Accordion solitaire rules

A worked example

Here is the row from the sketch, played out card by card:

1. 7♠ 4♥ Q♦ 9♥ 4♣: The 4♣ can go next door onto the 9♥? No: different suit and rank. Three to the left is the 4♥: same rank. Move it.
2. 7♠ 4♣ Q♦ 9♥: The 4♣ now sits on the 4♥, and the row closes up. Nothing else matches, so deal the next card: 9♠.
3. 7♠ 4♣ Q♦ 9♥ 9♠: 9♠ matches its neighbor 9♥ by rank. Move it onto the 9♥.
4. 7♠ 4♣ Q♦ 9♠: Deal the Q♣. Next door is the 9♠ (no match). Three to the left is the 4♣: same suit.
5. 7♠ Q♣ Q♦ 9♠: Now the Q♦ and the Q♣ are neighbors with the same rank, so the Q♦ pile can jump onto the Q♣ pile. One deal, two extra moves.
6. 7♠ Q♦ 9♠: Seven cards dealt, three piles left. (One step earlier, the 9♠ also had a choice: three places left was the 7♠, a suit match.) That chain reaction is what Accordion is about.

Notice that the 4♥ and 9♥ are still in the game, hidden under their piles. Only the top card of each pile counts for matching, so every move also decides which card stays visible.

Rules in detail

Use one deck without jokers. The traditional way is to deal one card at a time to the right end of the row, making any moves you like before the next card. Many players now lay all 52 cards out face up first and then start squeezing. The Wikipedia article notes both; the second is easier because you can plan around cards you would otherwise not have seen yet.

A move takes a pile (one card or many) and puts it on top of another pile, either its immediate neighbor on the left or the pile exactly three places left. The two top cards must share a suit or a rank. Count piles, not cards, when you count three. After the move, slide everything to the right of the gap over by one.

You are never forced to move, and when two moves are possible you choose. The game ends when the deck is dealt and no more moves are possible (or none you want to make).

Since one pile is so rare, many players keep score instead: the fewer piles left, the better. The bridge writer Alfred Sheinwold suggested counting five piles or fewer as a win (Wikipedia).

How winnable is Accordion?

Here the two kinds of figures are far apart, and the gap is interesting.

• With the whole deck laid out and perfect planning: about 99.9995% of deals can be squeezed into one pile, according to Blake and Gent's solver study (arXiv 1906.12314). Nearly every deal has a winning order of moves.
• In real play: the Wikipedia article quotes an estimate of about 1 game in 100.

The difference is search. Each move changes which cards are neighbors, so the number of possible move orders is enormous, and a computer can try them all where a person cannot. If you deal one card at a time, you also cannot plan for cards you have not seen.

Strategy

1. Look at what becomes neighbors. Every move closes a gap and creates a new pair of neighbors (and a new "three to the left" pair). Before moving, check whether the move sets up a follow-on match or breaks one.
2. Choose which card stays on top. When you can move A onto B or B onto something else, think about which top card is easier to match later. A card in a suit with many cards still to come is a better top than one whose suit is nearly used up.
3. Do not grab every match. Moves are optional. Sometimes holding a pile back keeps it in position for a three-left jump that clears two piles at once.
4. Lay the whole deck out if you want to win. Dealing all 52 first is allowed in modern rules and turns Accordion into a planning puzzle instead of a guess.