Ten Lanterns

You have ten lanterns, five of which are working, and five of which are broken. You are allowed to choose any two lanterns and make a test that tells you whether there is a broken lantern among them or not. How many tests do you need until you find a lantern you know for sure is working?

Remark: If the test detects that there are broken lanterns, it does not tell you which ones and how many (one or two) they are.

You need 6 tests:

(1, 2) → (3, 4) → (5, 6) → (7, 8) → (7, 9) → (8, 9)

If at least one of these tests is positive, then you have found two working lanterns.

It all of these tests are negative, then lantern #10 must be working. Indeed, since at least one lantern in each of the pairs (1, 2), (3, 4), (5, 6) is not working. Therefore, there are at least 2 working lanterns among #7, #8, #9, #10. If #10 is not working, then at least one of the pairs (7, 8), (7, 9), or (8, 9) must yield a positive test, which is a contradiction.

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Responses

1. This is an incorrect number of tests. The problem starts with know values of 5 working and 5 broken.
Trial 1:
Probability of picking a broken lantern from the initial draw is 5/10
Probability of picking a broken lantern on the second draw is 4/9
Trial 2:
Probability of picking a broken lantern on the third draw is 3/8
Probability of picking a broken lantern on the fourth draw is 2/7
Trial 3:
Probability of picking a broken lantern on the fifth draw is 1/6
Probability of picking a broken lantern on the sixth draw is 0/5

This means that in 3 trials you are guaranteed to get one working lantern. If you want to get a pair, then you only need one more trial.

1. Hi James. According to the problem, you only know whether there is a broken lantern in the chosen pair, but not how many and which ones. Your task is to find at least one lantern that is sure to be working.