Video on the same topic by Matt Parker: https://youtu.be/VTDKqW_GLkw
There is Møbee, a new, similar game, that has a common symbol between any three cards. The mathematics of that deck is based on finite Möbius planes.
Couple of additional comments a few weeks ago https://news.ycombinator.com/item?id=41608496
Here you can find a Javascript function (and a good explanation) for creating a Dobble deck: https://github.com/Darkseal/dobble
Ha! We decided to make a version of these for kids at our microschool (with their faces on the cards) for an end of year gift. Didn't realise the math was going to tricky. Found a page with code to generate combinations. Luckily, being a microschool meant we had the right number of kids plus educators for a small pack that covered all of them equally. Was fun diving into it and the kids loved the cards.