Daily dosage

Question:

A patient has fallen very ill and has been advised to take exactly one pill of medicine A and exactly one pill of medicine B each day. The two pills, which are indistinguishable, must be taken together. If they are not, the patient will die. The patient has bottles of A pills and B pills. She puts one of the A pills in her hand. Then while tilting the bottle of B pills, two B pills accidentally fall out.

Now there are three pills in her hand. Because all the pills look identical, she cannot tell which two pills are type B and which is type A. Since the pills are extremely expensive, the patient does not wish to throw away the ones in her hand. How can she save the pills in her hand and still maintain a proper daily dosage?

Solution: The key insight required here is that there must be a 1:1 ratio between the active ingredients of A pills and B pills. Don’t get hung up about the physical shape that the pills currently have, don’t assume the pills have to be manipulated intact, and let go of the assumption that you need to be able to label the pills. There are dozens of solutions. Here are two analyses, both of which have the same effect:

  1. Take an extra pill out of the A bottle to make two pills of each type of medication. It is not important to know which is which to know that a 1:1 ratio exists. Now cut each pill in half, being careful to keep the halves separate. For the first night, take half of one pill from each pile and you will be guaranteed exactly one A and one B pill. Save the other halves for tomorrow.
  2. Add one A pill to the three pills. The pile now includes two A and two B pills. Again, it is not important to identify which is which. Crush all four pills with a mortar and pestle, making sure the compound is mixed thoroughly. Now take half of the powder today and the rest of the powder the next day. No pill is wasted and the patient gets the right ratio of medication.

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