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Running total race

BrainteasersEasy

$2$ players build a running total that starts at $0$ . On each turn the player to move adds a whole number from $1$ to $6$ to the total. The player who makes the total exactly $50$ wins, and no move may take the total above $50$ . You move first and both players play optimally. What number should you add on your first turn to guarantee a win?

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Hint

Work out which totals you want to leave after each of your turns, starting from $50$ .

Solution

If you leave a total of $43$ , your opponent adds between $1$ and $6$ , reaching $44$ to $49$ , and you can then reach $50$ exactly.
The same argument shows the safe totals step down by $7$ : $50$ , $43$ , $36$ , and so on down to $1$ , because $50 - 7 \times 7 = 1$ .
From $0$ you can reach the safe total $1$ at once, and after that you answer each opponent move of $a$ with $7 - a$ .
So you add $1$ on your first turn.

Answer: $1$

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