Two Shepherd Brothers
1. A story.
A long time ago in
the
2. Questions.
(1) Could they divide their herd fairly? If it is possible, show how.
(2) Do you think that their idea of a fair
division (the same number of animals, and the same total value) was right? Are there any other ways to make a fair
division? (Example: Each gets 14 goats, 7 sheep, and 2 cows. And then they flip a coin to decide who gets
the remaining cow and who gets the remaining sheep.)
(3) Where is
(4) What was a thaler?
(See below)
(5) Could they resolve their differences without
dividing their herd? How?
3. An answer to the first question.
The choice of
variables can often decide between success and failure, because the complexity
of the computation heavily depends on it. We will show two ways to solve the problem.
First approach.
There are 28 goats, 15
sheep, and 5 cows, so there are 48 animals. Each brother will get half, or 24. A goat costs 1/3 thaler, a sheep 1/2 thaler, and a cow 2 1/2 thalers. The total value of the herd is 28/3 +15/2 + 25/2
= 29 1/3 thalers, so each brother should get animals
worth 14 2/3 thalers.
Let g, s, and c =
the number of goats, sheep, and cows one brother gets. Then
g + s + c = 24 (1)
1/3g + 1/2 s + 5/2 c = 14 2/3 (2)
Multiplying
equation (2) through by 2, we have
2/3g + s + 5 c = 29
1/3 (3)
Subtracting (1)
from (3):
2/3g + s + 5 c = 29
1/3 (3)
g + s + c = 24 (1)
-1/3g + 4c = 5 1/3
So
-g + 12c = 16
g = 12c -16.
The brother will
get 0, 1, 2, 3, 4, or 5 cows. He will
also get a whole number of goats (no fractions!). Let's see what happens.
| c | g = 12c - 16 |
| 0 | -16 not possible |
| 1 | -4 not possible |
| 2 | 8 possible |
| 3 | 20 possible |
| 4 | 32 not possible |
| 5 | 44 not possible |
Brother 1 may
receive 2 cows, 8 goats, and 24-10 = 14 sheep.
His animals are
worth 2*2.5 + 8*1/3 + 14*1/2 = 14 2/3 thalers.
Brother 2 may
receive 3 cows, 20 goats, and 1 sheep.
His animals are
worth 3*2.5 + 20*1/3 + 1*1/2 = 14 2/3 thalers.
Second approach.
Variables.
We choose G, S, and
C, to be the DIFFERENCES between the numbers of goats, sheep, and cows that
Hans and Karl will get. These numbers
can be positive, negative or zero, depending on which brother has more animals
of that kind. But G, S, and C must be
integers (they cannot be fractions), and their absolute values must be within
the appropriate range, |G| ≤ 28, |S| ≤ 15,
and |C| ≤ 5.
Equations.
If they get the
same number of animals, then
G + S + C = 0. (Why?)
If their shares
have the same value, then
1/3*G + 1/2*S + 2 1/2*C = 0. (Why?)
Multiplying the
second equation by 6, and the first by -2, and adding them, we have
2*G + 3*S + 15*C = 0
-2*G + -2*S + -2*C = 0
S + 13*C = 0
Thus,
S = -13*C.
Substituting this
value for S in the first equation, we get G + -13*C + C = 0.
Thus,
G = 12*C.
So we see that the
brother who gets more cows has also more goats, and less sheep.
Using
the constraints.
Remember: G, C, and S are integers.
- Because the
number of cows is odd, C ≠ 0.
- Because |S| =
|-13*C| ≤ 15, C = 1 or C = -1.
We may look from
the point of view of the brother (either Hans or Karl) who gets more cows. Therefore we may take
C = 1,
and so we have
S = -13,
and
G = 12.
More
variables and equations.
Variables.
We will use, and
reuse, X and
Y for the number of animals of a given
kind owned by the two brothers. X will
always be for the brother who has more cows.
Equations
for the number of cows.
X + Y = 5 (total number of cows)
X - Y = 1 (difference
in the number of cows)
Solution: X = 3, Y = 2.
Equations
for the number of sheep.
X + Y = 15 (total
number of sheep)
X - Y = -13 (difference in the number of sheep)
Solution: X = 1, Y = 14.
Equations
for the number of goats.
X + Y = 28 (total number of goats)
X - Y = 12 (difference in the number of goats)
Solution: X = 20, Y = 8.
Checking
the answer.
One brother gets
3 cows, 1 sheep,
and 20 goats, a total of 24 animals.
They are worth
3*2 1/2 + 1/2 + 20*1/3 thalers = 14 2/3 thalers.
The other one gets
2 cows, 14 sheep, and
8 goats, a total of 24 animals.
They are worth
2*2 1/2 + 14*1/2 + 8*1/3 thalers = 14 2/3 thalers.
(3)
Map of
(4)
Thalers are numerous silver coins that served as a unit of currency in certain Germanic countries between the 15th and 19th centuries.