The Pharaoh's Will

Date: 05/16/2000 at 09:26:53
From: George Balalis
Subject: The Pharaoh's Will

As he lay dying, the first Pharaoh of Ufractia proclaimed: "I bequeath 
to my oldest child one third of my estate; to the next oldest child 
one quarter of my estate; to the next oldest child one fifth of my 
estate; and to each succeeding child, except the youngest, the next 
unit fraction of my estate; and to the youngest the remainder."

When the pharaoh died and his estate was divided, the youngest child 
received the smallest share, which was worth 27,000 gold ingots.

A) What was the value of the oldest child's share?

The pharaoh's successor was so impressed with this method that he 
proclaimed: "In future, a Pharaoh's estate will be divided according 
to these rules:

   1) Each surviving child, except the youngest, will be bequeathed a 
      unit fraction of the estate.

   2) The oldest child will be bequeathed the biggest fraction.

   3) If a child is bequeathed 1/n of the estate, the next oldest 
      child, except the youngest, will be bequeathed 1/(n+1) of the 
      estate.

   4) The remainder is to be at most the next unit fraction and is to 
      be bequeathed to the youngest child."

B) The second pharaoh had eight children living when he died. What 
fraction of his estate was bequeathed to the oldest child?

C) When the third Pharaoh died, he had more surviving children than 
the first Pharaoh did, but less than eleven surviving children. It was 
found that the rules could not be followed precisely. How many 
children might have survived the third Pharaoh?

I would like, if at all possible, to see your answer as soon as 
possible, as it would help me greatly in my efforts.

Thank you.

Date: 05/17/2000 at 11:48:12
From: Doctor TWE
Subject: Re: The Pharaoh's Will

Hi George - thanks for writing to Dr. Math.

This is an interesting problem. Let's take a look at what we know. The 
first Pharaoh of Ufractia divided his estate as follows:

     Child   Fraction
     -----   --------
      1st      1/3
      2nd      1/4
      3rd      1/5
       :        :

Let's work out each child's share, beginning with the oldest. The 
oldest child received 1/3 of the estate, leaving:

     1 - 1/3 = 3/3 - 1/3 = 2/3

So 2/3 of the estate is left for the remaining children. Our 
information said, "to the youngest the remainder" and "the youngest 
child received the smallest share," so there must be at least two more 
children. If there were only one more child, (s)he would receive 2/3 
of the estate, which is more than the oldest child received.

     Child   Fraction   Remaining
     -----   --------   ---------
      1st      1/3         2/3
       :        :           :

The next oldest child received 1/4 of the estate, leaving:

     2/3 - 1/4 = 8/12 - 3/12 = 5/12

So 5/12 of the estate is left for the remaining children. Since the 
5/12 remaining is more than the 1/4 (= 3/12) that the second-oldest 
child received, there must be at least two more children.

     Child   Fraction   Remaining
     -----   --------   ---------
      1st      1/3         2/3
      2nd      1/4         5/12
       :        :           :

The third oldest child received 1/5 of the estate, leaving:

     5/12 - 1/5 = 25/60 - 12/60 = 13/60

So 13/60 of the estate is left for the remaining children. Since the 
13/60 remaining is more than the 1/5 (= 12/60) that the third-oldest 
child received, there must be at least two more children.

     Child   Fraction   Remaining
     -----   --------   ---------
      1st      1/3         2/3
      2nd      1/4         5/12
      3rd      1/5        13/60
       :        :           :

The fourth oldest child received 1/6 of the estate, leaving:

     13/60 - 1/6 = 13/60 - 10/60 = 3/60 (= 1/20)

So 1/20 of the estate is left for the remaining children. That is less 
than the 1/6 that the fourth-oldest child received, so there could 
possibly be only one more child.

     Child   Fraction   Remaining
     -----   --------   ---------
      1st      1/3         2/3
      2nd      1/4         5/12
      3rd      1/5        13/60
      4th      1/6         1/20
       :        :           :

Could there be two more children? For the Pharaoh's pattern to 
continue, the 5th child should receive 1/7 of the estate, but only 
1/20 (< 1/7) remains, so we can't continue and we can conclude that 
the Pharaoh had five children. The youngest would receive the 
remaining share, which is 1/20 of the estate. From this, and the fact 
that his share was worth 27,000 gold ingots, can you figure out what 
the original estate was worth?

Follow a similar method to find the answers to parts B and C. You'll 
have to try other fractions for the oldest child's share, because we 
saw that 1/3 only worked for five kids. See if you can find a way to 
figure out the oldest child's share (based on the total number of 
children) other than trial-and-error.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/18/2000 at 04:19:54
From: GeorgeB
Subject: Re: The Pharaoh's Will

Thanks for that, it helped a lot. The next thing is, I tried to come 
up with a way other than trial-and-error, but there isn't really 
another way. Well, there must be... but I can't find it. That also 
applies to part (C). (C) is a bit different, and I can see a relation 
but nothing of use.

Please write back with your opinion, i.e. your way of answering it. 

Thank you.

Date: 05/18/2000 at 13:09:42
From: Doctor TWE
Subject: Re: The Pharaoh's Will

Hi again George - thanks for writing back.

What I did (I guess this is really a variant on trial-and-error) was 
to add unit fractions 1/n + 1/(n+1) + 1/(n+2) + ... until the sum was 
greater than 1. I then counted the number of fractions added - this 
would be the number of children for which it works with 1/n as the 
starting fraction.

Next, I subtracted the last unit fraction (so that the sum was less 
than 1 again) and subtracted that result from 1 to see what the 
youngest child's share was. (I must confess, I did this on a 
spreadsheet to save time.) For example, with n = 4 I got:

     1/4       + 1/5  = 9/20
     9/20      + 1/6  = 37/60
     37/60     + 1/7  = 319/420
     319/420   + 1/8  = 743/840
     743/840   + 1/9  = 2509/2520
     2509/2520 + 1/10 = 2761/2520 > 1

        1 - 2509/2520 = 11/2520

That's 7 unit fractions, so n = 4 works for seven children, and the 
youngest child gets 11/2520 of the estate (about 1/229th.)

With n = 3 we got five children, with n = 4 we get seven children. 
That means that the system won't work with six children. If we start 
with the oldest getting 1/3, there's not enough to give the fifth 
child the 1/7 as mandated by the formula. If we start with the oldest 
getting 1/4, the youngest child would get 97/840 - more than the 1/9 
that is the next unit fraction.

If you construct a table comparing n and C (the number of children 
that n works for), you'll see it "skips" certain values. The value 
that is skipped between your answer for part B and 11 is the answer to 
part C.

I hope this helps. If you have any more questions, write back again.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/