IBM Ponder This #313 · May 2024

This problem was suggested by Lorenzo Gianferrari Pini and Radu-Alexandru Todor - thanks!

Dobble® is a card game in which each card contains eight symbols, and any two Dobble cards in the deck share exactly one symbol. A standard Dobble deck consists of 55 cards and 57 symbols. However, it is possible to have a deck of 57 cards and 57 symbols.

In general, given a number $N$, which is a prime power, it is possible to find a Dobble-like deck where each card contains $N+1$ symbols, and there are $N^2+N+1$ cards and symbols in the deck. Dobble itself corresponds to the case $N=7$.

Your goal: For $N=4$, find the number of all possible Dobble-like decks with $N^2+N+1$ cards and symbols, and $N+1$ symbols per card, where each pair of cards has exactly one common symbol.

A Bonus "*" will be given for finding the number of all possible Dobble-like decks for $N=8$.

Solution

Best opened after a real attempt

To be added.