Exactly-One Symbol Card Deck Generator

In a symbol-matching card game any two cards have one symbol in common, and the first to spot it wins. Getting a deck right by hand is hard: with eight symbols a card, 57 cards need every one of their 1,596 pairs to share exactly one symbol. This builds the deck from the mathematics that guarantees it — a finite projective plane — fills it with your own words or emoji, and checks every pair before you print.

Symbols on each card

Cards

2 to 13. Any subset of the full deck keeps the exactly-one rule.

Deck code (optional)

Leave empty for a fresh one. The same code and settings make the same result again while the generator is unchanged. The code is a seed: deterministic, not cryptographic, and not a secret.

Symbols, one per line

Words or emoji. 13 different ones are needed for this card size; extras are left out at random. They stay in your browser.

Paper size

Print size

Margin

Millimetres of blank edge, 5 to 40.

Turn the paper sideways

How to print this, or save it as a PDF

There is no PDF to download, and there does not need to be. Open the print command in your browser and choose “Save as PDF” as the destination. Your browser decides whether the page address and date print across the top and bottom; if they appear, turn off “Headers and footers” in the print dialog.
Set the scale to 100%, not “fit to page”. This sheet is already the exact size of the paper you chose, so anything that resizes it makes every measurement on it wrong.
Leave the paper size in the dialog set to the same one you chose here.

The cards will appear here.

What this generator does

Builds the deck from a finite projective plane of order 2, 3, 5 or 7, where each pair of cards shares exactly one symbol, then places your symbols on it. The memory matching cards are pairs to find face down; here every card matches every other.

How to use this tool

Choose how many symbols go on each card; the most cards that size allows is filled in.
Type your own symbols one per line, or keep the list given, and press Make the deck.
Print the cards, cut them out and play: turn two over and race to name the shared symbol.

Understanding the controls

Symbols on each card: 3, 4, 6 or 8 symbols a card, allowing up to 7, 13, 31 or 57 cards; each needs that many different symbols.
Cards: Any number up to the full deck. Leaving cards out never breaks the rule: the remaining cards still share exactly one symbol pairwise.
Symbols: Your words or emoji, one per line. Duplicates are removed; if there are more than needed, a random choice is used.
Deck code: Optional. The same code, settings and symbols make the same deck again while the generator is unchanged; the code is a seed, not a secret.

Worked examples

Code harbour, 4 symbols a card, 13 cards, the given list: First card: spoon, sun, zip, snail. Second card: lamp, teapot, flower, snail — they share snail, and nothing else.
8 symbols a card: 57 cards from 57 symbols: 1,596 pairs, each sharing exactly one.

Common use cases

A vocabulary game with words a class is learning
A party game with in-jokes or names
A maths-club demonstration of a projective plane

How this generator works

For a prime n, the plane has n² + n + 1 points and as many lines, n + 1 points on every line, and any two lines meeting in exactly one point. Cards are lines and symbols are points: the cards are the lines y = mx + c (mod n) for every slope m and intercept c, each with a point at infinity for its slope; the vertical lines x = constant with their own point at infinity; and one line made of all the points at infinity. Your symbols are assigned to the points at random, the cards shuffled, and the symbols on each card shuffled. Afterwards every pair of cards is compared directly by set intersection, without the construction, and the deck is shown only if each pair shares exactly one symbol and no card repeats a symbol.

Randomness and fairness

The choice of symbols, the card order and the layout come from the browser's secure generator, or a seeded one when you give a code. A code repeats a deck while the generator is unchanged. Codes are deterministic, not cryptographic.

For how randomness is produced across the whole site, see how Generate Random works.

Assumptions this tool makes

Orders are primes (2, 3, 5, 7); 6 symbols a card means order 5.
Words are shown as text; emoji print as your system draws them.

Limitations and good to know

Symbols are words or emoji placed in a circle, not pictures; no artwork is supplied.
Sizes are fixed by the mathematics: a full deck has 7, 13, 31 or 57 cards.
Emoji appearance depends on the printer's fonts.

Common mistakes

Adding a hand-made card to a printed deck: A new card almost always shares two symbols or none with some card. Make the deck again with more cards instead.
Using near-identical words: Symbols that look alike (cat, cart) slow the game down for the wrong reason.

Practical tips

Six symbols a card (31 cards) is a good size for younger players.
Print on card or thick paper so the symbols do not show through.

Privacy and your data

Decks are built in your browser. With analytics consent, a count records how many cards a deck had; your symbols and any code are never sent.