A lateral-thinking puzzle
I came up with a nice little lateral-thinking exercise.
Note that puzzles like these are inherently unfair, because you have to think outside the box, without breaking out of the (ill-defined and completely unrealistic) world model. But usually, once you see the solution, you'll know that it's the correct one.
Beware of spoilers in the comments.
Here we go:
You are a medical doctor working under rather chaotic conditions. Somebody turns up with three unconscious patients and their papers, and promptly disappears again. The patients have lost a lot of blood, and you realise that they're all going to die soon unless they get blood transfusions. From the papers, you can see that one patient has blood type A, another has blood type B, and the last one has blood type O. However, the papers are all mixed up, and you cannot tell which patient has which blood type.
If you do nothing, all three are going to die. Thus, you find it morally justifiable to take any risk that improves the odds of survival of at least one patient.
To clarify the rules of the game:
- Patient A will live after receiving a transfusion of Patient O's blood.
- Patient B will live after receiving a transfusion of Patient O's blood.
- Patient A's blood will instantly kill Patient B and/or Patient O.
- Patient B's blood will instantly kill Patient A and/or Patient O.
- The only way you can learn something about the blood type of a patient is by giving them a transfusion and seeing whether they die.
For instance, if we refer to the patients as 1, 2 and 3, one strategy might be to perform the transfusions 1→2 and 1→3. With 1/3 probability, Patient 1 has blood type O, and Patients 2 and 3 live. Otherwise, everybody dies. The expected number of people saved is thus 1/3 * 2 + 2/3 * 0 = 2/3.
Your own blood is of type AB and therefore useless.
Find the best strategy.
(If your strategy is expected to save 1 person, then good job, but there's an even better approach!)
Posted Saturday 19-Sep-2015 09:35
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Sat 19-Sep-2015 14:22
Linus Åkesson
Sun 20-Sep-2015 14:03
Mon 21-Sep-2015 14:12
Tue 22-Sep-2015 03:00
Valentino Miazzo
Tue 29-Sep-2015 08:09
Solution: Pick two patients randomly. Spill from the receiver a transfusion and keep the blood aside. Transfuse from donor to receiver. 1/3 of times the donor is O and receiver doesn't die immediately. Restore the receiver blood and do another transfusion from the donor to the other patient. You saved 2 people.
2/3 of times the donor is A or B and the receiver dies immediately. Reverse the transfusion.
In 1/2 of times the receiver was O. The corpe contains compatible blood and the reverse transfusion saves the original donor. The blood aside is O and useful to save the other patient.You saved 2 people.
In the other 1/2 of times the receiver is not O and all the patients die.
Therefore: 1/3*2 + 2/3*1/2*2 + 2/3*1/2*0=4/3
By the way, your projects are marvelous and very inspiring. Thanks!
Linus Åkesson
Thu 1-Oct-2015 21:13
Vale wrote:
Solution: Pick two patients randomly. Spill from the receiver a transfusion and keep the blood aside. Transfuse from donor to receiver. 1/3 of times the donor is O and receiver doesn't die immediately. Restore the receiver blood and do another transfusion from the donor to the other patient. You saved 2 people.2/3 of times the donor is A or B and the receiver dies immediately. Reverse the transfusion.
In 1/2 of times the receiver was O. The corpe contains compatible blood and the reverse transfusion saves the original donor. The blood aside is O and useful to save the other patient.You saved 2 people.
In the other 1/2 of times the receiver is not O and all the patients die.
Therefore: 1/3*2 + 2/3*1/2*2 + 2/3*1/2*0=4/3
That is exactly the solution I had in mind! Well done!
Thank you!
Mon 30-Nov-2015 20:37
Linus Åkesson
Tue 1-Dec-2015 23:00
Thu 10-Dec-2015 16:49
Define patients 1, 2 and 3. Draw and store blood from each of them. Give blood of patient 1 to 2, and if he survives, also to patient 3. If patient 2 does not survive, give his stored blood to patient 3. If patient 3 does not survive, give his stored blood to patient 1, but if patient 3 survives, give patient 2's blood also to patient 1.
List of scenarios with (1,2,3) marking blood types of each patient:
(o,a,b),(o,b,a): Patients 2 and 3 both survive, 1/3 cases.
(a,b,o),(b,a,o): Patient 1 survives, 1/3 cases.
(a,o,b),(b,o,a): Patients 1 and 3 both survive, 1/3 cases.
1/3*2+1/3*1+1/3*2=5/3
Linus Åkesson
Fri 11-Dec-2015 14:38
Fri 11-Dec-2015 14:54
You're right about giving leftover blood back to donor, and the needed assumption I missed to state is actually based on original rules:
Patient A's blood will instantly kill Patient B and/or Patient O.
Patient B's blood will instantly kill Patient A and/or Patient O.
So, we can assume that virtually a drop of blood is enough to see the potential negative effect, and transfusion can be stopped without consuming any significant amount of blood. We need to draw blood from everyone just to avoid contaminating patient O's blood with even a single drop of wrong type.
-Pets
Linus Åkesson
Fri 11-Dec-2015 16:40