Calcudoku Solving Techniques

Five arguments, in the order to reach for them. The hardest one a square forces on you is that square’s difficulty — on this site, and in the code that measures it.

Ordered by what it costs a person to see

The ladder is an order to try things in rather than a set of tricks to collect, and its order is not the one a machine would choose. Listing the ways a two-cell cage can be filled is trivial for a computer and takes a person a moment of concentration; hunting a row for a digit with nowhere left to go is the reverse. This ladder is a model of attention, not of processor time, which is why the cage arithmetic sits below the sweeping.

Each rung is also made cheaper by the rungs beneath it having been exhausted first. A cage considered while all its cells are wide open may have eleven arrangements; the same cage after one more digit has gone into its row may have two. Anybody who describes this puzzle as an arithmetic grind is almost certainly doing the expensive rung while a free one was still available.

The order pays off in the grading as well. Because the solver always takes the cheapest move it can find, the hardest move it was ever forced into is a genuine statement about the board: it means nothing simpler existed at that instant, rather than that the program failed to notice something.

• Rung 1Free cells

  A one-cell cage is its own answer, and a row with one gap left has no choice about it.

• Rung 2Cage combinations

  List every way a small cage can be filled and keep whatever all of them agree on.

• Rung 3Singles

  A cell with one digit left, or a digit with one cell left in its row or column.

• Rung 4Unit totals

  Every row adds up to the same known number, so the cages inside it price the rest.

• Rung 5Cage interaction

  What a group of cells can hold between them, whether they share a cage or only a row.

• Rung 6Assumption

  Write a digit, follow it until something breaks, and rub it out again.

How a square gets its label

Every published board goes through the same solver the hint button uses. It applies the cheapest argument that fits, over and over, until every cell is written, and it records the highest rung it ever stood on. Easy stops at the free cells. Medium reaches the singles. Hard reaches the row totals. Expert reaches the argument that holds a cage and a line in mind together.

Nothing is graded by cage count and nothing is relabelled afterwards. The carve that cuts the cages is capped at the level’s hardest allowed argument and stops merging the moment the next merge would outrun it — so a board can come out easier than the level asked for and can never come out harder. When it comes out easier, it ships under the level it measured. Ask for expert on a 4×4 and a hard board is a perfectly ordinary answer.

There is a sixth rung and nothing here is allowed to stand on it. Trial — write a digit, follow it until something breaks, rub it out — will finish any board whatsoever, which is exactly why it cannot be part of a difficulty scale. Its page explains what it is and why every board that needs it is discarded during generation.

Common questions

More Calcudoku pages

• CalcudokuUnlimited boards
• Daily calcudokuA new square every day
• Calcudoku rulesThree rules, shown
• Solving a squareA 6×6, worked
• Free cellsTechnique — easy
• Cage combinationsTechnique — easy
• SinglesTechnique — medium
• Row totalsTechnique — hard
• Cage interactionTechnique — expert
• Guessing at cagesTechnique — refused here too
• Printable calcudokuFor paper
• Calcudoku archiveEvery past square