MathDB

A weird calculator has a numerical display and only two buttons,

\boxed{D\sharp}

and

\boxed{D\flat}

. The first button doubles the displayed number and then adds

1

. The second button doubles the displayed number and then subtracts

1

. For example, if the display is showing

5

, then pressing the

\boxed{D\sharp}

produces

11

. If the display shows

5

and we press

\boxed{D\flat}

, we get

9

. If the display shows

5

and we press the sequence

\boxed{D\sharp}

\boxed{D\flat}

\boxed{D\sharp}

\boxed{D\sharp}

, we get a display of

87

.
[*] Suppose the initial displayed number is

1

. Give a sequence of exactly eight button presses that will result in a display of

313

. [*] Suppose the initial displayed number is

1

, and we then perform exactly eight button presses. Describe all the numbers that can possibly result? Prove your answer by explaining how all these numbers can be produced and that no other numbers can be produced.

Problem Statement