Problem 8

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Binarnia, a remote country in Mathmagic land, has the strangest currency. There are only two types of notes: 1 dollar notes, and ``doubler" notes. Everything is paid for using stacks of money composed of these two notes. The value of a stack is determined by reading all notes from the top down, adding $1$ to the value of the stack whenever a $1$ dollar note is read, and doubling the value whenever a "doubler" note is read. So for example, a stack reading $1, doubler, doubler, 1, doubler$ from the top would be worth $((1 \cdot 2)\cdot 2+1) \cdot 2=10$ Binarnian dollars. In Binarnia, everything needs to be paid for exactly, no change is given.

When the ambassador of Optimalia visited Binarnia, he made sure to take enough Binarnian notes so that he could pay any amount (exactly) from $1$ to $3000$. He took the least amount of $1$-dollar notes possible, as these cost more Mathmagicial Pounds than doublers, and he took no more notes than necessary. However, when he got off the train in the capital, he had to spend a dollar on a bus ticket to his hotel, and then another dollar on a new bus ticket because he got the wrong bus the first time. How many different amounts between $1$ and $3000$ can he still pay for with his remaining notes?

To deter guessing without thinking, we ask that you also solve the following simple arithmetic problem before checking your answer:

What is nine plus six, plus two?

This problem was first solved on Wed 6th March at 5:02:47pm