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The whole numbers from $1$ to $15$ are written on a whiteboard. A move erases any $2$ numbers $a$ and $b$ and writes $a + b + 2$ in their place. Moves continue until a single number remains. What is that number?

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Hint

Track how the sum of the board changes with each move.

Solution

Each move removes $a + b$ from the sum and adds $a + b + 2$ , so the sum grows by exactly $2$ .
The starting sum is $1 + 2 + \dots + 15 = 120$ .
Each move cuts the count of numbers by one, so there are $15 - 1 = 14$ moves.
The final number is $120 + 14 \times 2 = 148$ .

Answer: $148$

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