Mind the Gap

Full rules

1. Overview
2. Card values
3. The pyramid
4. Setup
5. Your turn
6. Dead ends
7. No talking
8. Winning and losing
9. Example
10. Difficulty
11. Tips for beginners
12. Variants

Overview

Mind the Gap is a cooperative puzzle for one to three players: you form one team and win or lose together. Together you build a pyramid of ten cards in four rows – four at the bottom, then three, two and one. Every card that rests on two others must be their difference: the higher card minus the lower one. The card at the top, the target, is turned up at the start, so the whole pyramid has to add up to it. The catch: each of you holds a few secret cards, and you may not talk about them. Every card you throw away costs a coin, and when the coins run out, time is short.

• Players: 1–3, best with 2.
• You need: one standard 52-card deck without jokers, and coins (up to 12, depending on the level – buttons or matches will do).

Mind the Gap is a new game written for this site and not yet playtested. It turns an old mathematical puzzle, the difference triangle, into a card game. The number of coins is the first thing to tune: if your games are won or lost too easily, change it before anything else (see "Difficulty").

Card values

Ace = 1, 2–10 count their number, Jack = 11, Queen = 12, King = 13. Suits don't matter at all.

The difference of two cards is the higher value minus the lower one. The difference of a 4 and a King is 13 − 4 = 9, so a 9 may rest on them. The difference of a 7 and a 3 is 4, whichever of the two lies on the left.

The pyramid

The pyramid has ten places in four rows. The target lies alone at the top. Below it lie two places, below those three, and at the bottom four. The rows are offset like the bricks of a wall, so every card above the bottom row rests on two cards: the one below it to the left and the one below it to the right.

The difference rule: every card that rests on two cards must be their difference. The target at the top is the difference of the two cards in the second row, each of those is the difference of two cards in the third row, and so on.

Two things follow from this rule:

• No two cards of the same rank side by side. Their difference would be 0, and there is no 0 card. So you may never place a card next to one of the same rank in its row.
• The higher the row, the smaller the cards. A difference can be at most 13 − 1 = 12, so the third row can hold at most a Queen, and the two cards in the second row usually are small. Kings always belong at the bottom, Queens nearly always.

An empty place is a gap. You don't have to build from the bottom up: a card may go into any gap, in any row, at any time – as long as every card that now rests on two cards is their difference. A card placed early doesn't break the rule, but it decides what its gaps need. If the target is 5 and a 3 lies in the second row, the other card of that row must be an 8: the only card whose difference with 3 is 5 (3 − 5 is not a card).

Setup

1. Shuffle the deck. Turn up its top card: if it is an Ace to 8, it is the target. If it is a 9 or higher, put it under the deck and turn up the next card, until you find an Ace to 8. Lay the target face up at the top of the table, with room below it for three more rows.
2. Deal the hands, one card at a time: 5 cards each with two players, 4 each with three, 7 if you play alone. Look at your own cards, but keep them hidden from your partners.
3. Put the rest of the deck face down beside the pyramid as the stock. Leave room next to it for a face-up discard pile.
4. Put the coins for your level in the middle: 6 on Normal (see "Difficulty"). They belong to the whole team.

Decide who begins; then take turns clockwise.

Your turn

On your turn you do exactly one of these:

• Place a card. Put one card from your hand face up into any gap of the pyramid, following the difference rule and never next to a card of the same rank. Then draw the top card of the stock. This costs nothing.
• Discard a card. Pay one coin (take it out of the game), put one card from your hand face up on the discard pile and draw the top card of the stock. If you can't or don't want to place a card, you must do this.
• Take a card back. Pay one coin and move any card of the pyramid except the target to the discard pile. You don't draw. Do this when the pyramid has become impossible to finish (see below) or when a card blocks every plan.
• Pass. Only if the coins are all gone and you can't place a card.

Spread the discard pile so that everyone can see all its cards: which ranks are used up is important.

If someone places a card that breaks the rules, it goes straight back into their hand and they take their turn again.

Dead ends

A pyramid can reach a point where it can never be finished:

• a gap would need a 0, or a card higher than a King (13), or
• a gap needs a rank whose four cards all lie in the pyramid or on the discard pile.

Example: The target is 4. In the third row lie a 9 on the left and a 2 in the middle, with a gap to the right. The card on the 9 and the 2 must be 7, and the target 4 then needs a 3 or an 11 next to the 7. A 3 is fine: the right-hand gap of the third row then needs a 5 (a 2 and a 5 have the difference 3). But suppose someone has put a Queen into the second row next to the gap for the 7: the Queen and a 7 have the difference 5, not 4, so the gap for the 7 can never be filled. The team must pay a coin and take the Queen back.

No talking

From Normal upwards you may not say anything about your hand or your plans (on Easy you may talk freely, see "Difficulty"): no "I've got a King", no "I need an Ace", no "put a 4 there", no pointing and no sighing when a card is placed. What your partner holds, you have to read from where they place their cards and from what they discard.

You may talk about the rules, whose turn it is and how many coins are left, and you may check each other's arithmetic – including what the pyramid already forces ("that gap must be a Queen now"). That is on the table for everyone to see.

Winning and losing

• You win together as soon as all ten places are filled and every card follows the difference rule.
• You lose when the pyramid can no longer be finished and there is no coin left to take a card back – or when all of you pass in a row.

If the stock runs out (which hardly ever happens), keep playing without drawing.

Example

Anna and Ben play on Normal, with 6 coins. The target is the 5♣. Anna holds 3♥ 9♠ 4♣ 10♦ 7♠, Ben holds 8♦ 6♣ K♥ 2♠ J♣.

1. Anna places the 3♥ in the second row, on the left. Now the other card of that row must be an 8: 8 − 3 = 5. She draws.
2. Ben has the 8♦ and places it: the top is complete, 8 − 3 = 5. He draws.
3. Anna places the 9♠ in the middle of the third row. The 3 above it on the left needs a 6 or a 12 beside the 9; the 8 on the right needs a 1 beside it (17 is not a card).
4. Ben places the 6♣ on the left of the third row: 9 − 6 = 3. He draws.
5. Anna places the 4♣ second from the left in the bottom row. Now the corner to its left must be a 10 (10 − 4 = 6), and the place to its right must be a King (13 − 4 = 9). She holds the 10 herself.
6. Ben has understood – and holds the K♥: he places it right of the 4.
7. Anna places the 10♦ in the left corner and draws the Q♠. Two gaps are left: the right end of the third row must be an Ace (9 − 1 = 8), and the right corner at the bottom must be a Queen (13 − 12 = 1).

8. Ben holds neither an Ace nor a Queen. He pays a coin, discards the J♣ and draws – the A♦.
9. Anna places the Q♠ in the right corner: 13 − 12 = 1 for the gap above it.
10. Ben places the A♦ into the last gap: K − Q = 1 below it, 9 − 1 = 8 above it. The pyramid is complete, with 5 coins left.

Difficulty

On Expert, no rank may appear twice in the pyramid, the target included – so the pyramid holds ten of the thirteen ranks. Turn up targets only from Ace to 7.

The budget. The pyramid needs nine cards. Placing is free, because you draw a new card each time; only discards and take-backs cost coins, and every coin buys exactly one new card. Two players on Normal see at most 10 + 8 + 6 = 24 cards before the last gap is filled. A brute-force count of all pyramids shows how tight that is: for a target of Ace there are 4588 different finished pyramids (by rank), for an 8 only 160, for a 10 just 8 – which is why the target stops at 8. Twenty-four random cards contain a complete pyramid for the target in about 95% of deals, 20 cards (Hard) in about 89%, and on Expert, where far fewer pyramids exist, 19 cards only in about half of all deals. So even a perfect team loses about one Normal deal in twenty: the right cards simply don't come.

In our simulation of 5000 deals per level, each player placed the card that kept the most finished pyramids possible (counting which ranks were still out there) and discarded only when nothing fitted. Two players won about 77% on Easy (simulated with open hands), 57% on Normal, 37% on Hard and 28% on Expert; three players won about 82%, 59%, 39% and 33%; one player alone about 73%, 53%, 34% and 25%. A high target is harder: on Normal, targets from Ace to 3 were won about 64% of the time, an 8 only about 43%. The simulated players never read each other's cards, but they were perfect at arithmetic. The game hasn't been playtested: if you win easily, take away a coin; if you keep losing, add one.

Tips for beginners

• Small cards are gold. The upper rows need small cards, and the bottom row takes the big ones. Aces, 2s and 3s are needed in almost every pyramid; a King fits only at the bottom, next to a card whose difference with it is needed above.
• Build from the top down. The two cards of the second row decide a lot. A small pair such as 2 and 7 for the target 5 leaves many ways to go on; 6 and 11 is risky, because an 11 needs an Ace and a Queen or a 2 and a King below it.
• Place, then read. Every card you place tells your partner what the gaps need now. If you place a card that forces a rank you hold yourself – like Anna's 4 in the example – you keep the plan in your own hands.
• Keep a gap with two chances. A gap at the end of the bottom row usually has two possible cards (the neighbour plus or minus the card above); a gap in the middle is forced by both sides. Leave the ends for last.
• Count the discards. When three 6s are gone, don't build a plan that needs two more.

Variants

All optional – agree before you start.

• Small pyramid: only three rows (six places, five cards to place) – a quick first game. In our simulation two players won about 77% on Normal.
• Jokers as 0: add both jokers to the deck. A joker counts 0: it may rest on two cards of the same rank, and a card resting on a joker and another card equals that other card. In our simulation this was slightly harder (about 53% on Normal), because jokers fit almost nowhere.
• Series: play five pyramids in a row. A won pyramid scores 1 point plus the coins left, a lost one 0. Try to beat your best total.