Question 71
Let $p$ be a prime number. There are several cards on the table. Numbers $0$, $1$, $\dots$, $p-1$ are written on them such that $0$ is written once and each of $1$, $2$, $\dots$, $p-1$ is written at most once. For any two cards with numbers $x$ and $y$ on them, respectively, we determine the *winner* card such that;
i. If card $x$ wins over card $y$ and card $y$ wins over card $z$, then card $x$ wins over card $z$;
ii. If card $x$ won't be defeated by card $y$ and card $z$ won't be defeated by card $t$, then if the cards $x+z$ and $y+t$ both are existing, then card $x+z$ won't be defeated by the card $y+t$. (We do the addition modulo $p$.)
*Note.* We say a card *won't be defeated by the other card* if it either wins over it or is identical to the other card.
Determine the maximum number of cards that such a rule-based game would become possible.
i. If card $x$ wins over card $y$ and card $y$ wins over card $z$, then card $x$ wins over card $z$;
ii. If card $x$ won't be defeated by card $y$ and card $z$ won't be defeated by card $t$, then if the cards $x+z$ and $y+t$ both are existing, then card $x+z$ won't be defeated by the card $y+t$. (We do the addition modulo $p$.)
*Note.* We say a card *won't be defeated by the other card* if it either wins over it or is identical to the other card.
Determine the maximum number of cards that such a rule-based game would become possible.