Question

A certain calculator has only two keys [ $+1$ ] and [ $\times 2$ ]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed " $9$ " and you pressed [ $+1$ ], it would display " $10.$ " If you then pressed [ $\times 2$ ], it would display " $20.$ " Starting with the display " $1,$ " what is the fewest number of keystrokes you would need to reach " $200$ "?

$8$

$9$

$10$

$11$

$12$

Solution:

Work backward from $200$ . When the number is even, undo $\times2$ by dividing by $2$ ; when it is odd, undo $+1$ by subtracting $1$ .

$200,100,50,25,24,12,6,3,2,1$ uses $9$ reverse steps, so $9$ keystrokes are enough.

Eight keystrokes are not enough: the largest value below $200$ reachable in $8$ keystrokes is obtained by $\times2,+1$ followed by six doublings, giving $3\cdot2^6=192$ . The next larger possibilities overshoot to at least $256$ , so $200$ cannot be reached in $8$ keystrokes.

Thus, B is the correct answer.

2005 AMC 8 Problem 24

Solution:

Problem 24 in Other Years