Python - Spot it! Cards

Question

I am trying to simulate deck of "Spot it!" cards with Python. For those who do not know what "Spot it!" is- it is a popular card game in which there is 55 cards in a deck and each card has 8 random symbols on it (for example a ball, a wave, etc). And each card has exactly 1 symbol in common with any other card. Basically, players draw a card and compare it with the one "common card" (which is visible for everyone) and the player who spots the symbol he/she has in common with the "common card" first, wins.

But I am trying only to generate the deck of cards, not the game itself.

So, let's say I have these items:

a list of symbols:

symbols = ["pelikan", "palma", "rukavnik", "morskyKon", "mraznicka", "kormidlo", "lehatko", "drink", "vedro", "Aloha", "koral", "nanuk", "opalovaciKrem", "Splash!", "plutva", "morskaPanna", "meduza", "krab", "stopa", "Wipeout", "pistol", "zralociaPlutva", "plachetnica", "raketa", "slnko", "sandala", "melon", "harpuna", "stolicka", "zachranneKoleso", "lietajuciTanier", "ryba", "hrad", "kalamar", "fotoaparat", "osuska", "naramok", "sarkan", "kotva", "fakla", "kamene", "snehoveGule", "motyka", "kosatka", "plavky", "odtlacokMusle", "klobuk", "ponorka", "morskyJez", "more", "musla", "slnecnik", "potapacskeOkuliare", "slnecneOkuliare", "majak", "vlna", "lopta"]

a deck of (for now, 6) cards, where each key of the dictionary represents a card and a value of each key is a list of 8 symbols, which is empty for now:

cardDeck = {1: [], 2: [], 3: [], 4:[], 5: [], 6:[]}

generating random symbols for each card so that there are no duplicates

import random
for card, symboly in cardDeck.items():
    symboly = []
    for i in range(8):
        symbol = random.choice(symbols)
        while symbol in symboly:
            symbol = random.choice(symbols)
        symboly.append(symbol)
    cardDeck.update({card : set(symboly)})

And this is where I am stuck - I know what to do, but I don't know if it's possible to do this in Python and if yes, how. I realise that I need to do combinations (every possible pairs) from all cards and then for each combination keep randomly picking symbols until it has exactly 1 common symbol (i.e. the length of intersection of each combination is 1). From what I know, combinations can be easily made by itertools.combinations(cardDeck, 2). Then I was trying execute what I wrote about above, but I somehow dont know how to write it in Python. (I am still more-less learning it and I thought this would be interesting to try in Python.)

Any ideas very much appreciated guys. Thanks in advance.

I'm wondering whether a random approach might not be the way to go. It seems highly likely that there is, as gboffi suggests, only one way to do this. All possible decks may well be simple remappings of the "alphabet" of symbols. OP, have you studied combinatorics at all? Careful analysis might reveal an efficient, deterministic solution. — Colin P. Hill, Dec 15 '14 at 21:31
A solution has already been posted at stackoverflow.com/questions/6240113/what-are-the-mathematical-computational-principles-behind-this-game#6240662 — Hugh Bothwell, Dec 15 '14 at 22:21
@Malnormalulo I was not suggesting anything, I was just shooting in the dark... from what I have read in stackoverflow.com/questions/6240113, referenced by Hugh Bothwell, there are 57*56 possible solutions. — gboffi, Dec 15 '14 at 22:43

Hugh Bothwell · Accepted Answer · 2014-12-22T02:42:37.610

Ok - just did a whole lot of reading on this, and here's the gist:

The game can be understood as the physical expression of a finite projective plane of order 7; 57 cards and 57 symbols, where every pair of cards shares a symbol and every pair of symbols shares a card. For unknown reasons, the manufacturer decided to arbitrarily discard two cards.

(More generally, you can* produce a finite projective plane of any order N == p ** n where p is a prime number and n is a positive integer; the result will have N ** 2 + N + 1 cards, and the same number of symbols, and N+1 symbols will appear on each card.) *This is strongly conjectured, but has only been definitely proven for N <= 11.

What you want to generate is a 57 * 57 incidence matrix - which symbols occur on which cards - which obeys the following rules:

Each row contains exactly 8 1s;
Each column contains exactly 8 1s;
Each pair of rows has exactly 1 1 in common;
Each pair of columns has exactly 1 1 in common.

Edit:

I wrote an algorithm which I calculated would take about 150 days to produce the needed result - it's trickier than it looks ;-)

I then did a bit more searching, and found out the the SAGE math package will do exactly what you need (and MUCH faster):

import sage

sol = sage.combinat.designs.block_design.projective_plane(7)

for row in sol.incidence_matrix().rows():
    print(row)

which takes half a second to produce a solution:

(1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
(1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
(0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0)
(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0)
(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0)
(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0)
(0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0)
(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0)
(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0)
(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0)
(0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0)
(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0)
(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)
(0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)
(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)
(0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)
(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0)
(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)
(0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0)
(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)
(0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)
(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)
(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0)
(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0)
(0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0)
(0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0)
(0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1)
(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)
(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)
(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)
(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1)
(0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1)
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1)

It is then pretty straight-forward to convert this to the card-game you want:

symbols = sorted(
    [
        "pelikan", "palma", "rukavnik", "morskyKon", "mraznicka", "kormidlo",
        "lehatko", "drink", "vedro", "Aloha", "koral", "nanuk", "opalovaciKrem",
        "Splash!", "plutva", "morskaPanna", "meduza", "krab", "stopa", "Wipeout",
        "pistol", "zralociaPlutva", "plachetnica", "raketa", "slnko", "sandala",
        "melon", "harpuna", "stolicka", "zachranneKoleso", "lietajuciTanier", "ryba",
        "hrad", "kalamar", "fotoaparat", "osuska", "naramok", "sarkan", "kotva",
        "fakla", "kamene", "snehoveGule", "motyka", "kosatka", "plavky",
        "odtlacokMusle", "klobuk", "ponorka", "morskyJez", "more", "musla",
        "slnecnik", "potapacskeOkuliare", "slnecneOkuliare", "majak", "vlna", "lopta"
    ],
    key = str.lower
)

cards = [
    [symbol for symbol, use_it in zip(symbols, row) if use_it]
    for row in sorted(sol.incidence_matrix().rows())
]

which produces

[
    ['Aloha', 'drink', 'fakla', 'fotoaparat', 'harpuna', 'hrad', 'kalamar', 'kamene'],
    ['Aloha', 'klobuk', 'koral', 'kormidlo', 'kosatka', 'kotva', 'krab', 'lehatko'],
    ['Aloha', 'lietajuciTanier', 'lopta', 'majak', 'meduza', 'melon', 'more', 'morskaPanna'],
    ['Aloha', 'morskyJez', 'morskyKon', 'motyka', 'mraznicka', 'musla', 'nanuk', 'naramok'],
    ['Aloha', 'odtlacokMusle', 'opalovaciKrem', 'osuska', 'palma', 'pelikan', 'pistol', 'plachetnica'],
    ['Aloha', 'plavky', 'plutva', 'ponorka', 'potapacskeOkuliare', 'raketa', 'rukavnik', 'ryba'],
    ['Aloha', 'sandala', 'sarkan', 'slnecneOkuliare', 'slnecnik', 'slnko', 'snehoveGule', 'Splash!'],
    ['Aloha', 'stolicka', 'stopa', 'vedro', 'vlna', 'Wipeout', 'zachranneKoleso', 'zralociaPlutva'],
    ['drink', 'klobuk', 'lietajuciTanier', 'morskyJez', 'odtlacokMusle', 'plavky', 'sandala', 'stolicka'],
    ['drink', 'koral', 'majak', 'mraznicka', 'pelikan', 'rukavnik', 'Splash!', 'zralociaPlutva'],
    ['drink', 'kormidlo', 'melon', 'naramok', 'opalovaciKrem', 'potapacskeOkuliare', 'snehoveGule', 'zachranneKoleso'],
    ['drink', 'kosatka', 'morskaPanna', 'motyka', 'pistol', 'plutva', 'slnko', 'vedro'],
    ['drink', 'kotva', 'lopta', 'nanuk', 'osuska', 'ryba', 'slnecnik', 'stopa'],
    ['drink', 'krab', 'meduza', 'morskyKon', 'plachetnica', 'raketa', 'slnecneOkuliare', 'Wipeout'],
    ['drink', 'lehatko', 'more', 'musla', 'palma', 'ponorka', 'sarkan', 'vlna'],
    ['fakla', 'klobuk', 'morskaPanna', 'nanuk', 'pelikan', 'potapacskeOkuliare', 'slnecneOkuliare', 'vlna'],
    ['fakla', 'koral', 'lopta', 'morskyKon', 'opalovaciKrem', 'plutva', 'sarkan', 'stolicka'],
    ['fakla', 'kormidlo', 'meduza', 'musla', 'pistol', 'ryba', 'sandala', 'zralociaPlutva'],
    ['fakla', 'kosatka', 'more', 'morskyJez', 'osuska', 'raketa', 'Splash!', 'zachranneKoleso'],
    ['fakla', 'kotva', 'lietajuciTanier', 'mraznicka', 'plachetnica', 'ponorka', 'snehoveGule', 'vedro'],
    ['fakla', 'krab', 'majak', 'naramok', 'palma', 'plavky', 'slnko', 'stopa'],
    ['fakla', 'lehatko', 'melon', 'motyka', 'odtlacokMusle', 'rukavnik', 'slnecnik', 'Wipeout'],
    ['fotoaparat', 'klobuk', 'more', 'mraznicka', 'opalovaciKrem', 'ryba', 'slnko', 'Wipeout'],
    ['fotoaparat', 'koral', 'lietajuciTanier', 'naramok', 'pistol', 'raketa', 'slnecnik', 'vlna'],
    ['fotoaparat', 'kormidlo', 'majak', 'motyka', 'osuska', 'ponorka', 'slnecneOkuliare', 'stolicka'],
    ['fotoaparat', 'kosatka', 'melon', 'nanuk', 'plachetnica', 'plavky', 'sarkan', 'zralociaPlutva'],
    ['fotoaparat', 'kotva', 'morskaPanna', 'morskyKon', 'palma', 'rukavnik', 'sandala', 'zachranneKoleso'],
    ['fotoaparat', 'krab', 'lopta', 'musla', 'odtlacokMusle', 'potapacskeOkuliare', 'Splash!', 'vedro'],
    ['fotoaparat', 'lehatko', 'meduza', 'morskyJez', 'pelikan', 'plutva', 'snehoveGule', 'stopa'],
    ['harpuna', 'klobuk', 'melon', 'morskyKon', 'pistol', 'ponorka', 'Splash!', 'stopa'],
    ['harpuna', 'koral', 'morskaPanna', 'musla', 'osuska', 'plavky', 'snehoveGule', 'Wipeout'],
    ['harpuna', 'kormidlo', 'lopta', 'morskyJez', 'plachetnica', 'rukavnik', 'slnko', 'vlna'],
    ['harpuna', 'kosatka', 'meduza', 'mraznicka', 'palma', 'potapacskeOkuliare', 'slnecnik', 'stolicka'],
    ['harpuna', 'kotva', 'more', 'naramok', 'odtlacokMusle', 'plutva', 'slnecneOkuliare', 'zralociaPlutva'],
    ['harpuna', 'krab', 'lietajuciTanier', 'motyka', 'pelikan', 'ryba', 'sarkan', 'zachranneKoleso'],
    ['harpuna', 'lehatko', 'majak', 'nanuk', 'opalovaciKrem', 'raketa', 'sandala', 'vedro'],
    ['hrad', 'klobuk', 'meduza', 'naramok', 'osuska', 'rukavnik', 'sarkan', 'vedro'],
    ['hrad', 'koral', 'more', 'motyka', 'plachetnica', 'potapacskeOkuliare', 'sandala', 'stopa'],
    ['hrad', 'kormidlo', 'lietajuciTanier', 'nanuk', 'palma', 'plutva', 'Splash!', 'Wipeout'],
    ['hrad', 'kosatka', 'majak', 'morskyKon', 'odtlacokMusle', 'ryba', 'snehoveGule', 'vlna'],
    ['hrad', 'kotva', 'melon', 'musla', 'pelikan', 'raketa', 'slnko', 'stolicka'],
    ['hrad', 'krab', 'morskaPanna', 'morskyJez', 'opalovaciKrem', 'ponorka', 'slnecnik', 'zralociaPlutva'],
    ['hrad', 'lehatko', 'lopta', 'mraznicka', 'pistol', 'plavky', 'slnecneOkuliare', 'zachranneKoleso'],
    ['kalamar', 'klobuk', 'majak', 'musla', 'plachetnica', 'plutva', 'slnecnik', 'zachranneKoleso'],
    ['kalamar', 'koral', 'melon', 'morskyJez', 'palma', 'ryba', 'slnecneOkuliare', 'vedro'],
    ['kalamar', 'kormidlo', 'morskaPanna', 'mraznicka', 'odtlacokMusle', 'raketa', 'sarkan', 'stopa'],
    ['kalamar', 'kosatka', 'lopta', 'naramok', 'pelikan', 'ponorka', 'sandala', 'Wipeout'],
    ['kalamar', 'kotva', 'meduza', 'motyka', 'opalovaciKrem', 'plavky', 'Splash!', 'vlna'],
    ['kalamar', 'krab', 'more', 'nanuk', 'pistol', 'rukavnik', 'snehoveGule', 'stolicka'],
    ['kalamar', 'lehatko', 'lietajuciTanier', 'morskyKon', 'osuska', 'potapacskeOkuliare', 'slnko', 'zralociaPlutva'],
    ['kamene', 'klobuk', 'lopta', 'motyka', 'palma', 'raketa', 'snehoveGule', 'zralociaPlutva'],
    ['kamene', 'koral', 'meduza', 'nanuk', 'odtlacokMusle', 'ponorka', 'slnko', 'zachranneKoleso'],
    ['kamene', 'kormidlo', 'more', 'morskyKon', 'pelikan', 'plavky', 'slnecnik', 'vedro'],
    ['kamene', 'kosatka', 'lietajuciTanier', 'musla', 'opalovaciKrem', 'rukavnik', 'slnecneOkuliare', 'stopa'],
    ['kamene', 'kotva', 'majak', 'morskyJez', 'pistol', 'potapacskeOkuliare', 'sarkan', 'Wipeout'],
    ['kamene', 'krab', 'melon', 'mraznicka', 'osuska', 'plutva', 'sandala', 'vlna'],
    ['kamene', 'lehatko', 'morskaPanna', 'naramok', 'plachetnica', 'ryba', 'Splash!', 'stolicka']
]

Do you know if it is also possible to do this using some other library/module? Since Sage is only compatible with Python 2.x version. Cheers — puk789, Dec 25 '14 at 17:47
Sage is the only one I came across; however, now that you have the incidence matrix above, you can just cut-and-paste it and apply it directly to your word-list ;-) — Hugh Bothwell, Dec 25 '14 at 17:51

score 1 · Answer 2 · answered Jul 22 '18 at 11:43

You can construct the incidence matrix for finite Desarguesian projective planes without trial-and-error. Here's some Python code I've written which has been tested in both Python 2.7 and Python 3.6:

def incidence_matrix(p = 7): # p can be any prime

    # Generate all p^3 - 1 points in affine space, minus the origin:
    points = [(a,b,c) for a in range(p) for b in range(p) for c in range(p)][1::]

    # Quotient by the equivalence relation to give the projective plane points:
    scale = lambda t, k : tuple([((k * v) % p) for v in t])
    canonise = lambda t : (sorted([scale(t, k) for k in range(1, p)])[0])
    points = list(set(map(canonise, points)))

    # We identify each point with its dual line by taking orthogonal complements.
    # A line is incident with a point if the pole of the line is orthogonal to
    # the point, which can be evaluated straightforwardly:
    innerprod = lambda x, y : (sum([a*b for (a, b) in zip(x, y)]) % p)
    imatrix = [[1 if (innerprod(x, y) == 0) else 0 for x in points] for y in points]

    # Output the incidence matrix:
    return imatrix

For projective planes of order-p^n, where n >= 2, it's similar but slightly more complicated: instead of working in the integers modulo p, you need to operate in the finite field F_(p^n). Fortunately for your case, 7 is prime, so the simpler code above will suffice.

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