FREE MOVES FIRST — SINGLE-CELL AND FORCED CAGES
Some cages require no work at all. Start with these every solve.
Single-cell cages: the cage target is the digit. Place it immediately.
Forced-result cages: some cage size + operator + target combinations have exactly one possible digit set in the grid size you're playing. In a 4×4 grid, a 2-cell cage of 4÷ must be {1, 4}. In a 6×6, a 3-cell cage of 6× must be {1, 2, 3}. Mark these on your first pass.
Pattern: extreme-low multiplication targets and extreme-high addition targets tend to lock the digit set. Division and subtraction cages are always pairs — see below.
SUBTRACTION AND DIVISION — ALWAYS PAIRS
The rules require subtraction and division cages to contain exactly two cells. This is a significant constraint.
A 2-cell subtraction cage of 3− in a 6×6 grid can only be one of: {1, 4}, {2, 5}, or {3, 6}. Three possible sets — better than six possible singles.
A 2-cell division cage of 3÷ can only be {1, 3} or {2, 6}. Two possible sets.
Cross-referencing with the row and column each cell belongs to usually reduces this to one set quickly — often within the first few moves.
ENUMERATION — LISTING CAGE CANDIDATES
For every cage that isn't immediately forced, list all possible digit sets that satisfy the cage target and operator, using each digit at most once within the cage.
Example: 6× in a 6×6 with 2 cells → possible sets: {1, 6}, {2, 3}. That's it — the digit 4 or 5 can't appear in either cell of that cage, regardless of the row/column state.
Write the union of all possible sets as candidates for each cell. Above: both cells get candidates {1, 2, 3, 6} — 4 and 5 are eliminated immediately, without touching the row or column.
ROW AND COLUMN ELIMINATION
Once you have cage candidates for each cell, apply the Latin-square rule: every digit must appear exactly once in each row and column.
If a digit already appears somewhere in a row, it's eliminated from every cage candidate in that row. Often this reduces a cell to one candidate and places it.
The intersection of cage elimination and row/column elimination is the core loop of Calcudoku. Neither alone solves most puzzles — together they almost always do.
THE SOLVING ORDER
- Place all single-cell cages immediately.
- Mark candidates for forced-set cages (subtraction, division, extreme-value addition/multiplication).
- Enumerate candidates for all remaining cages.
- Apply row and column elimination to all cage candidates.
- Place any cell with a single candidate remaining.
- Update all cages and rows/columns touching the placed cell.
- Go back to step 4 — each placement cascades.
This loop handles every easy and most medium Calcudoku grids. If you stall, look for a row or column where a digit can only go in one possible cage — the equivalent of a hidden single in Sudoku.
THE BEGINNER TRAP
Treating each cage independently and trying to solve it in isolation. Most cages aren't solvable alone — they need the row/column state to narrow the candidates. Skip the cage if you can't place it and come back once surrounding cells are filled in.
Calcudoku has exactly one solution and you can always reach it without guessing. If you feel stuck, the breakthrough is either an unapplied elimination from a placed digit or a cage candidate set you listed too broadly.