A surgeon has five patients, each with a different organ which has stopped working. Each is in need of having a working organ transplanted into them, and will die if they do not receive this. At this point, a person with a fully working body happens to enter the surgery. The surgeon could kill this person quickly and painlessly, and give each of the patients a working organ from this person's body. Thus, they would give up one life but would save five lives; utilitarianism implies that the surgeon must surely do this. But this course of action would be monstrous. Hence, utilitarianism is false.Most utilitarians are not too keen to bite the bullet by admitting that this would be the moral course of action, and tend to argue that there are advantages to a system where you can see a doctor without risk of being murdered. Non-utilitarians would respond by stipulating that absolutely no-one finds out about the murder, so that this system of cooperation is not disturbed. I don't know the utilitarian response to that, but basically it all gets very messy and unclear.
I have a better response to this thought experiment.
Killing the healthy person is stupid! What you need to do is kill one of the people who is already dying, and give their organs to the other people who are dying. Select this person at random, so there's no disincentive to go into the surgery - it's a choice between going in and facing a one-in-five chance of dying, and staying out and being sure to die. Five people survive, one person dies, and no-one can complain that their rights are being ignored. Just tell the patients: "You're all going to die if you don't take part in this raffle, because your organ will give out completely and you won't get a transplant. If you take part, then you may end up being killed right now but providing someone else loses, you will get one of their organs and you will live." Far cleaner, with no need to unilaterally violate the rights of an innocent.
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