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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:

  1. Patient A will live after receiving a transfusion of Patient O's blood.
  2. Patient B will live after receiving a transfusion of Patient O's blood.
  3. Patient A's blood will instantly kill Patient B and/or Patient O.
  4. Patient B's blood will instantly kill Patient A and/or Patient O.
  5. 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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Anonymous
Sat 19-Sep-2015 14:22
(No spoilers) I've got 4/3. Is a better solution possible?

7 more comments hidden. Click to show all.

Anonymous
Thu 10-Dec-2015 16:49
I reached 5/3, assuming that drawing blood enough to give transfusion to 2 persons does not kill patient immediately.

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
lft
Linus Åkesson
Fri 11-Dec-2015 14:38
That's an interesting approach! But I'm not entirely convinced about the third case. Let's assume (a,o,b). In the beginning, each patient lacks 1 transfusion's worth of blood. After the initial drawing, each patient lacks 3 units of blood. You use one of the stored units of Type A blood, giving it to Patient 2. Then you give one unit of Type O blood to Patient 3. Finally, you give the remaining unit of Type O blood to Patient 1. At this point, I assume that any leftovers are given back to the respective patient, so the two stored units of Type B blood are given to Patient 3. But you only have one unit left of Type A blood, so Patient 1 would still end up lacking one unit of blood. You'd have to transfuse some of the blood from Corpse 2 back to Patient 1. That may or may not work in real life (please don't try it), but it's lateral thinking all right. If it was part of the intended solution, then I think it would have been good to spell it out. Otherwise; did I miss something?
Anonymous
Fri 11-Dec-2015 14:54
Hi Linus,

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
lft
Linus Åkesson
Fri 11-Dec-2015 16:40
All right, then I follow you. Good work there!