Back in college in '82 one cold bitter winter I had a 1976 Honda Civic that did exactly as you described....determined to be a cracked block, something like $1,000+ to repair (current car value was only like $1,500 tops)....but since it stopped if I ran it for about 20-30 minutes.....I found a nice used '77 Celica on a lot (covered in ice & snow)....drove my little put-put down, let her idle in the parking lot while I negotiated the Celica. When their appraiser dude took it for a drive, he came back and offered me $1,500.....He commented it was so cold, it was nice of me to let it run and keep it warm inside.

Ummmm....I took the offer, bought the Celica and ran! (yeah, same winter & the same Celica that the hood hinge subsequently broke off when it was -19 degrees F in Pittsburgh)