My kids have this fun game called Spot It! The game constraints (as best I can describe) are:
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:
What are the minimum number of different pictures required to meet these criteria and how would you determine this?
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:
Finite Projective Geometries
The axioms of projective (plane) geometry are slightly different than the Euclidean geometry:
Now, add "finite" into the soup and you have the question:
Can we have a geometry with just 2 points? With 3 points? With 4? With 7?
There are still open questions regarding this problem but we do know this:
Q
points, then Q = n^2 + n + 1
and n
is called the order
of the geometry.n+1
points in every line.n+1
lines.Total number of lines is also Q
.
And finally, if n
is prime, then there does exists a geometry of order n
.
What does that have to do with the puzzle, one may ask.
Put card
instead of point
and picture
instead of line
and the axioms become:
Now, lets take n=7
and we have the order-7
finite geometry with Q = 7^2 + 7 + 1
. That makes Q=57
lines (pictures) and Q=57
points (cards). I guess the puzzle makers decided that 55 is more round number than 57 and left 2 cards out.
We also get n+1 = 8
, so from every point (card), 8 lines pass (8 pictures appear) and every line (picture) has 8 points (appears in 8 cards).
Here's a representation of the most famous finite projective (order-2) plane (geometry) with 7 points, known as Fano Plane, copied from Noelle Evans - Finite Geometry Problem Page
I was thinking of creating an image that explain how the above order-2 plane could be made a similar puzzle with 7 cards and 7 pictures, but then a link from the math.exchange twin question has exactly such a diagram: Dobble-et-la-geometrie-finie