Par and hints

A solution is not the shortest solution

Level 18 can be solved in five moves, as Working backwards shows. It can also be solved in six:

Solved in 6 moves. Par is 5.

The extra move is C. It slid down one square as soon as F had left its column, then had to slide again once G had moved. If G had gone first, C would have dropped three squares in one move. A slide counts as one move however far the vehicle travels, so the order of the moves matters more than their length.

Stars follow from the gap to par. Three stars need par exactly. Two stars allow the larger of two spare moves and 30% of par, rounded. One star is any other solve.

Wasted moves tend to have one of three shapes, and they are worth checking for after every solve:

• The same vehicle twice in a row. Two slides of one vehicle with nothing else moving in between can always be replaced by one slide, so no shortest solution contains them. In level 1, sliding C, then D, then the hero one square and then the hero again takes four moves. The hero’s two slides merge, and par is 3.
• A vehicle moved part of the way, then again later. The six-move line above. Move the deepest link first and each vehicle moves once.
• A vehicle that was never in a chain. Moving it changes nothing the hero needs.

How the game knows par

Par is not an estimate. The solver starts from the board as it is, tries every legal slide, then every legal slide from each of those positions, layer by layer, and never visits the same position twice. The first layer that contains a solved position gives the fewest moves. If the search finishes without finding anything shorter, nothing shorter exists.

The search is small enough to run in a moment. Level 1 has 56 reachable positions, level 18 has 137 and level 26 has 91. The largest level in the game has 37,740, which is more than anyone can try by hand and trivial for a computer.

Because every slide can be undone, the reachable positions form a web that you can walk in either direction, and par is the distance from the start to the nearest solved position in it. The level generator picks, from each web, the position farthest from every solution (how the levels are made has the details). All 200 levels start at such a point. In practice that means no first move can take you farther from a solution than you already are. A level offers about six first moves on average, and 2.9 of them, about half, begin a shortest solution. In 25 levels exactly one first move does.

What a hint is, and how to spend one

A hint lights up one vehicle and shows where to slide it. It is the first move of a shortest solution from the board as it stands now, not from the start. The game solves the position again for every hint, so a hint is right after any number of wrong turns. It also has no plan to show you: it knows the next move and nothing beyond.

• You start with 3 hints and each hint costs one.
• When they run out, a short video you choose to watch adds 3. In the apps you can also buy hint packs, or unlimited hints.
• Challenges have no hints, so their leaderboards stay fair.
• A hint never changes par or stars: the game keeps only your moves. One achievement does care, Unaided, for solving a level of 30 moves or more in par without a hint.

Used well, a hint is a check on your reading, not a replacement for it.

• Make a plan first. With the chain worked out, as in Reading a jam, a hint tells you whether your first move is right.
• Ask why that vehicle. The vehicle a hint names is the next link of the chain. Work out what it frees, and the rest of the chain usually follows.
• It may take your last move back. If that move made things worse, the shortest way on can begin by undoing it.
• Remember what it costs. Following hints from the start spends one per move of par. Three free hints are a nudge, not a solution.