Number Slide Puzzle

This board has been mapped, position by position

Eight tiles and a gap can sit in 362,880 arrangements. Half of them cannot be reached from a finished board by sliding, which leaves 181,440 that can – few enough that a computer can walk outwards from the finished position and record the exact distance of every last one of them in a couple of seconds. That produces numbers to rely on rather than estimate:

• the worst positions are 31 moves from home, and there are exactly two of them
• the average across all 181,440 positions is a little under 22 moves
• the commonest distance is 24 moves, shared by 24,047 positions
• only about 40 per cent of positions are 21 moves or closer

A scramble that takes you thirty-odd moves is therefore not a poor performance. It is close to the hardest thing three rows are capable of asking, and there are precisely two arrangements in existence that could have asked more.

Every hint here is a shortest move

On a larger frame, proving a route is the shortest one costs enough that the board gives up and follows a longer plan instead. Three rows are small enough that the search always finishes, so the tile wearing the dashed outline is always the opening move of a shortest solution and the count in brackets is exact. Follow it and the count falls by one every single time – a scrambled board played on hints alone went 22, 21, 20 and on down to 1 without once moving sideways. That is the difference between a hint and a nudge: this one is answerable for the number it prints.

Three rows is one row, one column and a spin

The layered method is the one the four-row frame needs, worked out there tile by tile, and on three rows it is over almost before it has begun: tile 1 into its corner, then 2 and 3 brought home together in the paired manoeuvre rather than one after the other, then 4 and 7 the same way down the left column. That is the whole of the planned part of a three-row solve. Two moves, and the live part of the frame is four cells out of the original nine.

What is left is three tiles in a two-by-two box, and the proportions are what make this board worth practising on. On four rows that rotation is the last handful of moves in a long solve and nobody thinks about it. Here it is most of the endgame, and it is where beginners give up, because a rotation that is not yet helping looks identical to a position that cannot be finished. It is not stuck; three tiles in a two-by-two box can always be brought round by turning consistently in one direction, and the other direction is sometimes shorter and never necessary. If it feels stuck you are changing your mind about which way to turn.

The arrow keys name the tile, not the gap

Press the right arrow and the tile to the left of the gap moves right into it. Both readings of an arrow key are defensible – you could just as fairly say the gap went left – but the thing that visibly moves goes the way the key points, and that is the one your eye is already following. The frame is a single tab stop rather than nine separate buttons, and the focus marker travels with the tile you have just moved, so the next press concerns the part of the board you are looking at. The Picture button cuts those same eight tiles out of a drawing built inside the page instead of numbering them; across three rows the slices are large enough to read, and each one keeps its number in the corner regardless.