(the missing condition is that every hand is beatable)
Each hand beats 2 distinct hands
With 3 distinct hands you can cover up to 6 distinct hands which is a superset of all possible hands
3 * 2 > 5
A --> D | E B --> D | E C --> D | E D --> C | E E --> D | C
A --> B | C B --> C | D C --> D | E D --> E | A E --> A | B
Paper: 5-letters (Mod4 == 1)
Spock: 5-letters (Mod4 == 1)
Lizard: 6-letters (Mod 4 == 2)
Scissors: 8-letters (Mod 4 == 0)
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There are no 3-letter or 7-letter words (aka: Blah % 4 != 3 given the inputs). So they just left the 3rd index (4th word) empty.