Competition problem — Combinatorics

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Combinatorics Difficulty 5.7 AIME, harder Prove it United States

Problem:
A Mystic Four Calculator has a four-digit display and four buttons. The calculator works as follows: Pressing button 1 replaces the number in the display with $1$ ; Pressing button 2 divides the number in the display by $2$ ; Pressing button 3 subtracts $3$ from the number in the display; Pressing button 4 multiplies the number in the display by $4$ .

Initially the display shows $0$ . Any operation yielding a negative, fractional, or five-digit answer is ignored.

a. Can $2007$ appear in the display?

b. Can $2008$ appear in the display?

Solution

Solution:

a. No. Notice that if the number on the display is not divisible by $3$ , then none of the operations can have as a result a number divisible by $3$ . At the start, $1$ is the only button producing a result, so we are required to press it at some point. After that the number will never be divisible by $3$ again, and $2007$ is divisible by $3$ .

b. Yes. For instance, press $1$ so that the display shows $1$ , press $4$ six times so that the display shows $4096$ , and press $3$ $3696$ times so that the display shows $2008$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.