Max heads when you may reflip coins two at a time

Question

You toss four fair coins and see the results. Afterwards you may retoss coins as often as you like, but only two at a time, never a single coin by itself. Playing optimally, what is the expected number of heads you finish with?

Short answer

$4$ . Any tail can be retossed together with another coin, and each retoss has positive probability of producing two heads, so by repeating you reach all heads with probability $1$ .

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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