The rope or bungee or even the belt off your pants trick is great. The next thing to do is mix up a bottle of soapy water in a sprayer and wet down the rim-to-tire joint and look for tiny bubbles. Look for punctures in the tread also, but usually it is not the problem.
If it has these micro-leaks, it will go flat soon enough.
The fix for that is to deflate, clean the rim of all grime or rust, and apply a coating of transmission fluid to the tire bead in that area. it softens the rubber just a bit and allows for a good seal again. Reassemble with your favorite band-aid and check again.