Is there a word for a person who looks absent-minded, but in fact notices everything carefully?

I'm coming out from the word absent-mindedness:

Absent-mindedness is where a person shows inattentive or forgetful behaviour. It can have three different causes:

• a low level of attention ("blanking" or "zoning out")
• intense attention to a single object of focus (hyperfocus) that makes a person oblivious to events around him or her;

I'm wondering whether there is a word for a person who only looks to behave like this, but in fact listens to everything around them.

I do this a lot. Whenever I'm in a group of people, I either look like I'm zoned-out thinking about my work stuff, or listening to only one person, but in fact I'm carefully taking in everything other people are speaking.

Another good example would be the character Columbo from the show of the same name. He always looked like he was ignoring everyone, totally absent-minded, while in fact he was doing this on purpose to keep his suspects from being too careful around him.

What is the word for such a person?

Playing dumb

Based on your description this seems to encompass the sentiment.

http://idioms.thefreedictionary.com/play+dumb

Alternatively:

Deceptively absent-minded

Idiot-savant comes to mind, but that's neither accurate nor indeed specific enough. I've looked up some synonyms: no cigar. The problem with defining such people is there's more to their patterns of behavior than meets the eye. They're whimsical. One never knows when they're:

1. Pretending to be absentminded to throw everybody off
2. Pretending to be absentminded because they're bored with what the others are discussing or doing
3. Being genuinely absentminded
4. Pretending to be absentminded because they don't know how to answer a question

5. Pretending to be absentminded just to spite everybody

   • and so forth.

Also, you never know how much of it is deliberate, and how much instinctive. Like that sleepy Texas sheriff. You know:

He's parked behind a tree, with the radar pointed in the right direction. He's dozing off. Cars keep zooming by on the highway. He couldn't care less. A whole bunch of cars rolls by, all of them most definitely over the speed limit. 80 mph. 85. 90. 85 again. 80. 92. 93. 85. And so forth. Nothing: no reaction from the sheriff. All of a sudden this souped-up Porsche flies by, doing 140! The sheriff opens one eye, looks at the radar, and goes, "Wow, I'll be f***!" The next moment, the engine is running, the lights and siren are on, and he swings onto the highway in pursuit of the offending Porsche.

He wasn't pretending to be absentminded. He was actually being absentminded. Which didn't stop him from acting quickly when it became really necessary to do his duty. Something like that.

Maybe you could say "deceptively attentive."