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Coins flipped in rounds

BrainteasersEasy

$10$ coins numbered $1$ to $10$ lie heads up in a row. In round $i$ , for $i$ from $1$ to $10$ , every coin with a number of at least $i$ is flipped over. How many coins show tails after round $10$ ?

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Hint

Count how many rounds touch coin $j$ .

Solution

Coin $j$ is flipped in rounds $1$ to $j$ , so it is flipped $j$ times.
A coin ends tails up exactly when it is flipped an odd number of times, so when $j$ is odd.
The odd numbers from $1$ to $10$ are $1$ , $3$ , $5$ , $7$ and $9$ .
So $5$ coins show tails.

Answer: $5$

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