Binary Patterns: A Puzzle

Imagine a world in which there are only 2 digits. We are used to having 10 digits, from 0 to 9, but with just 2 of them, we’d be limited to 0 and 1. We could count in that world just like we do in this one: start at 0, and next comes 1. But going farther is tricky because we’ve run out of digits. So, like an odometer showing a bunch of 9s, we reset those places back to 0s and increment the next-left digit. In this case, we reset the 1 to a 0, but then tack on a 1 to the left, getting 10. That number still represents the successor of 1, what we call 2, but it looks like 10. So, to avoid confusion, we’ll write it as $10_2$ to indicate that it is the number written as “one zero” in 2-world. We say that $10_2$ is the binary representation of the number 2 (in contrast with the 10-digit decimal representations).

We can find the binary representations of other numbers using the increment-and-reset algorithm described above.

Many people are familiar with binary representations, if only insofar as that computers use them. And that’s sort of where our problem comes from. One of the programmers I currently work with is a member of the San Diego Derby Dolls roller derby team, where she wears the number 1011. Any programmer worth her salt would read that as “number 11”, since $1011_2=11$.

[Photo Credit]

What’s curious about that is that $1011_2$ has an $11$ in it. Several other numbers do the same thing: their binary representations end with their decimal representations.

So here is the question:

How many numbers can you find whose binary representations end with their decimal representations?

Before you get too busy cranking away at the binary representations, allow me to suggest using WolframAlpha. You can type something in like “12 in binary” and it will give you what you’re looking for. So, you can let the computer crunch all the numbers and you can spend your time looking for patterns and such.