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# Steiner(5,6,12) playing cards: games needed

Ages ago, I owned a set of cards, made mid-1997, each with one hexad from a Steiner(5,6,12) system. That rather crude set had been laser printed onto white paper; laminated; and guillotined. It, and many odd socks, have disappeared into the ether.

What is such a system?
It has 132 members, each of which contains six letters from the first twelve of the alphabet (A … L), with the cute quality that if five letters are chosen, one card and only one card contains all five.
Self-evidently, this is the sort of thing that can excite mathematicians.
Games are needed.
What games work well with such a set of cards?

Designed and soon to be printed is a beautiful set of 144 cards (link has images).
It’s almost ready: doing one more test print, to choose better shades of red and blue.

• There are 132 Steiner(5,6,12) cards, in six suits of twenty-two cards. (No, not orbits of a subgroup of M₁₂; math.stackexchange question 3818982 helped with the split.)
• Eleven jokers: one of each suit; one of each colour; and two generic (one ‘light’ and one ‘dark’).
• One explanation card, asking that good games for these cards be published, tagged #SteinerKirkmanCards.

Obvious question: what good games can be played with such a deck?

The “what good games?” question is largely unconstrained. What follows are not requirements, merely mind food: facts and suggestions and possibilities.

• There are six suits, to which it is easy to assign an ordering: the circle is zero; one spike on the kite; two cheeks on the heart; three sides on the triangle; four arms on the cross; and five points on the star. Suits colours are black red blue blue red black.

• Each suit has 11 pairs of card-and-its-complement, arrangeable into a chain, with each card overlapping the cards of adjacent pairs in exactly three letters.

• Rules of any particular game could exclude all or some jokers.

• Rules of any particular game could exclude some suits.

• Jokers could be allowed to represent any hexad (of same suit or colour of joker, as appropriate), even if this deck does not include that hexad, or includes it but in a different suit or of a different colour.

• Or, much tougher on those who haven’t learnt the details, and somewhat inconvenient to enforce, jokers must represent an actual card.

• Or jokers may represent any visible card of matching qualities. (This has cute possibilities in poker variants: I won’t know how good my hand is until I can see what my joker-Blue might be.)

As an opening gambit, to encourage others to do better, perhaps a poker variant.
Four cards in a ‘hand’.
Best hand is of same suit (obviously), any two cards either being complements (sharing zero letters), or sharing exactly three letters (so the hand comprises adjacent pairs in a suit’s chain).
Obviously less structured would be worse, as would less consistency of suits or of colours.
Cards are revealed simultaneously, jokers then being chosen from visible non-jokers.
Perhaps there could be constrained swapping allowed: taking a car from ‘flop’, and exchanging it for a card of a different suit overlapping in exactly three letters.

For completeness, math.stackexchange question 3870810 asks how I can inform group theorists of the existence of this set of cards.