Royal Ladder

How to play

1. Set your entry and press deal. Five cards are dealt from a fresh pack.
2. Tap any card you want to keep. Kept cards are outlined and marked.
3. Press replace. Every card you did not keep is exchanged once, and the final five are read against the pay table printed beside the board.
4. The pay table is fixed and is the published one for this variant. It is the whole of the game: the only decision is which cards to keep, and that decision is what separates the published long-run return from a worse one.

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.

Standard 9/6

The formula

What to keep, in order

Keep the highest entry your five cards contain · 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.

1. Top Run, Suited Run or Four Alike
2. Four cards to a Top Run
3. Three and Two, Same Suit or Run
4. Three Alike
5. Four cards to a Suited Run
6. Two Pairs
7. A high pair
8. Four cards to a Same Suit
9. Three cards to a Top Run
10. Four cards to an open Run
11. Two suited high cards
12. Three cards to a Suited Run
13. Two unsuited high cards, the lowest two
14. A suited ten with one high card
15. One high card
16. Keep nothing

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.

Other boards in this family

Open the board Scribe’s Cards