Those Efficient Nephite Coins

I once read, and I do not remember where, that the Nephite coin system was the most efficient coin system possible. I think it might have been an old FARMS paper or something. I am a bit skeptical of such a claim, because proving something like that would be quite a mathematical feat. And once one set up the equations or simulations they well may have stacked things in favor of the system in question.

But something that can be easily done is comparing the Nephite coin system with other systems in a brute force way. So I would like to compare the Nephite coin system to the American coin system in terms of efficiency in making change. Efficiency will be defined as using the least amount of coins to make a given amount of change. I will compare the number of coins needed for amounts from 1 to 25 cents. After that I believe patterns would begin repeating. I will only assume the Nephite coins are in amounts of 1, 2, 4, and 7 without corresponding 10, 20, 40, and 70 amounts. American coins have a value of 1, 5, 10, and 25. I will keep a tally of the number of coins for each system, and an accumulative advantage of the leading system.

Ready, set, go.

Amount        AM      NM      Advantage

     1                 1           1               E
     2                 2           1             NM+1
     3                 3           2             NM+2
     4                 4           1             NM+5
     5                 1           2             NM+4
     6                 2           2             NM+4
     7                 3           1             NM+6
     8                4            2            NM+8
     9                5            2            NM+11
   10                1            3            NM+9
   11                2            2            NM+9
   12                3            3            NM+9
   13                4            3            NM+10
   14                5            2            NM+13
   15                2            3            NM+12
   16                3            3            NM+12
   17                4            4            NM+12
   18                5            3            NM+14
   19                6            4            NM+16
   20               2             4            NM+14
   21                3             3           NM+14
   22               4              4           NM+14
   23                5            4            NM+15
   24               6              5           NM+16
   25               1             4            NM+13

Well, there you have it. For smaller amounts of money the Nephite system clearly wins. But if this continues, the powerful quarter and dime will even things up. So the early advantage of the Nephite system will not last very long if we do not add larger coins like the 10, 20, 40, and 70 which may not have existed.

But even if large coin values did exist, and the Nephite coin system were the most efficient system, would that be meaningful to anybody? Would this imply that God had revealed the monetary system? Would this lend credibility to a translation by Joseph Smith? Would this make the Book of Mormon more true?

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14 Responses to “Those Efficient Nephite Coins”

  1. 1 Ben January 22, 2008 at 10:13 pm

    Were you aware the Nephites… didn’t have coins? :)

  2. 2 Justin January 23, 2008 at 11:04 am

    In a May 1954 Improvement Era article entitled “The Nephite Monetary System,” Richard Pearson Smith similarly compared the 1-2-4-7 system with the 1-2-4-8 and the 1-2-5-10 systems.

  3. 3 C Jones January 23, 2008 at 11:30 am

    Very interesting, Eric. I’m loving all the BofM posting going on this year.

    And I hope that 3 posts in one week means that BofJ is back in business :-)

  4. 4 Eric Nielson January 23, 2008 at 1:30 pm


    If the Nephite had pieces of metal that they used like money, may I call them coins?


    Thanks for the info. What? No links? It is surprising that the 1954 Improvement Era is not online.


    Thanks. Time will tell.

  5. 5 Justin January 23, 2008 at 2:21 pm

    Here’s the Gospelink version (figures are missing; formatting is a mess):

    The Nephite Monetary System
    by Richard Pearson Smith


    Table 1

    Comparison of Some Systems

    Amount of Number of Coins Required

    Purchase “1-2-4-8″ system “1-2-4-7″ system “1-2-5-10″ system
    1 1 1 1
    2 1 1 1
    3 2 2 2
    4 1 1 — 2
    5 2 2 — 1
    6 2 2 2
    7 3 —1 — 2
    8 1 —2 — 3
    9 2 2 — 3
    10 2 —3 — 1
    11 3 —2 2
    12 2 —3 — 2
    13 3 3 3
    14 3 —2 —3
    15 4 — 3 —2

    AN INTERESTING indication that the Book of Mormon was not written with mere human knowledge during the nineteenth century is to be found in an examination of the monetary system devised by the Nephites. This study is particularly interesting in that no acquaintance with scholarly works is necessary, nor do obscure references need to be quoted. Our considerations rest upon information to be found in the Book of Mormon, with only supplementary material which is well-known and undisputed.

    The monetary system used by the Nephites in about 82 B. C. is described in Alma II. Alma mentions that the system in use at that time was the result of a long series of changes, “according to the minds and the circumstances of the people,” and then outlines the system, summarized here in Figure 1. The numbers on the coins represent their values in terms of the senine of gold or its equivalent, the senum of silver, which, we are told in verse 3, was the daily pay for a judge. It is interesting to compare and contrast this system with the current United States system. (Figure 2.)

    The Nephite system was a peculiarly efficient one. The selection of 1, 2, 4, and 7 for the values of the larger coins seems particularly wise and is what intelligent people who were willing to have “… altered their reckoning and their measure … in every generation…” (Alma 11:4) might be expected to have worked out. This point is illustrated in Table 1, where the Nephite system is compared with two other possible systems. If the major coins had denominations 1, 2, 4, and 8, then three coins (1, 2, and 4) would be required for a purchase costing 7 of the basic units, while only one (the 7) would be required in the “1-2-4-7″ system; hence the “1-2-4-7″ system is more efficient here by two coins, as the long arrow in the table indicates. Shorter arrows indicate differences of one coin in efficiencies. Here the “1-2-4-7″ system is further compared with the “1-2-5-10″ system, and any other system could be compared in a similar manner. In every case it turns out that the “1-2-4-7″ system has an edge over the other systems from the standpoint of number of coins required for a purchase. Comparing the “1-2-4-8″ and “1-2-4-7″ systems, for example, we see that for some purchases the one system would be better, for others the other, but that over all, when we consider that smaller purchases will occur more frequently than larger ones, the “1-2-4-7″ system has great over-all efficiency.

    The more systematic 1, 2, 4, 8 series is almost as good as the 1, 2, 4, 7 series. A further reason for using 7 rather than 8, however, may have been that less gold or silver would be tied up in the smaller 7 coin, and this coin represented a fairly large sum of money—several days’ pay for a judge.

    The three subdivisions of the unit—1/2, 1/4, and 1/8—make it possible to build up any number of eighths of the unit with not more than three coins, and these subdivisions make a natural extension of the 1-2-4-7 system. The 11/2 coin, also, is useful for purchases between 1 and 2, which would be common; corresponding coins are found in many monetary systems.

    The “1-2-4-7″ system appears on a common type of punched card. (Figure 3.) This is an index card, for card files, holes being punched around the edge for classification. The cards are classified by punching out appropriate holes and are sorted on the basis of what holes are punched out. The holes are marked off in groups of four, and within each group are numbered 1, 2, 4, 7. “By punching various combinations of the four holes marked, respectively, 7, 4, 2, and 1, one may code any number from zero (no punching) up to and including fourteen (all holes punched) …. This code is a modification of the 1, 2, 4, 8, 16 … series; 7 is used instead of 8, so that with four positions any digit may be indicated by punching out not more than two holes.” 1 Thus the numbers 1, 2, 4, and 7 are used here for the same basic reason of efficiency that would be expected to apply to a well-designed monetary system.

    The Nephite system, being a slight modification of a binary system, where each coin would have twice the value of the next smaller one, is further interesting on historical grounds. Egyptian mathematics, which may have carried over into the Nephite culture in view of the background of Lehi and his people, was based largely on the binary system. 2 This system makes its appearance to some extent in many ancient system—for example, Alexander the Great established, in Macedon, a series of gold coins having values of 2, 11/2, 1/4 and 1/8 starters. 3 Other systems, mostly later, are based on the decimal system—1, 10, 100, etc. Remnants of both decimal and binary systems are found in our system (Figure 2), as well as in many others, ancient and modern, though the “1-2-4-7″ modification does not seem to have been recorded elsewhere in history.

    In conclusion, the Nephite system described in the book of Alma is an ingenious system which an intelligent group of people, with a willingness to change their system as improvements suggested themselves, could be expected to develop.

  6. 6 Eric Nielson January 23, 2008 at 4:06 pm

    Justin, you are amazing. Thanks for providing this.

  7. 7 Jettboy January 28, 2008 at 8:36 am

    I have one more question about this coinage. How many days worth of pay was Alma tempted by Zeezrom to deny his faith? Reading that section the other day it occured to me that the money was explained to express the amount of the temptation. From my own cursory examination it appears the amount offered was enough to be tempting for the poor, but slight enough to be an insult. I can’t help but think of the amount given to Judas at Jesus’ arrest.

  8. 8 Eric Nielson January 28, 2008 at 10:35 am

    Excellent point Jettboy. I believe you are right.

  9. 9 Emily Gillespie January 30, 2008 at 3:52 pm

    This is an amazing compilation of coinage wisdom. You guys just helped me explain this idea to my logics professor. Cheers!

  10. 10 vr6stress May 29, 2008 at 7:01 pm

    According to we had a half cent, 2 cent and 3 cent coins at one time or another. The last two coming after Joseph Smith’s death.

    I’m trying to make a point here, but I lost my train of thought. I think it had to do with, Joseph wouldn’t have gained an insight to the usefulness of a 2 or 3 cent coin (the Nephites would have won better with a 3 cent) because they weren’t in use at the time…

    I think that was my point…

  11. 11 helaman May 29, 2008 at 7:02 pm

    Sorry, that was me…stupid multiple log ins…

  1. 1 Amulek « ankylodoxy Trackback on July 4, 2008 at 4:02 pm

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