Puzzling Souls

We’re a little out of steam right now. Some big posts are planned, but it’s difficult to gather our strength to write them. In the meantime, we’ll keep things going with a few light and easy posts.

A while back we posted some (still unanswered) puzzles by Tony Gardiner, as well as the excellent article by Tony from which they came. Exploring Gardiner’s writing a little further, we stumbled upon a hilarious problem, from long ago.

John Bonnycastle was a mathematics teacher and textbook writer in the late 18th and early 19th centuries. (No, we hadn’t heard of him either.) His Scholar’s Guide to Arithmetic was first published in 1780 and went through many editions; the eighteenth, 1851 edition can be downloaded here, and other editions are readily available (behind academic paywalls). Bonnycastle’s Arithmetic is of course fascinating to browse, and it seems serviceable, if standardly wordy for the time and not particularly notable. One problem, however, which Gardiner used in slightly modified form, we just had to share:

19 Replies to “Puzzling Souls”

  1. There are several possible answers, depending on how the combined praying is defined, as well whether one of the fouls was a Maths Mangler or a lapdog or a Daft Curriculum writer (unlikely because they’re all eternally damned).

    The obvious and ‘expected’ answer is \displaystyle \frac{315}{176} hours (derivation available upon request). A bit longer than the time it would take the Pope, by himself, to pray out 3 fouls.

    Now I’ll issue a challenge: Who can come up with the ‘best’ ‘unexpected’ answer …?

    1. Does it not depend how the fouls combine themselves…? We are assuming it is an additive process, but one may expect such things to behave in some multiplicative form, each encouraging the other to more foulness.

    1. Hi Lancelot. I avoided mentioning rates (which might confuse Yr 7’s. Or some Yr 12’s …) by finding the fraction of a foul saved in 1 hour by each member of the clergy. Then I added those fractions together to get the amount of foul saved in 1 hour by all members praying together. Then got the time for 1 foul from this (if x fouls are saved in 1 hour then 1 foul is saved in 1/x hours) and multiplied by 3.

      Which boils down to what you did but without explicitly using rates.

      These sorts of ‘logic’ questions are very common on IQ and SAT tests (if 3 painters take 5 days to paint 2 houses, how long is a piece of string?)

      1. Hello John. Indeed, rates are very confusing. Your method sounds more intuitive but elegant.

        I remember when I was learning this idea 20+ years ago in primary school, learning division and fractions alongside this type of problem often involved rates problem.

        Later on, I learnt the idea of “total time = total amount of work/working efficiency” as a formula about year 3 or year 4, and this algebraic method works quite well on me since then.

        Now, when I am teaching this idea, I usually need diagrams to facilitate students visualising and understanding the problem. I find that many kids in the current era struggle to figure out which is which – when applying these sorts of formulae. But when it comes to a diagram/shape/picture, they like it more.

      2. There is possibly a link somewhere there to Harmonic Mean vs Arithmetic Mean vs whatever else.

        Definitely nice problems if the audience is properly chosen and the delivery properly timed.

        1. Hi RF. I had \displaystyle exactly the same thought (and we can toss the geometric mean into the pray fray). But I couldn’t think of how to incorporate the idea of an ‘average’ into the problem. Average time it takes a random member of the clergy to save a foul? Average number of fouls saved by a random member of the clergy?

          The question is also quite interesting from an English language perspective.

          1. Here is an idea (with absolutely no thought beyond what will be written here).

            There are some scholars reading this manuscript a century later, and just like they realise “foul” should be taken as “soul”, they don’t know what is meant by “average”.

            So when a bishop and a priest work together, they average X etc…

            Determine from this information alone which definition of “average” each of the prayers was using.

            Perhaps a bit simplistic, but might be good for a lesson..?

    2. Without algebra:

      Find LCM of 1, 3, 5 and 7 = 105.

      Now find how many souls each can pray out of purgatory in 105 hours:

      Cardinal 3 x 5 x 7 = 105
      Bishop 5 x 7 = 35
      Priest 3 x 7 = 21
      Friar 3 x 5 = 15

      Total 176 souls in 105 hours

      To pray out 3 souls need 105 * 3 / 176 = 315/176 hours (Rule of Three Direct).

  2. Thanks everyone. My understanding is that Asian students do a lot of these kinds of problems, starting in upper primary school. The approach then would be along the lines John is indicating, avoiding algebra and explicit mention of rates. Of course, my favourite is the following:

    1. Re: Asia.
      Indeed. The ‘technique’ is simple and can be easily rote learnt, but it provides a fun context for making good friends with the basic arithmetic of fractions and several other concepts (such as least common multiple etc).

  3. Harmonic mean will give time required for 1 soul saved per each, which is 4 souls. So 3/4 of harmonic needed.
    Nor that I started this way, long time no use of it.
    But worthy extention when dealing with such a problem. I am sure I mentioned this to y11s, a runner , 2 different speeds over the same dist or the same time, finding av speed.

  4. Close to it is the effective resistance of resistors in parallel setting. So it is not completely “foreign”.

  5. Hi all. Maybe this was a running gag but did you all really mean “fouls”. Back then they printed a “s” like an “f” without the cross piece so it is “souls”. Maybe that just doesn’t fit in todays PC world…

      1. Soles would be a running joke …

        (And I suppose it also means that the various Daft curriculums are fhit for purpose)

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