Wolves and sheep
$begingroup$
All the sheep were living peacefully in the Land of Shewo. But suddenly they were struck by a danger. A few wolves dressed up as sheep entered the territory of Shewo and started killing the sheep one by one.
To find a solution to this misery, the king of Shewo called upon all of his sheep to the palace hall. He made the following announcement:
From my secret sources, I came to know that the total number of 'sheep' (including the wolves) now present in my kingdom is 100. Among which 5 are wolves. Our doctors have come up with a very expensive blood test which could be used to differentiate the wolves and sheep.
Each test costs 1000$ and we don't have enough funds to test all the 100 'sheep'.
I discussed with our ministers and came to know that the tests can be done on pooled bloodsamples. i.e., I can collect bloods from any number of 'sheep' and mix them. Then if I test the mixture, I will get a positive result if the mixture contain blood from any wolf. I will get a negative result if all the samples are from actual sheep.
One caveat is that the test results are available to you after all the tests are done!
Now , I am looking for ideas where I can find ALL the wolves in minimum number of pooled tests. I request the brilliant young minds of this land to come up with a testing strategy.
Can you help the king by devising a strategy?
strategy combinatorics algorithm
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
$endgroup$
add a comment |
$begingroup$
All the sheep were living peacefully in the Land of Shewo. But suddenly they were struck by a danger. A few wolves dressed up as sheep entered the territory of Shewo and started killing the sheep one by one.
To find a solution to this misery, the king of Shewo called upon all of his sheep to the palace hall. He made the following announcement:
From my secret sources, I came to know that the total number of 'sheep' (including the wolves) now present in my kingdom is 100. Among which 5 are wolves. Our doctors have come up with a very expensive blood test which could be used to differentiate the wolves and sheep.
Each test costs 1000$ and we don't have enough funds to test all the 100 'sheep'.
I discussed with our ministers and came to know that the tests can be done on pooled bloodsamples. i.e., I can collect bloods from any number of 'sheep' and mix them. Then if I test the mixture, I will get a positive result if the mixture contain blood from any wolf. I will get a negative result if all the samples are from actual sheep.
One caveat is that the test results are available to you after all the tests are done!
Now , I am looking for ideas where I can find ALL the wolves in minimum number of pooled tests. I request the brilliant young minds of this land to come up with a testing strategy.
Can you help the king by devising a strategy?
strategy combinatorics algorithm
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
$endgroup$
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
4 hours ago
1
$begingroup$
First of all, does the government have enough funds to test 99 of the sheep? Because that would work, at a cost of $99,000. Congrats, you just saved 1,000 bucks.
$endgroup$
– user45266
2 hours ago
1
$begingroup$
Alternatively, you know the location of all 5 wolves. Take initiative and slaughter all 100. Now you have no more wolves, and food for a good while to come.
$endgroup$
– user45266
2 hours ago
add a comment |
$begingroup$
All the sheep were living peacefully in the Land of Shewo. But suddenly they were struck by a danger. A few wolves dressed up as sheep entered the territory of Shewo and started killing the sheep one by one.
To find a solution to this misery, the king of Shewo called upon all of his sheep to the palace hall. He made the following announcement:
From my secret sources, I came to know that the total number of 'sheep' (including the wolves) now present in my kingdom is 100. Among which 5 are wolves. Our doctors have come up with a very expensive blood test which could be used to differentiate the wolves and sheep.
Each test costs 1000$ and we don't have enough funds to test all the 100 'sheep'.
I discussed with our ministers and came to know that the tests can be done on pooled bloodsamples. i.e., I can collect bloods from any number of 'sheep' and mix them. Then if I test the mixture, I will get a positive result if the mixture contain blood from any wolf. I will get a negative result if all the samples are from actual sheep.
One caveat is that the test results are available to you after all the tests are done!
Now , I am looking for ideas where I can find ALL the wolves in minimum number of pooled tests. I request the brilliant young minds of this land to come up with a testing strategy.
Can you help the king by devising a strategy?
strategy combinatorics algorithm
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
$endgroup$
All the sheep were living peacefully in the Land of Shewo. But suddenly they were struck by a danger. A few wolves dressed up as sheep entered the territory of Shewo and started killing the sheep one by one.
To find a solution to this misery, the king of Shewo called upon all of his sheep to the palace hall. He made the following announcement:
From my secret sources, I came to know that the total number of 'sheep' (including the wolves) now present in my kingdom is 100. Among which 5 are wolves. Our doctors have come up with a very expensive blood test which could be used to differentiate the wolves and sheep.
Each test costs 1000$ and we don't have enough funds to test all the 100 'sheep'.
I discussed with our ministers and came to know that the tests can be done on pooled bloodsamples. i.e., I can collect bloods from any number of 'sheep' and mix them. Then if I test the mixture, I will get a positive result if the mixture contain blood from any wolf. I will get a negative result if all the samples are from actual sheep.
One caveat is that the test results are available to you after all the tests are done!
Now , I am looking for ideas where I can find ALL the wolves in minimum number of pooled tests. I request the brilliant young minds of this land to come up with a testing strategy.
Can you help the king by devising a strategy?
strategy combinatorics algorithm
strategy combinatorics algorithm
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
asked 5 hours ago
Jyotish RobinJyotish Robin
515112
515112
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
4 hours ago
1
$begingroup$
First of all, does the government have enough funds to test 99 of the sheep? Because that would work, at a cost of $99,000. Congrats, you just saved 1,000 bucks.
$endgroup$
– user45266
2 hours ago
1
$begingroup$
Alternatively, you know the location of all 5 wolves. Take initiative and slaughter all 100. Now you have no more wolves, and food for a good while to come.
$endgroup$
– user45266
2 hours ago
add a comment |
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
4 hours ago
1
$begingroup$
First of all, does the government have enough funds to test 99 of the sheep? Because that would work, at a cost of $99,000. Congrats, you just saved 1,000 bucks.
$endgroup$
– user45266
2 hours ago
1
$begingroup$
Alternatively, you know the location of all 5 wolves. Take initiative and slaughter all 100. Now you have no more wolves, and food for a good while to come.
$endgroup$
– user45266
2 hours ago
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
4 hours ago
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
4 hours ago
1
1
$begingroup$
First of all, does the government have enough funds to test 99 of the sheep? Because that would work, at a cost of $99,000. Congrats, you just saved 1,000 bucks.
$endgroup$
– user45266
2 hours ago
$begingroup$
First of all, does the government have enough funds to test 99 of the sheep? Because that would work, at a cost of $99,000. Congrats, you just saved 1,000 bucks.
$endgroup$
– user45266
2 hours ago
1
1
$begingroup$
Alternatively, you know the location of all 5 wolves. Take initiative and slaughter all 100. Now you have no more wolves, and food for a good while to come.
$endgroup$
– user45266
2 hours ago
$begingroup$
Alternatively, you know the location of all 5 wolves. Take initiative and slaughter all 100. Now you have no more wolves, and food for a good while to come.
$endgroup$
– user45266
2 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Here's my shot at it:
Pool 50 of the sheep. Of those 50, if the result comes back positive (wolf detected), split in half again, testing 25 pooled together. If this comes back positive, either 12 or 13 ofthe positive group. You see where this is going. Any negative results rule out all sheep in that group. Worst case scenario, it takes 54 tests (see image for explanation). Best case scenario, you get 14 tests. Price range: 14,000 - 54,000 dollars.
Not sure yet of the expected average, but I know this is better than 99 tests on average.
Diagram:
If this isn't the optimal solution, then my best bet on how to improve it would be to:
split into 1/5 the size of each group.
answered 2 hours ago
user45266user45266
32514
$endgroup$
add a comment |
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$begingroup$
Here's my shot at it:
Pool 50 of the sheep. Of those 50, if the result comes back positive (wolf detected), split in half again, testing 25 pooled together. If this comes back positive, either 12 or 13 ofthe positive group. You see where this is going. Any negative results rule out all sheep in that group. Worst case scenario, it takes 54 tests (see image for explanation). Best case scenario, you get 14 tests. Price range: 14,000 - 54,000 dollars.
Not sure yet of the expected average, but I know this is better than 99 tests on average.
Diagram:
If this isn't the optimal solution, then my best bet on how to improve it would be to:
split into 1/5 the size of each group.
answered 2 hours ago
user45266user45266
32514
$endgroup$
add a comment |
$begingroup$
Here's my shot at it:
Pool 50 of the sheep. Of those 50, if the result comes back positive (wolf detected), split in half again, testing 25 pooled together. If this comes back positive, either 12 or 13 ofthe positive group. You see where this is going. Any negative results rule out all sheep in that group. Worst case scenario, it takes 54 tests (see image for explanation). Best case scenario, you get 14 tests. Price range: 14,000 - 54,000 dollars.
Not sure yet of the expected average, but I know this is better than 99 tests on average.
Diagram:
If this isn't the optimal solution, then my best bet on how to improve it would be to:
split into 1/5 the size of each group.
answered 2 hours ago
user45266user45266
32514
$endgroup$
add a comment |
$begingroup$
Here's my shot at it:
Pool 50 of the sheep. Of those 50, if the result comes back positive (wolf detected), split in half again, testing 25 pooled together. If this comes back positive, either 12 or 13 ofthe positive group. You see where this is going. Any negative results rule out all sheep in that group. Worst case scenario, it takes 54 tests (see image for explanation). Best case scenario, you get 14 tests. Price range: 14,000 - 54,000 dollars.
Not sure yet of the expected average, but I know this is better than 99 tests on average.
Diagram:
If this isn't the optimal solution, then my best bet on how to improve it would be to:
split into 1/5 the size of each group.
answered 2 hours ago
user45266user45266
32514
$endgroup$
Here's my shot at it:
Pool 50 of the sheep. Of those 50, if the result comes back positive (wolf detected), split in half again, testing 25 pooled together. If this comes back positive, either 12 or 13 ofthe positive group. You see where this is going. Any negative results rule out all sheep in that group. Worst case scenario, it takes 54 tests (see image for explanation). Best case scenario, you get 14 tests. Price range: 14,000 - 54,000 dollars.
Not sure yet of the expected average, but I know this is better than 99 tests on average.
Diagram:
If this isn't the optimal solution, then my best bet on how to improve it would be to:
split into 1/5 the size of each group.
answered 2 hours ago
user45266user45266
32514
answered 2 hours ago
user45266user45266
32514
answered 2 hours ago
user45266user45266
32514
answered 2 hours ago
user45266user45266
32514
32514
add a comment |
add a comment |
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$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
4 hours ago
1
$begingroup$
First of all, does the government have enough funds to test 99 of the sheep? Because that would work, at a cost of $99,000. Congrats, you just saved 1,000 bucks.
$endgroup$
– user45266
2 hours ago
1
$begingroup$
Alternatively, you know the location of all 5 wolves. Take initiative and slaughter all 100. Now you have no more wolves, and food for a good while to come.
$endgroup$
– user45266
2 hours ago