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.
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 1 2 3 4 5 6 1 cage adding to 19123456 123456 123456 123456 cage adding to 16123456 single cell, 2123456 2 123456 cage adding to 19123456 123456 123456 123456 123456 3 cage adding to 14123456 123456 123456 123456 123456 single cell, 3123456 4 123456 cage adding to 17123456 123456 single cell, 6123456 123456 cage adding to 10123456 5 123456 cage multiplying to 18123456 123456 cage with a difference of 4123456 123456 123456 6 123456 123456 123456 123456 cage with a difference of 4123456 123456 A moment later 1 2 3 4 5 6 1 cage adding to 1913456 13456 13456 13456 cage adding to 1613456 single cell, 22 2 123456 cage adding to 19123456 123456 123456 123456 13456 3 cage adding to 14123456 123456 123456 123456 123456 single cell, 313456 4 123456 cage adding to 17123456 123456 single cell, 6123456 123456 cage adding to 1013456 5 123456 cage multiplying to 18123456 123456 cage with a difference of 4123456 123456 13456 6 123456 123456 123456 123456 cage with a difference of 4123456 13456 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 1 2 3 4 5 6 1 cage adding to 1913456 13456 13456 1345 cage adding to 1613456 single cell, 22 2 123456 cage adding to 19123456 123456 12345 123456 1456 3 cage adding to 1412456 12456 12456 1245 12456 single cell, 33 4 12345 cage adding to 1712345 12345 single cell, 66 12345 cage adding to 10145 5 123456 cage multiplying to 18123456 123456 cage with a difference of 412345 123456 1456 6 123456 123456 123456 12345 cage with a difference of 4123456 1456 A moment later 1 2 3 4 5 6 1 cage adding to 1913456 13456 13456 1345 cage adding to 1613456 single cell, 22 2 123456 cage adding to 19123456 123456 12345 123456 15 3 cage adding to 1412456 12456 12456 1245 12456 single cell, 33 4 1235 cage adding to 171235 1235 single cell, 66 1235 cage adding to 104 5 12345 cage multiplying to 1812345 12345 cage with a difference of 412345 12345 6 6 123456 123456 123456 12345 cage with a difference of 4123456 15 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 1 2 3 4 5 6 1 cage adding to 1913456 13456 13456 1345 cage adding to 1613456 single cell, 22 2 123456 cage adding to 19123456 123456 12345 123456 15 3 cage adding to 1412456 12456 12456 1245 12456 single cell, 33 4 1235 cage adding to 171235 1235 single cell, 66 1235 cage adding to 104 5 12345 cage multiplying to 1813 12345 cage with a difference of 415 15 6 6 136 1236 123456 12345 cage with a difference of 415 15 A moment later 1 2 3 4 5 6 1 cage adding to 1913456 1456 13456 1345 cage adding to 1613456 single cell, 22 2 123456 cage adding to 1912456 123456 12345 123456 15 3 cage adding to 1412456 12456 12456 1245 12456 single cell, 33 4 1235 cage adding to 17125 1235 single cell, 66 1235 cage adding to 104 5 24 cage multiplying to 183 24 cage with a difference of 415 15 6 6 136 126 123456 12345 cage with a difference of 415 15 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 1 2 3 4 5 6 1 cage adding to 1913456 1456 13456 1345 cage adding to 1613456 single cell, 22 2 123456 cage adding to 1912456 123456 12345 123456 15 3 cage adding to 1412456 12456 12456 1245 12456 single cell, 33 4 1235 cage adding to 17125 1235 single cell, 66 1235 cage adding to 104 5 24 cage multiplying to 183 24 cage with a difference of 415 15 6 6 136 126 123456 12345 cage with a difference of 415 15 A moment later 1 2 3 4 5 6 1 cage adding to 1913456 1456 13456 1345 cage adding to 1613456 single cell, 22 2 123456 cage adding to 1912456 123456 12345 123456 15 3 cage adding to 1412456 12456 12456 1245 12456 single cell, 33 4 1235 cage adding to 17125 1235 single cell, 66 1235 cage adding to 104 5 24 cage multiplying to 183 24 cage with a difference of 415 15 6 6 136 126 3456 2345 cage with a difference of 415 15 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 1 2 3 4 5 6 1 cage adding to 191456 1456 13456 1345 cage adding to 1613456 single cell, 22 2 12456 cage adding to 191456 123456 12345 123456 15 3 cage adding to 1412456 1456 12456 1245 12456 single cell, 33 4 125 cage adding to 1715 1235 single cell, 66 1235 cage adding to 104 5 24 cage multiplying to 183 24 cage with a difference of 415 15 6 6 3 2 46 cage with a difference of 415 15 A moment later 1 2 3 4 5 6 1 cage adding to 191456 1456 13456 135 cage adding to 1613456 single cell, 22 2 12456 cage adding to 191456 123456 1235 123456 15 3 cage adding to 1412456 1456 12456 125 12456 single cell, 33 4 125 cage adding to 1715 1235 single cell, 66 1235 cage adding to 104 5 24 cage multiplying to 183 24 cage with a difference of 415 15 6 6 3 2 4 cage with a difference of 415 15
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.
| cage adding to 194 | 5 | 1 | 3 | cage adding to 166 | single cell, 22 |
| 6 | cage adding to 194 | 3 | 5 | 2 | 1 |
| cage adding to 141 | 6 | 5 | 2 | 4 | single cell, 33 |
| 5 | cage adding to 171 | 2 | single cell, 66 | 3 | cage adding to 104 |
| 2 | cage multiplying to 183 | 4 | cage with a difference of 41 | 5 | 6 |
| 3 | 2 | 6 | 4 | cage with a difference of 41 | 5 |
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
- CalcudokuUnlimited boards
- Daily calcudokuA new square every day
- Calcudoku rulesThree rules, shown
- Calcudoku techniquesThe whole ladder
- 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