When we say “it was on the back of a napkin,” it means we did it quickly. It wasn’t formal. It makes you imagine the person was eating dinner, suddenly had an idea, and had to write it on the nearest paper, which was a napkin. 🙂
When a tutor uses a fast food premises to teach his student: instead of using his own paper for teaching to avoid being chased away, he uses the outlet’s napkins to explain the math concepts.
“back-of-the-envelope” is often used for the same thing. It implies that a calculation was done quickly and without full rigor but should be “close enough” to the correct answer for the current purpose.
A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as the actual back of an envelope. It is more than a guess but less than an accurate calculation or mathematical