Vizier’s Rule

How to play

1. Set your entry and press deal. You receive two cards and the house receives two, one of them face down.
2. Cards two through ten are worth their number, the jack, queen and king are worth ten, and an ace is worth eleven until that would take you past twenty-one, at which point it is worth one.
3. Twenty-one on the first two cards is a natural and pays three to two, unless the house also has one, in which case the round is a push.
4. While it is your turn you may draw another card, hold, double for one card only, or separate a pair into two hands. Doubling after separating is allowed on this board, and separated aces receive one card each and then stop.
5. When the house shows an ace you are offered cover for half your entry. Cover pays two to one if the house has a natural and is lost otherwise.
6. The house then draws until it reaches its stopping total, and each of your hands is settled against it independently.

The rules

Cards here carry numbers only, in four colour families, with no faces and no suits. On a higher-or-lower board you see one card and call the next; a correct call advances you a step and grows the running multiplier, a wrong call ends the round. You may collect at any point. On a memory board, turn two cards per move; a matching pair stays face up, and clearing the board within the move limit scores.

If this is your first board

Take the four-by-two memory board first. Eight cards, four pairs, five moves, and every mistake is visible immediately. On higher-or-lower, remember that collecting early is a real choice and usually the mathematically better one: the multiplier grows more slowly than the chance of the chain breaking.

Pay table

Every figure below is computed from this board's own settings, not typed in by hand.

The formula

Combinations and the response

This chart is the published response for this board's exact rule set. Following it produces the long-run return stated above; departing from it lowers it.

Hard totals

Soft totals (an ace counted as eleven)

Pairs

The maths behind it

On higher-or-lower with a top card of H, seeing a card of value v makes "higher" correct for (H − v) of the H values and "lower" correct for (v − 1). The best call is whichever side holds more, and against a low card that edge is large. Chaining calls multiplies the probabilities, which is why a five-step chain is far less likely than it feels: five calls at 70% each land together only about one time in six.

Where it comes from

Numbered cards are among the oldest portable randomisers, and the two mechanics here are the two oldest questions you can ask of them. Is the next one bigger or smaller than this one, and can you remember where you saw its twin. Both questions predate any formal theory of probability, and both were used as teaching devices long before they became pastimes.

If this is your first board

Take the four-by-two memory board first. Eight cards, four pairs, five moves, and every mistake is visible immediately. On higher-or-lower, remember that collecting early is a real choice and usually the mathematically better one: the multiplier grows more slowly than the chance of the chain breaking.