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 4 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 4 hours ago
Jyotish RobinJyotish Robin
515112
$endgroup$
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
3 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
1 hour 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
1 hour 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 4 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 4 hours ago
Jyotish RobinJyotish Robin
515112
asked 4 hours ago
Jyotish RobinJyotish Robin
515112
asked 4 hours ago
Jyotish RobinJyotish Robin
515112
asked 4 hours ago
Jyotish RobinJyotish Robin
515112
asked 4 hours ago
Jyotish RobinJyotish Robin
515112
515112
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
3 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
1 hour 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
1 hour ago
add a comment |
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
3 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
1 hour 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
1 hour ago
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
3 hours ago
$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
3 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
1 hour 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
1 hour 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
1 hour 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
1 hour ago
add a comment |
2 Answers
2
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 1 hour ago
user45266user45266
32514
$endgroup$
$begingroup$
I do not consider it much of a spoiler (there must be better algorithms) but the worst case scenario can be easily reduced to 35 tests. rot13(Qb ovanel frnepurf sbe bar jbys. Rnpu frnepu vf qbar ba nyy gur furrc zvahf jbysf sbhaq va gur cerivbhf frnepurf. Lbh jvyy arrq 5 frnepurf. Rnpu frnepu jvyy gnxr ng zbfg 7 grfgf. Fb gur jbefg pnfr fpranevb vf 35 grfgf).
$endgroup$
– ppgdev
56 mins ago
$begingroup$
Wait a minute, how would that work? Could you be a little more specific?
$endgroup$
– user45266
54 mins ago
$begingroup$
@ppgdev Like starting with 100, you split the group into 5 groups? What do you mean?
$endgroup$
– user45266
53 mins ago
$begingroup$
I moved my comment into an upper bound answer, with a more detailed explanation.
$endgroup$
– ppgdev
36 mins ago
add a comment |
$begingroup$
There must be better algorithms, but just to put an upper bound on a solution, the worst case scenario can be reduced to
35 tests. Do binary searches for one wolf. Each search is done on all the sheep minus wolfs found in the previous searches. So the first binary search is done on a set of all 100 'sheep'. You will find a wolf in no more than 7 tests. The next binary search is done on remaining 99 sheep. And so on. You will need 5 binary searches. Each search will take at most 7 tests. So the worst case scenario is 35 tests.
answered 38 mins ago
ppgdevppgdev
41516
$endgroup$
$begingroup$
Unfortunately that will not actually work. The problem states you will only get the results of the tests after all the tests have been done (maybe it takes a few days for the results to come through!), so you can’t base your second test on the results of the first one.
$endgroup$
– Amorydai
7 mins ago
$begingroup$
@Amorydai, you are right. I missed what was stated in bold font :-). Removing my answer and comments.
$endgroup$
– ppgdev
24 secs ago
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
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 1 hour ago
user45266user45266
32514
$endgroup$
$begingroup$
I do not consider it much of a spoiler (there must be better algorithms) but the worst case scenario can be easily reduced to 35 tests. rot13(Qb ovanel frnepurf sbe bar jbys. Rnpu frnepu vf qbar ba nyy gur furrc zvahf jbysf sbhaq va gur cerivbhf frnepurf. Lbh jvyy arrq 5 frnepurf. Rnpu frnepu jvyy gnxr ng zbfg 7 grfgf. Fb gur jbefg pnfr fpranevb vf 35 grfgf).
$endgroup$
– ppgdev
56 mins ago
$begingroup$
Wait a minute, how would that work? Could you be a little more specific?
$endgroup$
– user45266
54 mins ago
$begingroup$
@ppgdev Like starting with 100, you split the group into 5 groups? What do you mean?
$endgroup$
– user45266
53 mins ago
$begingroup$
I moved my comment into an upper bound answer, with a more detailed explanation.
$endgroup$
– ppgdev
36 mins ago
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 1 hour ago
user45266user45266
32514
$endgroup$
$begingroup$
I do not consider it much of a spoiler (there must be better algorithms) but the worst case scenario can be easily reduced to 35 tests. rot13(Qb ovanel frnepurf sbe bar jbys. Rnpu frnepu vf qbar ba nyy gur furrc zvahf jbysf sbhaq va gur cerivbhf frnepurf. Lbh jvyy arrq 5 frnepurf. Rnpu frnepu jvyy gnxr ng zbfg 7 grfgf. Fb gur jbefg pnfr fpranevb vf 35 grfgf).
$endgroup$
– ppgdev
56 mins ago
$begingroup$
Wait a minute, how would that work? Could you be a little more specific?
$endgroup$
– user45266
54 mins ago
$begingroup$
@ppgdev Like starting with 100, you split the group into 5 groups? What do you mean?
$endgroup$
– user45266
53 mins ago
$begingroup$
I moved my comment into an upper bound answer, with a more detailed explanation.
$endgroup$
– ppgdev
36 mins ago
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 1 hour 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 1 hour ago
user45266user45266
32514
answered 1 hour ago
user45266user45266
32514
answered 1 hour ago
user45266user45266
32514
answered 1 hour ago
user45266user45266
32514
32514
$begingroup$
I do not consider it much of a spoiler (there must be better algorithms) but the worst case scenario can be easily reduced to 35 tests. rot13(Qb ovanel frnepurf sbe bar jbys. Rnpu frnepu vf qbar ba nyy gur furrc zvahf jbysf sbhaq va gur cerivbhf frnepurf. Lbh jvyy arrq 5 frnepurf. Rnpu frnepu jvyy gnxr ng zbfg 7 grfgf. Fb gur jbefg pnfr fpranevb vf 35 grfgf).
$endgroup$
– ppgdev
56 mins ago
$begingroup$
Wait a minute, how would that work? Could you be a little more specific?
$endgroup$
– user45266
54 mins ago
$begingroup$
@ppgdev Like starting with 100, you split the group into 5 groups? What do you mean?
$endgroup$
– user45266
53 mins ago
$begingroup$
I moved my comment into an upper bound answer, with a more detailed explanation.
$endgroup$
– ppgdev
36 mins ago
add a comment |
$begingroup$
I do not consider it much of a spoiler (there must be better algorithms) but the worst case scenario can be easily reduced to 35 tests. rot13(Qb ovanel frnepurf sbe bar jbys. Rnpu frnepu vf qbar ba nyy gur furrc zvahf jbysf sbhaq va gur cerivbhf frnepurf. Lbh jvyy arrq 5 frnepurf. Rnpu frnepu jvyy gnxr ng zbfg 7 grfgf. Fb gur jbefg pnfr fpranevb vf 35 grfgf).
$endgroup$
– ppgdev
56 mins ago
$begingroup$
Wait a minute, how would that work? Could you be a little more specific?
$endgroup$
– user45266
54 mins ago
$begingroup$
@ppgdev Like starting with 100, you split the group into 5 groups? What do you mean?
$endgroup$
– user45266
53 mins ago
$begingroup$
I moved my comment into an upper bound answer, with a more detailed explanation.
$endgroup$
– ppgdev
36 mins ago
$begingroup$
I do not consider it much of a spoiler (there must be better algorithms) but the worst case scenario can be easily reduced to 35 tests. rot13(Qb ovanel frnepurf sbe bar jbys. Rnpu frnepu vf qbar ba nyy gur furrc zvahf jbysf sbhaq va gur cerivbhf frnepurf. Lbh jvyy arrq 5 frnepurf. Rnpu frnepu jvyy gnxr ng zbfg 7 grfgf. Fb gur jbefg pnfr fpranevb vf 35 grfgf).
$endgroup$
– ppgdev
56 mins ago
$begingroup$
I do not consider it much of a spoiler (there must be better algorithms) but the worst case scenario can be easily reduced to 35 tests. rot13(Qb ovanel frnepurf sbe bar jbys. Rnpu frnepu vf qbar ba nyy gur furrc zvahf jbysf sbhaq va gur cerivbhf frnepurf. Lbh jvyy arrq 5 frnepurf. Rnpu frnepu jvyy gnxr ng zbfg 7 grfgf. Fb gur jbefg pnfr fpranevb vf 35 grfgf).
$endgroup$
– ppgdev
56 mins ago
$begingroup$
Wait a minute, how would that work? Could you be a little more specific?
$endgroup$
– user45266
54 mins ago
$begingroup$
Wait a minute, how would that work? Could you be a little more specific?
$endgroup$
– user45266
54 mins ago
$begingroup$
@ppgdev Like starting with 100, you split the group into 5 groups? What do you mean?
$endgroup$
– user45266
53 mins ago
$begingroup$
@ppgdev Like starting with 100, you split the group into 5 groups? What do you mean?
$endgroup$
– user45266
53 mins ago
$begingroup$
I moved my comment into an upper bound answer, with a more detailed explanation.
$endgroup$
– ppgdev
36 mins ago
$begingroup$
I moved my comment into an upper bound answer, with a more detailed explanation.
$endgroup$
– ppgdev
36 mins ago
add a comment |
$begingroup$
There must be better algorithms, but just to put an upper bound on a solution, the worst case scenario can be reduced to
35 tests. Do binary searches for one wolf. Each search is done on all the sheep minus wolfs found in the previous searches. So the first binary search is done on a set of all 100 'sheep'. You will find a wolf in no more than 7 tests. The next binary search is done on remaining 99 sheep. And so on. You will need 5 binary searches. Each search will take at most 7 tests. So the worst case scenario is 35 tests.
answered 38 mins ago
ppgdevppgdev
41516
$endgroup$
$begingroup$
Unfortunately that will not actually work. The problem states you will only get the results of the tests after all the tests have been done (maybe it takes a few days for the results to come through!), so you can’t base your second test on the results of the first one.
$endgroup$
– Amorydai
7 mins ago
$begingroup$
@Amorydai, you are right. I missed what was stated in bold font :-). Removing my answer and comments.
$endgroup$
– ppgdev
24 secs ago
add a comment |
$begingroup$
There must be better algorithms, but just to put an upper bound on a solution, the worst case scenario can be reduced to
35 tests. Do binary searches for one wolf. Each search is done on all the sheep minus wolfs found in the previous searches. So the first binary search is done on a set of all 100 'sheep'. You will find a wolf in no more than 7 tests. The next binary search is done on remaining 99 sheep. And so on. You will need 5 binary searches. Each search will take at most 7 tests. So the worst case scenario is 35 tests.
answered 38 mins ago
ppgdevppgdev
41516
$endgroup$
$begingroup$
Unfortunately that will not actually work. The problem states you will only get the results of the tests after all the tests have been done (maybe it takes a few days for the results to come through!), so you can’t base your second test on the results of the first one.
$endgroup$
– Amorydai
7 mins ago
$begingroup$
@Amorydai, you are right. I missed what was stated in bold font :-). Removing my answer and comments.
$endgroup$
– ppgdev
24 secs ago
add a comment |
$begingroup$
There must be better algorithms, but just to put an upper bound on a solution, the worst case scenario can be reduced to
35 tests. Do binary searches for one wolf. Each search is done on all the sheep minus wolfs found in the previous searches. So the first binary search is done on a set of all 100 'sheep'. You will find a wolf in no more than 7 tests. The next binary search is done on remaining 99 sheep. And so on. You will need 5 binary searches. Each search will take at most 7 tests. So the worst case scenario is 35 tests.
answered 38 mins ago
ppgdevppgdev
41516
$endgroup$
There must be better algorithms, but just to put an upper bound on a solution, the worst case scenario can be reduced to
35 tests. Do binary searches for one wolf. Each search is done on all the sheep minus wolfs found in the previous searches. So the first binary search is done on a set of all 100 'sheep'. You will find a wolf in no more than 7 tests. The next binary search is done on remaining 99 sheep. And so on. You will need 5 binary searches. Each search will take at most 7 tests. So the worst case scenario is 35 tests.
answered 38 mins ago
ppgdevppgdev
41516
answered 38 mins ago
ppgdevppgdev
41516
answered 38 mins ago
ppgdevppgdev
41516
answered 38 mins ago
ppgdevppgdev
41516
41516
$begingroup$
Unfortunately that will not actually work. The problem states you will only get the results of the tests after all the tests have been done (maybe it takes a few days for the results to come through!), so you can’t base your second test on the results of the first one.
$endgroup$
– Amorydai
7 mins ago
$begingroup$
@Amorydai, you are right. I missed what was stated in bold font :-). Removing my answer and comments.
$endgroup$
– ppgdev
24 secs ago
add a comment |
$begingroup$
Unfortunately that will not actually work. The problem states you will only get the results of the tests after all the tests have been done (maybe it takes a few days for the results to come through!), so you can’t base your second test on the results of the first one.
$endgroup$
– Amorydai
7 mins ago
$begingroup$
@Amorydai, you are right. I missed what was stated in bold font :-). Removing my answer and comments.
$endgroup$
– ppgdev
24 secs ago
$begingroup$
Unfortunately that will not actually work. The problem states you will only get the results of the tests after all the tests have been done (maybe it takes a few days for the results to come through!), so you can’t base your second test on the results of the first one.
$endgroup$
– Amorydai
7 mins ago
$begingroup$
Unfortunately that will not actually work. The problem states you will only get the results of the tests after all the tests have been done (maybe it takes a few days for the results to come through!), so you can’t base your second test on the results of the first one.
$endgroup$
– Amorydai
7 mins ago
$begingroup$
@Amorydai, you are right. I missed what was stated in bold font :-). Removing my answer and comments.
$endgroup$
– ppgdev
24 secs ago
$begingroup$
@Amorydai, you are right. I missed what was stated in bold font :-). Removing my answer and comments.
$endgroup$
– ppgdev
24 secs ago
add a comment |
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$begingroup$
This is close to a covering design (Lotto wheel) problem.
$endgroup$
– Arnaud Mortier
3 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
1 hour 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
1 hour ago