Eight batteries, four good, one flashlight

Question

You have 8 batteries; exactly 4 are good and 4 are dead, but they look identical. Your flashlight needs two good batteries inserted together to turn on. One "test" means inserting some pair and seeing whether the flashlight lights. What is the fewest number of tests that guarantees you get the flashlight working?

Accepted Answer

Seven tests are enough, and this is the known minimum. A clean 7-test plan. Split the 8 batteries into three groups: two groups of 3, call them A = \{a1,a2,a3\} and B = \{b1,b2,b3\}, and one group of 2, C = \{c1,c2\}. Key fact (pigeonhole). There are 4 good batteries spread over groups of sizes 3, 3, 2. If every group had at most one good battery, that's at most 1+1+1 = 3 good batteries, but we have 4. So at least one group contains two good batteries. Test every pair inside each group. - Group A (3 batteries): its pairs are a1a2, a1a3, a2a3 → 3 tests. - Group B (3 batteries): another 3 tests…

Eight batteries, four good, one flashlight

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