Building an electronic dice is a piece of cake especially when MCU has sufficient number of I/O pins. But why waste powerful microcontroller for such simple task. It is possible to drive more LEDs than there are pins available. So using 5 attiny13 pins Mike specially connected 14 LEDs so they can be controlled like he wants.
Dice is triggered with single button and that’s it what you need. If you dig deeper you’ll find a nice description of working concept and of course ASM code that is quite rarely used these days.