sxltrex ( 198448 ) on Friday May 16, 2003 @11:52AM (#5973144)
        Think about a store that sells DVDs. Many of the items sold are going to be priced at $19.99. Maybe some at $15.99, and maybe some at $29.99, etc., but even then you've only got 4 or 5 amounts that are going to represent a large percentage of the sales at that store. I used to be a cashier, and you learn a few of the most common totals very quickly.
    • by bryane ( 614590 ) on Friday May 16, 2003 @11:21AM (#5972776)
      Turns out that featuring 18-cent is more for sensationalism than reality. The article says adding a 2-cent coin will accomplish the same goal of reducing the average number of coins below 4. It would also keep change-making easy.

      The article also assumes a uniform distribution of change between 1 and 99 - not likely, given how things work.
    • by tmasman ( 604942 ) <tmasman@ya h o o . com> on Friday May 16, 2003 @11:23AM (#5972789) Homepage
      Could you imagine trying to get change for a dollar?
      Customer :Can I get change for a dollar?
      Cashier :Would you like,
      • 3 x .29, 2 x .05, & 3 x .01
      • 2 x .29, 2 x .18, 1 x .05, & 1 x .01
      • Oh dear GOD!!!
      Yeah, I can see your local fastfood cashier doing that math in their head.
      But I'll give him this much... He'd save us 0.81 of a coin each time we got change!
      (I don't want 81% of a coin!!! It wouldn't roll or spin so well after you cut it!!!)
    • Re:I hate math... (Score:3, Insightful)

      by sporty ( 27564 )
      I took psych as a minor, and came across an interesting thing on IQ tests and math.

      A poor family, in a foreign country, that sends their child out to work selling stuff on the corner would do poorly on an IQ. They'd lack the skills you get from going to school necessary to actually take an IQ test. You learn basic things, like pattern matching and some minor analytical skills.

      Take that child, and ask that child who has to properly make change daily, and does it, could do math quiet well. Easily better
    • by PD ( 9577 ) * <> on Friday May 16, 2003 @11:45AM (#5973071) Homepage Journal
      Considering that the last dude who rang up my sale at WalMart had 9 fingers, I'd say that he'd probably find the 18 cent coin quite natural.
    • Re:I hate math... (Score:3, Interesting)

      by Mr Guy ( 547690 )
      And let's not forget cashiers aren't the only ones who need to do the math quickly. People also "over" pay in order to optimize the change returned. An example of this is when I don't have exact change for my coke, (1.26) so I pay a dollar, three dimes and a penny to get a solid nickle back. If you have 3 pennies, 4 dimes, 2 eighteens and a dollar, how much should you give the cashier to get the optimal highest single coinage back?

      I'd probably still go with 3 dimes and a penny, cause I can do that in my
  • by IpsissimusMarr ( 672940 ) * on Friday May 16, 2003 @10:57AM (#5972447) Journal
    Are you kidding me?!

    Have you ever gotten a bill for dinner for say $12.50 and you give the cashier $15 saying the tip is included?
    You would think 15.00 - 12.50 is doable right?

    HELL NO! The cashier pulls out a calculator to do the math so she can write it in for the waiter's tips!!!

    If people can't add things like this 18cent coins are out of the question.

    Although I would like to hear a cashier go,
    "That makes $0.88 change sir." Pick out two quarters then, ... *pause* .... and just stare blankly at the change drawer.
    • by ip_vjl ( 410654 ) on Friday May 16, 2003 @11:08AM (#5972602) Homepage
      One time at a grocery/conveneience store, I had a total come to something like $1.87

      Wanting to minimize some of the change in my pocket, I gave the clerk $2.00 in bills and 12 cents.

      The clerk tried to hand it back, saying "it's only $1.87"

      I said, "yes, but this way, I'll get a quarter back in change."

      He took the money, punched it into the cash register, and as he handed me back the quarter, he said "How did you know that?"

      It's funny (in a VERY sad way) that to him, the cash register was this magic oracle that told him what to do, and that it didn't occur to him that what he was doing was even knowable without its use.

      • Re:Yeah Right... (Score:5, Insightful)

        by gentgeen ( 653418 ) on Friday May 16, 2003 @11:37AM (#5972971) Homepage

        It's funny (in a VERY sad way) that to him, the cash register was this magic oracle that told him what to do, and that it didn't occur to him that what he was doing was even knowable without its use.

        I am a High School math teacher, and I can't agree with this statement enough. Somedays I laugh, somedays I cry, but it is always sad when I see a student need the calculator for the most BASIC of operations (And I am not even counting the OP's example as "basic", that would be "basic+")

        I think it all comes from the fact that students are allowed to use calculators at such an earlier point in thier schooling. I am only 29, but I was not allowed to use a calculator in school until somewhere around 11th grade. It really hones (sp?) those basic math skills. I'll step off my soap-box now Sorry ;-}

        • Re:Yeah Right... (Score:5, Insightful)

          by ip_vjl ( 410654 ) on Friday May 16, 2003 @11:53AM (#5973155) Homepage
          I don't blame people for not being able to do math in their head. I know some smart people that have trouble doing calculations without paper. My wife was a math major in college, and she sometimes has problems doing calculations in her head.

          The thing that struck me about this guy was that it wasn't even that he couldn't do the math in his head ... he wasn't even aware that there was something that he could do to arrive at the answer. It's as if he didn't know that math even existed.

      • by blahedo ( 24332 ) on Friday May 16, 2003 @11:56AM (#5973192) Homepage
        A few years back, my dad was paying for something, and paid an uneven amount in order to get even change. The clerk looked at the money, sort of shrugged, and punched it in and started counting out the change. The catch is---my dad misheard the amount. So when the clerk started counting out a bunch of pennies and nickels, my dad was like, "wait, what?" Had the clerk had *any idea* why my dad had given an uneven amount, she would have realised that he'd misheard the price. But she just punched it in and started counting it out....

        A few years after that, my sister (in 5th grade at the time) had a test with a miscalculated grade, and when my mom went in for a parent-teacher conference, she brought it up. In particular, she said she'd added up the number correct and divided by the total number of questions, and got a different percentage... the teacher looked down her nose at my mom and said, "that's *not* how it's calculated." How was it calculated? Well, you have these cardboard discs that you turn according to the total number of questions, and then you read the grade out of the little window corresponding to the number right.... This woman had only the vaguest notion that this grade was a percentage correct, and *no idea at all* that---as a percentage---it could also be calculated by dividing the numbers out. None.
  • D'oh! (Score:3, Funny)

    by aitala ( 111068 ) on Friday May 16, 2003 @10:57AM (#5972448) Homepage
    Is it too early in the morning or does this article not make sense? I have never seen an 18 cent piece in circulation n the US...
  • by Anonymous Coward on Friday May 16, 2003 @10:58AM (#5972454)
    My God...have I been playing Everquest that long?
  • Instead... (Score:5, Insightful)

    by frodo from middle ea ( 602941 ) on Friday May 16, 2003 @10:58AM (#5972463) Homepage
    Why not just get rid of silly prices like 99.99 and 4.37 and 1.49. ?
    Why not round prices to dimes ? Or even quarters ?
    • Re:Instead... (Score:3, Insightful)

      by Bazman ( 4849 )
      Because retailers would round _upwards_.

      • Re:Instead... (Score:3, Insightful)

        by xyzzy ( 10685 )
        Fine, who cares, let them! For god's sake, they're fooling no one. In addition, they should incorporate sales tax INTO the price so that the price you see is the price you pay. If an article is $4, it's $4! No change necessary!
        • Re:Instead... (Score:4, Insightful)

          by sehryan ( 412731 ) on Friday May 16, 2003 @11:13AM (#5972670)
          That would increase costs to the company. Take a department store. They can mass produce a sign for a certain item without any worries. Because sales tax can be different all the way to the county level, the company now has to make signs for every variation of price.

          I suppose they could just print out signs with an average price, and take a hit in certain areas, and make a much larger profit margin in others. It might balance out in some areas, but possibly not in others. And if you are a consumer in one of those higher profit margin areas, then you are getting screwed as well. It would work, but I don't think anyone would be too terribly happy with it
        • Re:Instead... (Score:3, Insightful)

          For god's sake, they're fooling no one.

          Who are you kidding? Of COURSE they're fooling someone. It's a proven fact that a given good will sell more units at $9.99 than at $10.00. YOU may not think you're being affected, but the truth it is works. Retailers price their goods at a level that will maximise sales.

        • Re:Instead... (Score:5, Insightful)

          by GospelHead821 ( 466923 ) on Friday May 16, 2003 @11:24AM (#5972814)
          They're not trying to fool anybody. At least, that was not the original intent of setting prices at $9.99 instead of $10.00. It was a technique that was intended to help keep cashiers honest. If an item cost exactly $10.00, the cashier didn't need to open the drawer to get change for the customer. At some point, it was determined that cashiers who did not have to open the drawer were statistically more likely to pocket the money themselves than to put it in the register. So prices were dropped by a penny to force the cashier to open up the cash drawer, to get change for the customer, thus increasing the statistical likelihood that company gets its money.
          • by Anonymous Coward on Friday May 16, 2003 @11:57AM (#5973202)
            Cecil [] has the right answer instead of this complete conjecture.
          • Re:Instead... (Score:3, Interesting)

            by ManxStef ( 469602 )
            I think it also had something to do with the way a transaction works at your average store.

            Consider the following typical purchase:
            • You walk into the store.
            • You find yourself a bag of CheesyPoofs, which cost $0.99USD (hey, they're that expensive thanks to the constant free advertising the get on SouthPark!), and head over to the counter.
            • The cashier looks up from their magazine (sneers) then rings up the price on the till, and asks for 99 cents.
            • You hand over a dollar, and grab your bag of CheesyPoofs
            • Th
        • Re:Instead... (Score:3, Interesting)

          Fine, who cares, let them! For god's sake, they're fooling no one. In addition, they should incorporate sales tax INTO the price so that the price you see is the price you pay. If an article is $4, it's $4! No change necessary!

          Give it up, you will never ever see logical pricing that doesn't require loose change.

          $19.99 may not fool you or me, but subliminally many people perceive it quite differently from $20. It begins with a "1". "In the teens" seems easier to justify psychologically than $20 for

      • Re:Instead... (Score:5, Informative)

        by simong_oz ( 321118 ) on Friday May 16, 2003 @12:09PM (#5973320) Journal
        In Australia copper coins (1c & 2c) were taken out of circulation in 1991 (I think). So everything is rounded to a multiple of 5c. The rules for the rounding (set out by law) are:

        For cash transactions:
        1 & 2 cents -- rounded DOWN to the nearest 10 cents
        3 & 4 cents -- rounded UP to the nearest 5 cents
        6 & 7 cents -- rounded DOWN to the nearest 5 cents
        8 & 9 cents -- rounded UP to the nearest 10 cents
        Rounding is on the total value of the bill. Individual items should never be rounded.

        And where a consumer pays by cheque, credit card or EFTPOS (electronic transaction) there is no need to round at all.

        So basically you win some and you lose some, but it evens out in the end. If you're really diligent, yes you can use it to your advantage, but most people have a life instead.
    • Re:Instead... (Score:5, Informative)

      by jmv ( 93421 ) on Friday May 16, 2003 @11:07AM (#5972584) Homepage
      In France (and probably other countries) most of the prices end in .00 and the taxes are already included (unlike Canada where I live). It's much simpler that way. If only there was a way to convince stores to do that in here...
      • Re:Instead... (Score:4, Informative)

        by pi radians ( 170660 ) on Friday May 16, 2003 @11:16AM (#5972696)
        If only there was a way to convince stores to do that in here...

        Its not up to the store, but the law. You must show the PST and GST on every sale in Canada. There was some debate a couple years ago about changing it to hidden costs, but that seems to have been quelled with recent wars and weed laws.
      • Re:Instead... (Score:3, Interesting)

        by Erik Hensema ( 12898 )

        In the netherlands (and most, if not all, of europe), consumer prices must always be advertised including VAT. This includes pricetags in the store itself and commercials on TV and such. It is illegal to advertise prices without VAT to consumers.

        Consumers never have to deal with prices without VAT included. The price isn't even mentionned anywhere.

        Because VAT is tax deductable when you buy a product for commercial use, you can get a receipt which shows how many VAT you have paid.

        It actually never occured

    • by YrWrstNtmr ( 564987 ) on Friday May 16, 2003 @11:23AM (#5972802)
      The US military overseas has done away with pennies (except at the post office)

      The ticket price still reflects .99 or whatever, but the price at the register is rounded down or up as needed.
      .08, .09, .00, .01, .02 = .00
      .03, .04, .05, .06, .07 = .05

      It averages out over time, especially when you buy more than one item.
    • Re:Instead... (Score:3, Interesting)

      by lightspawn ( 155347 )
      Why not just get rid of silly prices like 99.99 and 4.37 and 1.49. ?
      Why not round prices to dimes ? Or even quarters ?

      Because retailers use the fractional part to encode information. Did anybody here ever work at a Target? (or perhaps a wal-mart, etc?) Do you still remember what a price ending in ".97" means?
  • by charlieo88 ( 658362 ) on Friday May 16, 2003 @11:00AM (#5972483)
    I was at a conveinece store yesterday. The price came to $1.37. I tendered $2.12. The cashier's head almost exploded.
  • by Frightened_Turtle ( 592418 ) on Friday May 16, 2003 @11:02AM (#5972507) Homepage
    So- you have 7 18-cent coins, Susie gives you 13, and you give Bobbie 3. How many nickels must Daddy give you for your 18-cent coins...?

    Then, you get on a train in Boston traveling east at 300 MPH. In 30 minutes, will you really care about how many 18-cent coins you're carrying?
  • by MilesParker ( 572840 ) on Friday May 16, 2003 @11:02AM (#5972519)
    More proof of the ungoing schism between science and common sense.

    Me, I'm on the side of science.
  • by Violet Null ( 452694 ) on Friday May 16, 2003 @11:03AM (#5972533)
    You think it's bad enough when the cashier has to use the machine to figure out how much change to give, and in what denominations? You think it's bad when the little old lady in front of you in line starts counting, and then double-counting to make sure, the change she's going to hand over?

    You give them a 29 cent piece and see how fast things get.

    I'm willing to bet that most of the "coin cost" or whatever you want to call it comes from pennies, anyway -- if the dollar amounts are random, every 5 transactions are going to involve (0 + 1 + 2 + 3 + 4 = ) 10 pennies, or 2 pennies per transaction. Rounding prices to the nickel would be simpler, easier, and more efficient.
  • by citroidSD ( 517889 ) <citroidsd@y[ ] ['aho' in gap]> on Friday May 16, 2003 @11:03AM (#5972536)
    more efficient transaction costs?? Again the computer scientists are concentrating on the efficiency of the system, without any regard to the efficiency of the user. What is more important, handing back a smaller amount of change, or allowing for quicker mental computation of what coins to use in the transaction? For example when you are told something costs $3.48, it takes more time to calculate how to break up 48 cents into the available coin denominations, then it does to to actually exchange and identify the coins. The system is not what needs to be optimized, instead the user is what matters, and our mind work well in 10's and 5's.
  • by Viogression ( 231351 ) on Friday May 16, 2003 @11:03AM (#5972537)
    My mother went to the store to purchase something. The price on it was $20. It was also marked 25% off. It rang up as $18 instead of $15. My mother pointed this out, but the cashier would have none of it. "No, no, that sounds like 25% off."

    How the hell can we expect these people to handle 18 cent pieces when they can't even figure out what 25% of 20 is?
  • Or, even better ... (Score:3, Informative)

    by simong_oz ( 321118 ) on Friday May 16, 2003 @11:04AM (#5972544) Journal
    Do what Australia did a while back and round everything to the nearest 5c and get rid of 1c and 2c coins entirely (so now Australian coins are 5c, 10c, 20c, 50c, $1 and $2). I couldn't decide whether I liked it to start with, but after a little while you realise just how much shrapnel you carry around and have no intention of using except to empty it from your pocket/wallet at the end of every day. Every time I go to another country and have to again deal with 1/2 cent/euro cent/pence/etc I just appreciate this move even more.
  • by f97tosc ( 578893 ) on Friday May 16, 2003 @11:08AM (#5972598)
    It seems like the objective here was to minimize coin exchange. Ususally I try to minimize the number of coins in my pocket.

    If something costs 77c I give them 1.02 - and get a quarter back. In the US, the tellers stare at me blankly, but then dutifylly enter the amount I give them - and then smile in amazement at the simplicity of the exchange.

    In Japan, it is almost the other way around. The tellers come up with the most creative combinations that minimize my number of coins (and maximize theirs - this is in both of our interest).

    • The tellers come up with the most creative combinations that minimize my number of coins (and maximize theirs - this is in both of our interest).

      Just as a note here, its probably not in their best interest to get back as many coins as possible.

      I used to admin at a bank, and you are charged for "coin." This means the more you bring in which is unsorted, the more you are charged.

      Of course, if they roll their coin then this is not a problem.

      However, it would cost them more if they try to maximize their c
  • by Bingo Foo ( 179380 ) on Friday May 16, 2003 @11:08AM (#5972605)
    1. Some egghead thinks "optimal" means "fewest coins returned in change, on average."
    2. He recommends introducing 18 and 83 cent coins.
    3. The people who actually use coins laugh at this idiocy.
    Sheesh, "optimal" coinage denominations are those that make using coins easiest. That means quick mental calculations of change, manipulating them with your fingers, and passing them back and forth.

    The ivory tower academics are certainly earning their reputation for foolishness.

    • Some egghead thinks "optimal" means "fewest coins returned in change, on average."

      No no no. Academia don't have to think about definitions. We just define it that way.

      Be seriously, RTFA, people. The important part of this result is not that 18 or 83 cent recommendations. The author did it in jest in reference to the phrase "What this country needs is a good five cent cigar". (cited in the footnote of the paper). Just wait for /. to come along and rip everything out of context.

      The important part of this paper is the second half, the general analysis of methods for finding "optimal" denominations or "optimal" change returns (the first defined to minimize the number of coins returned on average, the second defined as given a set denomination, finding the best way to represent a given amount). It gives asymtotic results. It is more of a computer science excercise then anything else.

    • by dave_mcmillen ( 250780 ) on Friday May 16, 2003 @11:48AM (#5973100)
      There's another problem. Quote from the ScienceNews article:

      Assuming that each amount of change between 0 and 499 cents is equally likely, Shallit's calculations show that the average cost of making change would fall from 5.90 to 4.58 coins per transaction with the addition of an 83-cent coin.

      That's a pretty big assumption, isn't it? I'd assume that amounts of change would cluster around certain values. That was one thing that caught my interest, so I went to look at the article to find out how they evaluated that effect. Answer: apparently they didn't.

      To be fair, it's quite possible -- even probable -- that the original article was a light-hearted, tongue-in-cheek sort of piece, and that the author has been horrified to see it turned into a serious suggestion about actually changing the denominations of coins.

      In fact, the more I think about it, the more likely this seems. From TMI's site: "The Mathematical Intelligencer encourages authors to write in a relaxed, expository style and to include pictures and other graphics with articles. Opinion, mathematics, and historical comments can (and often should) be intermingled to make lively reading. Humor and controversy are welcome." So it was probably just a goofy abstract problem, written for entertainment value, not "serious" research. So I take it all back: let's give the guy a break, smile quietly, and move on.

    • by misterpies ( 632880 ) on Friday May 16, 2003 @01:01PM (#5973794)
      First, can't you tell a joke when you see one? (By joke I don't mean the maths is wrong, just that obviously the writer wasn't intending that we move to 18c coins).

      Second, what is easy is what comes with practice. Currencies, like most other measurement systems, were not originally decimal, but duodecimal (i.e. using base 12) and various multiples thereof. Right up to the 1970s, the UK used a currency system which had 12 pennies to a shilling and 20 shillings to a pound. The US and UK still use duodecimal for weights and measures (think pounds and feet) and the whole world uses it for time (12/24 hour systems) and angles (360 degrees is 30 times 12).

      Why were systems based on numbers like 6, 12, 24, 360 etc. so common, given that we tend to count in decimal? Well, they have large numbers of factors. In other words, while they might be harder to add and subtract in your head than decimal systems, they're much easier to do division with. And since division is much harder to do in mental arithmetic than addition, that's a big advantage.

      For example, with 12 ounces in a pound, I can take a half, a third, a quarter, a sixth or a twelth of a pound and still be dealing in whole ounces. With a decimal system, 10 has only 2 factors: 2 and 5. So to buy a quarter of something devised in a decimal system you end up with 2.5.

      Now that also has a knock-on effect when making change. Because of the limited factorisation of 10, most decimal systems divide things into 100s or 1000s.

      Result: in a decimal currency, you end up not with 10 cents per dollar, but with 100 cents. And that's the real reason you have so much change in your pocket. If we had 12 cents to the dollar (or euro), then by copying the old british system -- with a 1c, 2c, 3c and 6c coin -- you'd never need more than 4 coins to make change from a shilling.

      And would the cashier at WalMart be able to handle it? Well first off, maybe if as a result they had to think more as kids they'd be better off at maths to start with. And secondly, since they have to use a calculator now anyway, what would be the difference?
  • by MexicanMenace ( 673792 ) on Friday May 16, 2003 @11:10AM (#5972634)
    Don't y'all remember the SchoolHouse Rock about counting by 18?

    *taps foot*

    Eighteen is a magic number.
    Yes it is, it's a magic number.
    Somewhere in the ancient, mystic eighteenity
    You get eighteen as a magic number.
    The past and the present and the future,
    Faith and hope and charity,
    The heart and the brain and the body
    Give you eighteen.
    That's a magic number.

    18, 36, 54 . . .
    72, 90, 108 . . .
    126, 144, 162 . . .
  • by syle ( 638903 ) <`syle' `at' `'> on Friday May 16, 2003 @11:11AM (#5972646) Homepage
    An 18 cent coin is a good idea, but it's only a start. What we really need is one coin per possible amount of change. That way, when you pay with $1.00 for a $0.63 candy bar, you just get one 37-cent coin in return.

    This approach simplifies all transactions to one-coin change. Some people might argue that this is just too many coins to keep track of, but since no one keeps track of their change anyway, it wouldn't matter. It's easier to use the new change to pay as well: Instead of $0.67 being 2 quarters, a dime, a nickel, and 2 pennies, it can be paid in one coin. Or, you could use a 50-cent and a 17-cent piece. Or two 27s and a 13! The possibilities are endlessly easy!

    Some people say that it's a problem to differentiate the 99 different coins (95 new coins) by sight. There's a simple answer to this -- each coin would have a number of sides based on its amount. A 4-cent coin is a square, an 8-cent is an octogon, and so forth. So, remember, don't give them three quarters -- just reach into your pocket, feel for the coin with 75 sides, and hand it over.

    Oh, and if you can't tell a 99-sided coin from a 97-sided coin by sight, perhaps you should stick to smaller denominations.

    The new two-cent coins are easy to lose, so be careful.

  • why did we ... (Score:5, Interesting)

    by ( 410908 ) on Friday May 16, 2003 @11:14AM (#5972683) Journal
    Why did we fight against the Imperial System ?

    easy, look :

    Measures of length
    After 1959, the U.S. and the British inch were defined identically for scientific work and were identical in commercial usage (however, the U.S. retained the slightly different survey inch for specialized surveying purposes). A similar situation existed for the U.S. and the British mass unit pound, and many relationships, such as 12 inches = 1 foot, 3 feet = 1 yard, and 1760 yards = 1 international mile, were the same in both countries; but there were some very important differences.

    Measures of volume
    In the first place, the U.S. customary bushel and the U.S. gallon, and their subdivisions differed from the corresponding British Imperial units. Also the British ton is 2240 pounds, whereas the ton generally used in the United States is the short ton of 2000 pounds. The American colonists adopted the English wine gallon of 231 cubic inches. The English of that period used this wine gallon and they also had another gallon, the ale gallon of 282 cubic inches. In 1824, the British abandoned these two gallons when they adopted the British Imperial gallon, which they defined as the volume of 10 pounds of water, at a temperature of 62F, which, by calculation, is equivalent to 277.42 cubic inches. At the same time, they redefined the bushel as 8 gallons.

    In the customary British system the units of dry measure are the same as those of liquid measure. In the United States these two are not the same, the gallon and its subdivisions are used in the measurement of liquids; the bushel, with its subdivisions, is used in the measurement of certain dry commodities. The U.S. gallon is divided into four liquid quarts and the U.S. bushel into 32 dry quarts. All the units of capacity or volume mentioned thus far are larger in the customary British system than in the U.S. system. But the British fluid ounce is smaller than the U.S. fluid ounce, because the British quart is divided into 40 fluid ounces whereas the U.S. quart is divided into 32 fluid ounces.

    From this we see that in the customary British system an avoirdupois ounce of water at 62F has a volume of one fluid ounce, because 10 pounds is equivalent to 160 avoirdupois ounces, and 1 gallon is equivalent to 4 quarts, or 160 fluid ounces. This convenient relation does not exist in the U.S. system because a U.S. gallon of water at 62F weighs about 8 1/3 pounds, or 133 1/3 avoirdupois ounces, and the U.S. gallon is equivalent to 4 x 32, or 128 fluid ounces.

    1 U.S. fluid ounce = 1.041 British fluid ounces
    1 British fluid ounce = 0.961 U.S. fluid ounce
    1 U.S. gallon = 0.833 British Imperial gallon
    1 British Imperial gallon = 1.201 U.S. gallons

    Measures of weight and mass
    Among other differences between the customary British and the United States measurement systems, we should note that they abolished the use of the troy pound in England January 6, 1879, they retained only the troy ounce and its subdivisions, whereas the troy pound is still legal in the United States, although it is not now greatly used. We can mention again the common use, for body weight, in England of the stone of 14 pounds, this being a unit now unused in the United States, although its influence was shown in the practice until World War II of selling flour by the barrel of 196 pounds (14 stone). In the apothecary system of liquid measure the British add a unit, the fluid scruple, equal to one third of a fluid drachm (spelled dram in the United States) between their minim and their fluid drachm.

    In Great Britain, the yard, the avoirdupois pound, the troy pound, and the apothecaries pound are identical with the units of the same names used in the United States. The tables of British linear measure, troy mass, and apothecaries mass are the same as the corresponding United States tables, except for the British spelling "drachm" in the table of apothecaries mass. The table of British avoirdupois mass is the same as the United States table up to 1
  • by Chmarr ( 18662 ) on Friday May 16, 2003 @11:15AM (#5972692)
    One can reduce the amount of change simply by getting rid of the silly 1c piece, like Australia did (it got rid of both 1c and 2c pieces).

    Sure, people will bitch and moan for about 6 months, but then noone would ever consider going back.

    All you need to do is 2/3 round at the till. It's great! :)
  • by Graff ( 532189 ) on Friday May 16, 2003 @11:58AM (#5973212)
    Hey, I can beat this guy at this math thing. According to my calculations there are much more efficient combinations. For example, if you use the coins 1, 2, 4, 8, 16, 32, 64 it will take approximately 3.19 coins per transaction (this is simple binary arithmetic). That's way better than his system which takes 3.89 coins per transaction. The only problem is that the geeks will do just fine with these denominations but just try and ask the average waitress to make change using those coins. Go ahead and ask, I'm sure it will work out just fine! :)

    You know, if we mint 1 coin for every amount of change (like a 57 cent coin, a 58 cent one, etc.) then it will only take 1 coin per transaction. Of course then we have to worry about having 99 different coins, making them distinguishable from each other, etc.

    The current United States system of currency works just fine. Denominations of 1, 5, 10, 25 are easy enough to calculate and efficient enough for all intensive purposes. Sure this proposed new system may be 17% "more efficient" for a computer but real people need to use the system also.

    Some things are best off just left alone...
  • by jjohn ( 2991 ) on Friday May 16, 2003 @12:34PM (#5973541) Homepage Journal

    <voice style="school house rock">



    Multiply by 18 is like multiplying by 20 but subtracting multiples of 2. So 18*3 is really like 20*3 - 2*3. That's just 60 - 6, or 54! Let's do it again!

    18...36...54...72...90...108...126...144...162.. .180!

    Ready or not, here I come!


    no, I didn't use a calculator. I sure hope the math is right.

  • Once Again... (Score:3, Interesting)

    by Hank Reardon ( 534417 ) on Friday May 16, 2003 @12:45PM (#5973648) Homepage Journal
    Once again, we have a novel solution to a problem that doesn't exist. Isn't life fun?

    I believe that what the researcher failed to take into account is the way that the human mind works. Adding 1's, 5's, 10's and 25's is definately easier than adding 1.37's or 83's for us.

    Sure, it may make the handing out of change more efficient by lowering the average amount of change given from 4.x to 3.x coins, but that efficiency will be more than lost when the clerks at the local mini-mart -- who already have problems giving out the correct change -- have to figure out that my $0.72 in change will be two 29-cent coins, two 5-cent coins and four 1-cent coins.

    Not to mention the increasing size of cash drawer shortages caused by less-than-mathematically-inclined clerks.

    Is it just me, or does it seem that the less "rounded" education becomes, the more one-dimensional "solutions" appear? Guess it is more true than ever: when all you have is a hammer, everything looks like a nail.


  • by radulovich ( 47127 ) on Friday May 16, 2003 @12:46PM (#5973656) Homepage
    This article is a complete waste of time. This might be a fun paper for a discussion about coinage, but it fails horribly when taken as practical advice.

    The US does not need another coin. Indeed, the *opposite* is true. If you get rid of the penny, you can increase efficiency tremendously, to only 2.75 coins per transaction, and a whopping 45% of transactions would require 2 or fewer coins!

    Many people oppose the elimination of the penny, but bear with me for a moment. Consider the following issues:

    - Pennies cannot be used in vending machines, and therefore are not as "spendable" as all the other coins.

    - Prices will not rise as people think they will; they will fall instead! Everything that is priced at $n.99 will now be $n.95 instead (marketers HATE to price in round dollars because it makes their prices look higher). All other numbers will be rounded to the nearest $n.n5.

    - The US government makes 12 billion pennies at a cost of $100 million each year (, which could be put to better use than filling up my coin jar.

    - Half of these pennies will disappear from circulation within a year! ( ws_and_views/straight_dope.html)

    - Counting out pennies costs the economy an estimated $20 billion in productivity annually (

    - The U.S. Mint loses $8 million a year manufacturing pennies. ( ws_and_views/straight_dope.html)

    Think about it - do you *really* want another coin in your pocket? Thank God that politicians don't listen to us all the time!

  • by Enonu ( 129798 ) on Friday May 16, 2003 @12:57PM (#5973761)
    * 2.79 coins average per transaction given that getting 5 cents is just as probable as getting 95 cents, and that no 50-cent pieces are used.

    * Counting in 5's, 10s, and 25's is a lot easier.

    * Saving pennies, rolling them up, going to the bank, and then driving home is a pain-in-the-ass, and honestly isn't worth my time, e.g. 2 hours of work to get $10 of pennies?!?!?. It's more economical to throw the friggen ugly coins in the trash, but I can't do that out of principle.

  • by Wansu ( 846 ) on Friday May 16, 2003 @01:07PM (#5973849)

    It doesn't matter what the denomination is. As long as change has to be made, some patrons will receive the wrong change.

    Lots of cashiers don't know how to make change. Many have been trained to do it wrong. The most common error is the cashier puts the large bill the customer just handed them into the drawer before giving the customer change and watching them count it. There used to be a little slot between the plastic guard and the metal cash register enclosure that was perfect for temporarily storing that large bill in customer sight. When the customer looks at you after counting his money, pause to see whether he questions it, then put the large bill in the drawer and close it.

    Adding this momentary delay before putting the customer's large bill in the drawer and closing it, protects the cashier and the customer from being short changed.

    I've seen managers put large bills in the drawer before I counted my change. One gave me change for $10 instead of change for a $20. I'm a creature of habit. When I hand a cashier a large bill, I always say, "outta twenty" or whatever the bill is. I'm sure I did that with this one. But she'd already put the bill in the drawer and insisted upon a recount of the drawer and by the time she did, my food was cold. That is not the way to do things. When I pointed out her mistake, she lost her temper. Then I lost mine.

    I was trained on older cash registers to do things this way by a store manager who was very particular about this. He's been in business for more than 30 years and says he's never had a dispute with a customer over incorrect change. Way back then, you had to actually count the coin change. Many of the newer cash registers do this for you. I wonder how many of today's cashiers could make change in their heads.

    What's my point? Most point of sale problems concerning change making are due to lack of skill and/or poor training of the cashier. Using more efficient denominations or pricing items to the nearest buck won't fix this.

  • by KlomDark ( 6370 ) on Friday May 16, 2003 @01:34PM (#5974093) Homepage Journal
    I'd be curious to see how the average distribution of change is affected by the dominance of Walmart and it's strange method of pricing where most things end up costing X dollars and 88 cents. With the significant percentage of money spent at Walmart, it seems that it would throw off his distribution method, so perhaps a different denomination coin would be more appropriate in dealing with the Walmartization of America.
  • by Theovon ( 109752 ) on Friday May 16, 2003 @01:48PM (#5974210)
    The average cachier over the age of 18 barely managed to graduate highschool and is very unlikely to be able to do simple arithmetic. The average engineer I know making $80k+ also cannot do simple arithmetic.

    It's hard enough when you have to deal in 5's and 10's, but as soon as you start asking a cachier to add or subtract 18 from ANYTHING, you're going to have trouble.

    The whole problem here is that the author doesn't realize that humans are (a) not computers, and (b) don't care about handing out one less coin. The system we have, as imperfect as it is, evolved this way through error and natural selection. Sure, perhaps no one considered printing an 18 cent coin, but that's likely because they knew people would have trouble dealing with them. Humans inherently have trouble with simple arithmetic, so a system evolved that was less ERROR-PRONE, completely ignoring minor improvements in efficiency.

    So, of course, one has to ask the question: Could we make the system less error-prone? Probably. Maybe our esteemed computer scientist should develop a system to determine which coins we need to have in order to make it more likely for a cachier to give back correct change.

    What's better, taking 2 seconds longer to give you correct change or two seconds less to give you incorrect change? I'll wait the extra 2 seconds.

    Or maybe I'll just use my credit card.

Whenever people agree with me, I always think I must be wrong. - Oscar Wilde