The Last Tile

Every placement in Murdoku, in the end, comes down to the same moment: a suspect who could once have stood in a dozen places has exactly one tile left. Nothing else is possible, so that is where they were. Solvers call it a naked single; in this game it is simply the last tile.

It sounds too obvious to need a name, and in a sense it is. The interesting part is everything that leads to it. No clue in Murdoku names a tile. A clue says “beside a window” or “north of Dana” or “in a room with no other woman”, and each of those leaves an area, not a point. The last tile is what you get when enough areas have been cut away from one person that only a single cell survives the cuts.

Keep a Count for Every Suspect

The habit that makes the last tile appear early is counting. When you first read a case, work out for each suspect how many tiles their own clues leave them. On a good case those numbers vary wildly — one suspect might have forty possibilities, another three. The suspect with three is where the case begins, because every tile you cross out for any reason has a far bigger effect on them.

Then, whenever you cross something out — because a row is taken, because a room is full, because two people cannot share a line — check whether it touched one of the short lists. Most of the time a placement is not found; it is noticed, because a list you were watching ran out.

What a Placement Gives Back

A placement is never only about the person placed. Because every person has a row and a column of their own, putting someone on a tile clears that whole row and that whole column for everybody else. On a nine-by-nine board that is seventeen tiles gone at once, and those tiles are very often the ones that were keeping another suspect’s list long.

This is why the game crosses out the row and column for you when you place someone: it is the single largest elimination on the board, and it happens every time. Look at the cleared lines straight away, before you read another clue. More often than not, the next last tile is sitting in them.

When It Does Not Come

On harder cases the last tile is deliberately held back. A well-made hard case lets you cross out a great deal while nobody is yet certain, so that the first placement comes many steps in. If you have been working for a while and nothing has dropped to one, that is the case doing its job, not you missing something. Switch from people to lines: the next three techniques are about rows and columns, and they are usually what breaks the deadlock.

A Worked Example

From the case Tutorial — 4×4, Tutorial, House & Home.

C1C2C3C4
R1T
R2Tv
R3B
R4BP
The floor plan of “Tutorial”. Letters mark objects: B Bed, P Plant, T Table, Tv TV.

The clues in play:

This step comes after the first suspects are placed, so it reveals part of the solution — skip it if you want to solve “Tutorial” cold.

At step 4 of the solve, the reasoning reads:

Carmen must be on R3/C4 – the only possible cell.

Cases That Need It

142 of the 142 cases in the casebook use this technique at least once. A selection, smallest board first:

Other Techniques

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