[Algorithm] What are the mathematical/computational principles behind this game?


Answers

So there are k=55 cards containing m=8 pictures each from a pool of n pictures total. We can restate the question 'How many pictures n do we need, so that we can construct a set of k cards with only one shared picture between any pair of cards?' equivalently by asking:

Given an n-dimensional vector space and the set of all vectors, which contain exactly m elements equal to one and all other zero, how big has n to be, so that we can find a set of k vectors, whose pairwise dot products are all equal to 1?

There are exactly (n choose m) possible vectors to build pairs from. So we at least need a big enough n so that (n choose m) >= k. This is just a lower bound, so for fulfilling the pairwise compatibility constraint we possibly need a much higher n.

Just for experimenting a bit i wrote a small Haskell program to calculate valid card sets:

Edit: I just realized after seeing Neil's and Gajet's solution, that the algorithm i use doesn't always find the best possible solution, so everything below isn't necessarily valid. I'll try to update my code soon.

module Main where

cardCandidates n m = cardCandidates' [] (n-m) m
cardCandidates' buildup  0  0 = [buildup]
cardCandidates' buildup zc oc
    | zc>0 && oc>0 = zerorec ++ onerec
    | zc>0         = zerorec
    | otherwise    = onerec
    where zerorec = cardCandidates' (0:buildup) (zc-1) oc
          onerec  = cardCandidates' (1:buildup) zc (oc-1)

dot x y = sum $ zipWith (*) x y
compatible x y = dot x y == 1

compatibleCards = compatibleCards' []
compatibleCards' valid     [] = valid
compatibleCards' valid (c:cs)
  | all (compatible c) valid = compatibleCards' (c:valid) cs
  |                otherwise = compatibleCards'    valid  cs

legalCardSet n m = compatibleCards $ cardCandidates n m

main = mapM_ print [(n, length $ legalCardSet n m) | n<-[m..]]
  where m = 8

The resulting maximum number of compatible cards for m=8 pictures per card for different number of pictures to choose from n for the first few n looks like this:

This brute force method doesn't get very far though because of combinatorial explosion. But i thought it might still be interesting.

Interestingly, it seems that for given m, k increases with n only up to a certain n, after which it stays constant.

This means, that for every number of pictures per card there is a certain number of pictures to choose from, that results in maximum possible number of legal cards. Adding more pictures to choose from past that optimal number doesn't increase the number of legal cards any further.

The first few optimal k's are:

Question

My kids have this fun game called Spot It! The game constraints (as best I can describe) are:

  • It is a deck of 55 cards
  • On each card are 8 unique pictures (i.e. a card can't have 2 of the same picture)
  • Given any 2 cards chosen from the deck, there is 1 and only 1 matching picture.
  • Matching pictures may be scaled differently on different cards but that is only to make the game harder (i.e. a small tree still matches a larger tree)

The principle of the game is: flip over 2 cards and whoever first picks the matching picture gets a point.

Here's a picture for clarification:

(Example: you can see from the bottom 2 cards above that the matching picture is the green dinosaur. Between the bottom-right and middle-right picture, it's a clown's head.)

I'm trying to understand the following:

  1. What are the minimum number of different pictures required to meet these criteria and how would you determine this?

  2. Using pseudocode (or Ruby), how would you generate 55 game cards from an array of N pictures (where N is the minimum number from question 1)?

Update:

Pictures do occur more than twice per deck (contrary to what some have surmised). See this picture of 3 cards, each with a lightning bolt:







Here's Gajet's solution in Python, since I find Python more readable. I have modified it so that it works with non-prime numbers as well. I have used Thies insight to generate some more easily understood display code.

from __future__ import print_function
from itertools import *

def create_cards(p):
    for min_factor in range(2, 1 + int(p ** 0.5)):
        if p % min_factor == 0:
            break
    else:
        min_factor = p
    cards = []
    for i in range(p):
        cards.append(set([i * p + j for j in range(p)] + [p * p]))
    for i in range(min_factor):
        for j in range(p):
            cards.append(set([k * p + (j + i * k) % p
                              for k in range(p)] + [p * p + 1 + i]))

    cards.append(set([p * p + i for i in range(min_factor + 1)]))
    return cards, p * p + p + 1

def display_using_stars(cards, num_pictures):
    for pictures_for_card in cards:
        print("".join('*' if picture in pictures_for_card else ' '
                      for picture in range(num_pictures)))

def check_cards(cards):
    for card, other_card in combinations(cards, 2):
        if len(card & other_card) != 1:
            print("Cards", sorted(card), "and", sorted(other_card),
                  "have intersection", sorted(card & other_card))

cards, num_pictures = create_cards(7)
display_using_stars(cards, num_pictures)
check_cards(cards)

With output:

***      *   
   ***   *   
      ****   
*  *  *   *  
 *  *  *  *  
  *  *  * *  
*   *   *  * 
 *   **    * 
  **   *   * 
*    * *    *
 * *    *   *
  * * *     *
         ****



Others have described the general framework for the design (finite projective plane) and shown how to generate finite projective planes of prime order. I would just like to fill in some gaps.

Finite projective planes can be generated for many different orders, but they are most straightforward in the case of prime order p. Then the integers modulo p form a finite field which can be used to describe coordinates for the points and lines in the plane. There are 3 different kinds of coordinates for points: (1,x,y), (0,1,x), and (0,0,1), where x and y can take on values from 0 to p-1. The 3 different kinds of points explains the formula p^2+p+1 for the number of points in the system. We can also describe lines with the same 3 different kinds of coordinates: [1,x,y], [0,1,x], and [0,0,1].

We compute whether a point and line are incident by whether the dot product of their coordinates is equal to 0 mod p. So for example the point (1,2,5) and the line [0,1,1] are incident when p=7 since 1*0+2*1+5*1 = 7 == 0 mod 7, but the point (1,3,3) and the line [1,2,6] are not incident since 1*1+3*2+3*6 = 25 != 0 mod 7.

Translating into the language of cards and pictures, that means the card with coordinates (1,2,5) contains the picture with coordinates [0,1,1], but the card with coordinates (1,3,3) does not contain the picture with coordinates [1,2,6]. We can use this procedure to develop a complete list of cards and the pictures that they contain.

By the way, I think it's easier to think of pictures as points and cards as lines, but there's a duality in projective geometry between points and lines so it really doesn't matter. However, in what follows I will be using points for pictures and lines for cards.

The same construction works for any finite field. We know that there is a finite field of order q if and only if q=p^k, a prime power. The field is called GF(p^k) which stands for "Galois field". The fields are not as easy to construct in the prime power case as they are in the prime case.

Fortunately, the hard work has already been done and implemented in free software, namely Sage. To get a projective plane design of order 4, for example, just type

print designs.ProjectiveGeometryDesign(2,1,GF(4,'z'))

and you'll obtain output that looks like

ProjectiveGeometryDesign<points=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20], blocks=[[0, 1, 2, 3, 20], [0,
4, 8, 12, 16], [0, 5, 10, 15, 19], [0, 6, 11, 13, 17], [0, 7, 9, 14,
18], [1, 4, 11, 14, 19], [1, 5, 9, 13, 16], [1, 6, 8, 15, 18], [1, 7,
10, 12, 17], [2, 4, 9, 15, 17], [2, 5, 11, 12, 18], [2, 6, 10, 14, 16],
[2, 7, 8, 13, 19], [3, 4, 10, 13, 18], [3, 5, 8, 14, 17], [3, 6, 9, 12,
19], [3, 7, 11, 15, 16], [4, 5, 6, 7, 20], [8, 9, 10, 11, 20], [12, 13,
14, 15, 20], [16, 17, 18, 19, 20]]>

I interpret the above as follows: there are 21 pictures labeled from 0 to 20. Each of the blocks (line in projective geometry) tells me which pictures appears on a card. For example, the first card will have pictures 0, 1, 2, 3, and 20; the second card will have pictures 0, 4, 8, 12, and 16; and so on.

The system of order 7 can be generated by

print designs.ProjectiveGeometryDesign(2,1,GF(7)) 

which generates the output

ProjectiveGeometryDesign<points=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56], blocks=[[0, 1, 2, 3, 4, 5, 6,
56], [0, 7, 14, 21, 28, 35, 42, 49], [0, 8, 16, 24, 32, 40, 48, 50], [0,
9, 18, 27, 29, 38, 47, 51], [0, 10, 20, 23, 33, 36, 46, 52], [0, 11, 15,
26, 30, 41, 45, 53], [0, 12, 17, 22, 34, 39, 44, 54], [0, 13, 19, 25,
31, 37, 43, 55], [1, 7, 20, 26, 32, 38, 44, 55], [1, 8, 15, 22, 29, 36,
43, 49], [1, 9, 17, 25, 33, 41, 42, 50], [1, 10, 19, 21, 30, 39, 48,
51], [1, 11, 14, 24, 34, 37, 47, 52], [1, 12, 16, 27, 31, 35, 46, 53],
[1, 13, 18, 23, 28, 40, 45, 54], [2, 7, 19, 24, 29, 41, 46, 54], [2, 8,
14, 27, 33, 39, 45, 55], [2, 9, 16, 23, 30, 37, 44, 49], [2, 10, 18, 26,
34, 35, 43, 50], [2, 11, 20, 22, 31, 40, 42, 51], [2, 12, 15, 25, 28,
38, 48, 52], [2, 13, 17, 21, 32, 36, 47, 53], [3, 7, 18, 22, 33, 37, 48,
53], [3, 8, 20, 25, 30, 35, 47, 54], [3, 9, 15, 21, 34, 40, 46, 55], [3,
10, 17, 24, 31, 38, 45, 49], [3, 11, 19, 27, 28, 36, 44, 50], [3, 12,
14, 23, 32, 41, 43, 51], [3, 13, 16, 26, 29, 39, 42, 52], [4, 7, 17, 27,
30, 40, 43, 52], [4, 8, 19, 23, 34, 38, 42, 53], [4, 9, 14, 26, 31, 36,
48, 54], [4, 10, 16, 22, 28, 41, 47, 55], [4, 11, 18, 25, 32, 39, 46,
49], [4, 12, 20, 21, 29, 37, 45, 50], [4, 13, 15, 24, 33, 35, 44, 51],
[5, 7, 16, 25, 34, 36, 45, 51], [5, 8, 18, 21, 31, 41, 44, 52], [5, 9,
20, 24, 28, 39, 43, 53], [5, 10, 15, 27, 32, 37, 42, 54], [5, 11, 17,
23, 29, 35, 48, 55], [5, 12, 19, 26, 33, 40, 47, 49], [5, 13, 14, 22,
30, 38, 46, 50], [6, 7, 15, 23, 31, 39, 47, 50], [6, 8, 17, 26, 28, 37,
46, 51], [6, 9, 19, 22, 32, 35, 45, 52], [6, 10, 14, 25, 29, 40, 44,
53], [6, 11, 16, 21, 33, 38, 43, 54], [6, 12, 18, 24, 30, 36, 42, 55],
[6, 13, 20, 27, 34, 41, 48, 49], [7, 8, 9, 10, 11, 12, 13, 56], [14, 15,
16, 17, 18, 19, 20, 56], [21, 22, 23, 24, 25, 26, 27, 56], [28, 29, 30,
31, 32, 33, 34, 56], [35, 36, 37, 38, 39, 40, 41, 56], [42, 43, 44, 45,
46, 47, 48, 56], [49, 50, 51, 52, 53, 54, 55, 56]]>