At one day a miller, the miller o'drone, was ordered by the bakers guild to supply 80 sacks of flour, a hundredweight each, to the bakers in the city. The miller orderd his hand to load the cart with the sacks of flour he had set ready and then to harness the horse and haul the load to the city. As the miller's mother lived in the city and had a coop with chickens, the miller also had put some grinding residue in a sack for her to feed the chickens.

The grinding residue consisted of sand, small chips of the grinding stones and some coarse ground flour or even whole or broken grain. Due to the fact it contained sand and stone chips, the sack was a little heavier then the sacks with flour.

But however much the miller tried he could not tell which one sack contained the grinding residue, but he could not risk supplying a bakery with it. So he gave his hand a 1 shilling coin and told him to have the sacks checked for weight at the weigh-house. Doing so would even guarantee his proper trade and professional practices, a nice and welcome boon. Every time the balance was lifted it would cost a thropence and the miller hoped the he had given his hand enough money to do the job, although he had some doubt.
Anyway, he joyfully told his hand that if there was some money left that he was allowed to have a pint of stout beer at the tavern as he had earned it. A pint would cost him 2 pence.

Question: If there was any money left, howmany pints could the hand have had, given that a shilling is 20 pence and a thropence stands for 3 pence?

Please answer per PM, so as not to spoil the joy for others.

Have a nice X-mas and a properous new year!

OldFart

changed 'flower' into 'flour' 4 times...
I stand corrected

Last edited by OldFart on 01-Jan-2016 at 08:46 AM.
Last edited by OldFart on 29-Dec-2015 at 03:20 PM.

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@OldFart

No idea, but i find it strange that a miller sells sacks of flower to bakeries...

Nice hobby...
Did he pick them by himself?
Do the bakeries need them to decorate their shops?

Or should that have been flour?

@OldFart

Tricky one as a shilling is 12 pence not 20 and there is no such thing as a thropence, there is a threepence that is pronounce thruppence though...

It was 20 shillings (bob) to a pound (240 pence) and 4 threepenny bits to a bob.

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We also need to know what basic type of scales is used: Is it an equal-arm balance capable of comparing two loads for weight equality (or one or the other being heavier), or is it a steelyard or Bismar type of thing where the weight of one load can be determined against a known fixed counterweight?

Best regards,

Niels

@Severin

Quote:

Tricky one as a shilling is 12 pence not 20 and there is no such thing as a thropence, there is a threepence that is pronounce thruppence though...

Ah, I see where you come from: 't ís England! But as you might have read in the first line, the story was about the miller o'drone, whose name and fame were eternalised by a tune called 'The Miller O'Drone', which is a strathspey for the Great Highland Bagpipe. So the story's setting is not in England but in Scotland and you are probably well aware of the fact that the Scotts maintain quite some vastly different cultural aspects. But then again the price for a pint of beer would probably have been expressed in 'bawbee's'...


OldFart

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@nbache

Quote:

We also need to know what basic type of scales is used: Is it an equal-arm balance capable of comparing two loads for weight equality (or one or the other being heavier), or is it a steelyard or Bismar type of thing where the weight of one load can be determined against a known fixed counterweight?

Nitpicking, are we? The store tells about a balance, which by tradition is an equal arm implement. The story's setting tells about grinding stones and a horsedrawn cart and even a miller's hand: all ingredients of a tale of yore, somewhere in medieval times, right after 'Danelaw'. Things were simple back then...


OldFart

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@OldFart

Thanks - it wasn't clear to me, possibly a language thing.

But in that case, cheers - and Happy New Year!

Best regards,

Niels

@All
Despite being 'only' a miller's hand, the hand actually was very smart and probably outwitted the miller. Here's how he did it.

Two facts are important:
a - 1 (and only 1) of the sacks is HEAVIER
b - the number of sacks is 80 + 1 = 81

When at the weigh-house the hand instructs the weigh-master to split the whole lot of 81 sacks into 3 lots of 27 each, of which 1 lot is placed on one of the balance's scale and 1 other lot at the other scale, after which the balance is hoisted.
Now 2 outcomes are possible:
1 -- the scales do not balance, thereby revealing the lot with the heavier sack
2 -- the scales DO balance, in which case the heavier sack is NOT on either scale and therefore has to be in he lot which is NOT on the scales

The lot with the heavier sack is again split into 3 lots of 9 sacks each and the above procedure is repeated, simmilarly identifying the lot of 9 sacks containing the heavier one.

The identified lot is again split into 3 lots, now of 3 sacks each, and the procedure is repated again, identifying the lot of 3 sacks containing the heavier one.

For the 4th and final time the lot of 3 sacks is split into 3 and the procedure is repeated again and this finally identifies the heavier sack.

So in just 4 weighings the heavier sack is identified.
4 Weighings at a thropence each, costed the hand 12 pence.
The shilling the miller had given him was 20 pence, so 8 pence remained for which he could have had 4 pints at the tavern!

Note well, that the heavier sack could have been identified even without ever having been weighed.

I wish you all a well-balanced, resourcefull and fortuneous 2016!

OldFart

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@OldFart

Damn, I knew I should have learnt my 27 times table...

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@Severin

Quote:

Damn, I knew I should have learnt my 27 times table...

Better still, you should have paid attention when 'powers' were taught, especially with base 3, as obviously applied here, for the saying goes: 'three is a crowd', so actually you missed out on the power of the crowd. Tsk, tsk, tsk...

Have a nice day!

OldFart

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@OldFart

Heh - that strategy will give you "the sissy's solution" - you're sure to spend 4 weighings to find the sack.

I was thinking of a solution where you could have a chance of finding it in the first weighing, and a worst case of spending six, still leaving money for at least one pint.

1. 40 on each, one left over. If balance, you've been lucky, the odd sack is the one you left off.
2. 20 on each (out of the 40 which were on the heaviest side in 1.).
3. 10 on each.
4. 5 on each.
5. 2 on each, leave the fifth one off. Again you might be lucky and see balance, then you have enough money left over for two pints (and a penny to keep).
6. The two from the heaviest side in 5. go on each side, and the heavier one will have been found.

Best regards,

Niels

@OldFart

And here's the source code...

/*
** The Miller's lemma
**
***********************/

#include 

int16 Identify_HeavierSack(uint16, uint16 *);

int main(int argc, char * argv[])
{
  int RetVal = RETURN_OK;

  uint16 Sacks[] = { 100, 100, 100, 100, 100, 100, 100, 100, 100
                   , 100, 100, 100, 100, 100, 100, 100, 100, 100
                   , 100, 100, 100, 100, 100, 100, 100, 100, 100
                   , 100, 100, 100, 100, 100, 100, 100, 100, 100
                   , 100, 100, 100, 100, 100, 100, 100, 100, 100
                   , 100, 100, 100, 100, 100, 100, 100, 100, 100
                   , 100, 100, 100, 100, 100, 100, 100, 100, 100
                   , 100, 100, 100, 101, 100, 100, 100, 100, 100
                   , 100, 100, 100, 100, 100, 100, 100, 100, 100
                   };

  IDOS->Printf("INFO : The heavier sack is #%ld\n", Identify_HeavierSack(81, Sacks));

  return RetVal;
}

int16 Identify_HeavierSack(uint16 LotSize, uint16 SackWeight[])
{
  int16 SackIndex = -1;

  uint16 LotOrdinal = 0,
         LotWeight[3],
         Lot,
         Sack;

  while (
         ( LotSize       > 1) &&
         ((LotSize % 3) == 0)
        )
   {
    LotSize /= 3;

    for (Lot = 0; Lot < 3; Lot++)
     {
      LotWeight[Lot] = 0;

      for (Sack = 0; Sack < LotSize; Sack++)
       {
        LotWeight[Lot] += SackWeight[LotOrdinal + (LotSize * Lot) + Sack];
       }
     }

    if (LotWeight[0] == LotWeight[1])
     {
      LotOrdinal += LotSize + LotSize;
     }
    else
     {
      if (LotWeight[0] < LotWeight[1])
       {
        LotOrdinal += LotSize;
       }
     }
   }

  if (LotSize == 1)
   {
    SackIndex = LotOrdinal + 1;
   }

  return SackIndex;
}

Last edited by OldFart on 04-Jan-2016 at 12:07 PM.

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