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Puzzle Quarry

How to Solve a Calcudoku

One real 6×6 with twelve cages, from an empty square to a finished one. Every move below is a move the solver made, in the order it made it.

The order that finishes squares

Start by writing in every single-cell cage, then sweep the board for cages with only one or two arrangements. A two-cell cage with a large sum or a large product on a small board is usually pinned outright: on a six, 11+ can only be five and six, and 30× can only be five and six as well. These cost nothing to spot and they are what opens a board.

Then, and only then, start crossing digits out. Every digit you write removes itself from the rest of its row and the rest of its column, and the value of doing that bookkeeping honestly is that it makes the next round of cage arithmetic much shorter. Solvers who skip it end up re-listing the same cage combinations four times and calling the puzzle tedious.

Adding a row up is a slower move and belongs later. It is genuinely worth doing when a run of free sweeps has produced nothing, because it uses information no cage carries — the fixed total of a complete line — but doing it early means computing sums that a free sweep was about to make irrelevant. The rhythm that finishes boards is: copy the given cages in, work the tight ones, cross out, sweep again, and only then reach for the arithmetic.

One square, five arguments

The square below is a 6×6 with twelve cages, graded expert — chosen not because a beginner should start there but because its opening happens to use all five arguments within a dozen moves, which no gentler board does. The five positions shown are the five moments where something new is being used; the twenty-eight moves in between are more of the same.

They do not arrive in ladder order, and that is worth expecting rather than being surprised by. The solver always takes the cheapest move available at that instant, so the sequence here runs first rung, second rung, fifth, fourth, third — the fifth-rung argument becomes available early on this board because two cages happen to sit inside a single row.

Read each pair of grids left to right. On the left is the position as it stood, with the cells the argument read marked and the cells it is about to decide outlined. On the right is the same position a moment later. Small digits in a cell are the candidates still standing there; watching one disappear is very often the whole content of a move, which is why they are drawn rather than left out.

  1. The cell in the top-right corner is a cage of its own, marked 2 with no sign beside it. That is the digit, it is written in without a moment’s thought, and it immediately takes the 2 out of the top row and the last column.

    The position
    123456
    1cage adding to 19 123456 123456 123456 123456cage adding to 16 123456single cell, 2 123456
    2 123456cage adding to 19 123456 123456 123456 123456 123456
    3cage adding to 14 123456 123456 123456 123456 123456single cell, 3 123456
    4 123456cage adding to 17 123456 123456single cell, 6 123456 123456cage adding to 10 123456
    5 123456cage multiplying to 18 123456 123456cage with a difference of 4 123456 123456 123456
    6 123456 123456 123456 123456cage with a difference of 4 123456 123456
    A moment later
    123456
    1cage adding to 19 13456 13456 13456 13456cage adding to 16 13456single cell, 22
    2 123456cage adding to 19 123456 123456 123456 123456 13456
    3cage adding to 14 123456 123456 123456 123456 123456single cell, 3 13456
    4 123456cage adding to 17 123456 123456single cell, 6 123456 123456cage adding to 10 13456
    5 123456cage multiplying to 18 123456 123456cage with a difference of 4 123456 123456 13456
    6 123456 123456 123456 123456cage with a difference of 4 123456 13456

    Free cells

  2. The 10+ cage down the right-hand side has two cells, so its digits are two of the six that add to ten: four and six. Row four already holds a 6 in its own single-cell cage, so the upper of the two cells cannot be the 6 — the pair is settled without either digit having been guessed at.

    The position
    123456
    1cage adding to 19 13456 13456 13456 1345cage adding to 16 13456single cell, 22
    2 123456cage adding to 19 123456 123456 12345 123456 1456
    3cage adding to 14 12456 12456 12456 1245 12456single cell, 33
    4 12345cage adding to 17 12345 12345single cell, 66 12345cage adding to 10 145
    5 123456cage multiplying to 18 123456 123456cage with a difference of 4 12345 123456 1456
    6 123456 123456 123456 12345cage with a difference of 4 123456 1456
    A moment later
    123456
    1cage adding to 19 13456 13456 13456 1345cage adding to 16 13456single cell, 22
    2 123456cage adding to 19 123456 123456 12345 123456 15
    3cage adding to 14 12456 12456 12456 1245 12456single cell, 33
    4 1235cage adding to 17 1235 1235single cell, 66 1235cage adding to 104
    5 12345cage multiplying to 18 12345 12345cage with a difference of 4 12345 123456
    6 123456 123456 123456 12345cage with a difference of 4 123456 15

    Cage combinations

  3. Row five now contains a 4− cage lying flat inside it, and that cage can only be a 1 and a 5. Both of those digits are therefore spoken for somewhere inside the cage, and every other cell of row five loses them — which leaves the cell in the second column with a single candidate.

    The position
    123456
    1cage adding to 19 13456 13456 13456 1345cage adding to 16 13456single cell, 22
    2 123456cage adding to 19 123456 123456 12345 123456 15
    3cage adding to 14 12456 12456 12456 1245 12456single cell, 33
    4 1235cage adding to 17 1235 1235single cell, 66 1235cage adding to 104
    5 12345cage multiplying to 18 13 12345cage with a difference of 4 15 156
    6 136 1236 123456 12345cage with a difference of 4 15 15
    A moment later
    123456
    1cage adding to 19 13456 1456 13456 1345cage adding to 16 13456single cell, 22
    2 123456cage adding to 19 12456 123456 12345 123456 15
    3cage adding to 14 12456 12456 12456 1245 12456single cell, 33
    4 1235cage adding to 17 125 1235single cell, 66 1235cage adding to 104
    5 24cage multiplying to 183 24cage with a difference of 4 15 156
    6 136 126 123456 12345cage with a difference of 4 15 15

    Cage interaction

  4. Nothing free is left, so the bottom row gets added up. It must total twenty-one, the cages lying in it account for a known share of that, and what remains prices the two cells belonging to a cage that reaches up out of the row. Nothing is written here; two cells simply lose several candidates each.

    The position
    123456
    1cage adding to 19 13456 1456 13456 1345cage adding to 16 13456single cell, 22
    2 123456cage adding to 19 12456 123456 12345 123456 15
    3cage adding to 14 12456 12456 12456 1245 12456single cell, 33
    4 1235cage adding to 17 125 1235single cell, 66 1235cage adding to 104
    5 24cage multiplying to 183 24cage with a difference of 4 15 156
    6 136 126 123456 12345cage with a difference of 4 15 15
    A moment later
    123456
    1cage adding to 19 13456 1456 13456 1345cage adding to 16 13456single cell, 22
    2 123456cage adding to 19 12456 123456 12345 123456 15
    3cage adding to 14 12456 12456 12456 1245 12456single cell, 33
    4 1235cage adding to 17 125 1235single cell, 66 1235cage adding to 104
    5 24cage multiplying to 183 24cage with a difference of 4 15 156
    6 136 126 3456 2345cage with a difference of 4 15 15

    Unit totals

  5. All that crossing-out has left one cell in the bottom row with exactly one candidate standing in it. No cage was consulted and no row was counted — the cell had run out of alternatives several moves ago and this is the first look that noticed.

    The position
    123456
    1cage adding to 19 1456 1456 13456 1345cage adding to 16 13456single cell, 22
    2 12456cage adding to 19 1456 123456 12345 123456 15
    3cage adding to 14 12456 1456 12456 1245 12456single cell, 33
    4 125cage adding to 17 15 1235single cell, 66 1235cage adding to 104
    5 24cage multiplying to 183 24cage with a difference of 4 15 156
    632 46 cage with a difference of 4 15 15
    A moment later
    123456
    1cage adding to 19 1456 1456 13456 135cage adding to 16 13456single cell, 22
    2 12456cage adding to 19 1456 123456 1235 123456 15
    3cage adding to 14 12456 1456 12456 125 12456single cell, 33
    4 125cage adding to 17 15 1235single cell, 66 1235cage adding to 104
    5 24cage multiplying to 183 24cage with a difference of 4 15 156
    632 4cage with a difference of 4 15 15

    Singles

what the reasoning read what it decides what it wrote

Twenty-eight more moves of the same five ideas, and every cage comes out exact. Here is the finished square.

The finished square
cage adding to 194513cage adding to 166single cell, 22
6cage adding to 1943521
cage adding to 1416524single cell, 33
5cage adding to 1712single cell, 663cage adding to 104
2cage multiplying to 1834cage with a difference of 4156
3264cage with a difference of 415

What to do with a stalled square

Re-sweep the cages nearest whatever you last wrote. By far the commonest cause of a stalled board at this level is a cage whose combinations you worked out five moves ago and have not revisited since a neighbouring digit went in. Cage arithmetic is not a one-time calculation; every digit written anywhere in a cage’s rows and columns shortens its list again.

If that fails, take a row and add it up. Find the cages lying wholly inside it, work out what they must contribute between them, and subtract from the row’s fixed total to price whatever is left over. A cage with all but one cell inside the row counts too, and that variant is the one that breaks most stuck boards, because it is the one nobody thinks to try.

And if a full sweep and a full round of row arithmetic both come up empty, look for a cage that traps a digit inside a line. What the board does not want, ever, is a guess — every square published here was checked to be finishable without one, so a board that will not move is a board with something still on it to find.

Common questions

What is the first move in a calcudoku?

Write in every single-cell cage, then look for two-cell cages whose target leaves only one pair — a sum or product near the top of the range for that board size usually has exactly one answer.

Should I write candidate digits in the cells?

Yes, once the obvious cages are done. Almost every argument above the second rung works by removing a candidate rather than by writing a digit, and a board with nowhere to record that makes you derive the same elimination repeatedly.

Why do I keep stalling two-thirds of the way through?

Usually because a cage worked out early has not been revisited. Its list of arrangements shrinks every time a digit is written in one of its rows or columns, and the shortened list is often what the board is waiting for.

More Calcudoku pages