There is a great video by Matt Parker, explaining the maths behind the game Dobble. If you haven’t checked it out yet, please do it now, its great fun.
The Dobble cards form a finite projective plane of order 7, and Matt visualize the plane by laying out the cards in a 7×7 square.
The only bad thing about the video is that he does not explain how to create such a satisfying square.
At first I thought ”How hard can it be?”, but it turned out to be way harder than I initially expected.
So here it is: The Dobble Square for dummies!
Select two random cards, one to be your column card and the other to be you row card. The symbols on the column card will form the vertical rows of the square, and the symbol on the row card will form the horizontal lines.
Note the symbol the two cards have in common, in my example the snowman, and take away all other cards containing that symbol. This symbol will represent the infinity line.
Pick a random card from the deck (after you put the snowmen away) to start with. Compare the card with the column card to find the symbol of the first column (sunglasses) and compare it with the row card to find the symbol of the first row (lips). Find all the other cards with lips and sunglasses and sort them out from the deck, and place them at the end of the first row and the firs column.
Now select a random card from the first row (lips) and the first column (sunglasses) and place them next to the first card. This will be the final random step, every card from now on has a decided place in the sqare.
By comparing the random cards with the column and row card, you will find the symbol of the second column (haircross) and second row (zebra).
You have now only one option for the fourth card: There is only one card containing the symbol of the second row and second column (the haircross-zebra card).
Find all the other cards with these symbols (zebra cards and haircross-cards) and place them at the end of the row and column.
Now, return to card 2 and card 3 and find the common symbol (scissors). Find a card in the pile of infinity line cards (snowman cards) that contain the symbol.
This card will be you diagonal card, and all the symbol will be found on lines going diagonally across the square.
When you have found the diagonal card, you can compare it to card 4 to find the symbol of the second diagonal (ghost).
Look for this symbol in the cards of the first row and first column, to find card 5 (next card in first row) and card 6 (next card in the first column).
By comparing card 5 with the row card you will find the symbol of the third column (spider), and by comparing card 6 with the column card you will find the symbol of the third row (pencil).
Complete the rows and columns by looking in the piles you already have, and sort out all cards containing these symbols from the deck. Not that there is only one card containing both symbols (pencil-spider) and that goes in the lower right corner of the 3×3 square that you now have formed.
Find the symbol of the third diagonal (four leaf clover), and add another card to the first row and first column.
Identify the symbol of the fourth column (lighning) and fourth row (clock) by comparing the new cards to the row card and column card. Complete the 4×4 square and find all cards with the new symbols in the deck.
Repeat again: Find the symbol for the next diagonal (g-clef) and continue first row and column.
Use the row card and column card to find the symbol of column 5 (anchor) and row 5 (dragon). Fill in the square.
Use the diagonal again to expand the first row and column one step more, and since you only have one card left you will also know the card of the last column and row.
Again, use the column and row cards to find the symbol of column 6 (clown), column 7 (stopsign), row 6 (bird) and row 7 (snowflake).
Take a moment to enjoy your Dobble sqare, and then make sure to watch Maths video for the details about all other diagonals.
Note that the selection of the snowman as the infinity line symbol was not as random at it seems. It was selected so that the two missing cards (again, watch Matts video!) did not ended up in the square!
