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Archimedes' Cattle (Posted on 2004-02-17) Difficulty: 4 of 5
The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown.

Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown.

Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle.

What was the composition of the herd?

See The Solution Submitted by DJ    
Rating: 3.5000 (8 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution full solution | Comment 1 of 6
a = 0.205728733062
b = 0.148058144818
c = 0.146024886899
d = 0.082346945713
e = 0.143014314093
f = 0.097109250770
g = 0.069773448145
h = 0.107944276500

Where:
a = the number of white bulls
b = the number of black bulls
c = the number of spotted bulls
d = the number of brown bulls
e = the number of white cows
f = the number of black cows
g = the number of spotted cows
h = the number of brown cows

and the numbers are the fraction of the herd that they make up (adding to 1)
_______________________________________________

This problem is actually much easier than one might think at first glance.

It's a straight forward solution of 8 linear equations.

(It is interesting to note that DJ didn't ask how many of each type there is in the herd; he asked what the composition is. This is because there isn't enough information to determine the size of the herd. Although, a related question might be... what is the minimum size of the herd?)

If is first key to realize that the second paragraph (about the bulls) refers to only the bulls... whereas, the third paragraph (about the cows) refers to the cattle referring to the bulls and cows.

If we assign the eight letters as described above. Then we need 8 equations to solve them all. The seven that DJ gives us are:

about the bulls
A = (1/2 + 1/3)B + D
B = (1/4 + 1/5)C + D
C = (1/6 + 1/7)A + D

about the cows
E = (1/3 + 1/4)(B + F)
F = (1/4 + 1/5)(C + G)
G = (1/5 + 1/6)(D + H)
H = (1/6 + 1/7)(A + E)

implied by question: must add up to one
A + B + C + D + E + F + G + H = 1

At this point, you have 8 equations and 8 unknowns... have at it!

(Okay.. you may find it a bit tedious... so I suggest you do what I did...)

Now solve by using matrices:

   6   -5    0   -6    0    0    0    0    0
0 20 -9 -20 0 0 0 0 0
-13 0 42 -42 0 0 0 0 0
0 -7 0 0 12 -7 0 0 0
0 0 -9 0 0 20 -9 0 0
0 0 0 -11 0 0 30 -11 0
-13 0 0 0 -13 0 0 42 0
1 1 1 1 1 1 1 1 1

This is an equivalent matrix for the system of equations above, and if you solve it, you'll find the answers above. I used, everyone's favorite tool... EXCEL!
  Posted by SilverKnight on 2004-02-17 21:28:47
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