Today, a fellow user was given a CAPTCHA that looks like this:

He wasn't sure how to solve the CAPTCHA, so he asked me: Are hamburgers considered sandwiches? Well, I couldn't figure it out, so I'm asking here!

I know that the hamburger has in the past been called a sandwich. Wikipedia even says:

A hamburger (also called a beef burger, hamburger sandwich, burger or hamburg) is a sandwich consisting of one or more cooked patties of ground meat, usually beef, placed inside a sliced bun.

It explicitly says that it's a sandwich. And I suppose someone must have called it a "hamburger sandwich" at one point, but it sounds very odd to me—like someone put a piece of beef between two hamburgers. Anyway, I don't think that term is in common use, and it's easy to find examples like the following in print:

And this seems to imply that hamburgers and sandwiches are different things. So I'm not sure I can take Wikipedia's word on this one.

Since I didn't know how to respond to the question, I brought it up in EL&U chat, where different people put forth different arguments. It was even suggested that the American English / British English difference might be relevant. But so far I don't think we've come to a conclusion, so I decided to ask on the main site.

I'm not sure what to think. All I have is speculation. Maybe hamburgers were once considered sandwiches by everyone, but now the terms are diverging and speakers have started to disagree. Or maybe it just depends on where you're from. Or maybe hamburgers exist in an interdeterminate state of quasi-sandwichhood.

So I ask you all:

Is a hamburger a sandwich?