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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
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.

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
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/
```
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