How to play
Calcudoku is a number puzzle that mixes logic with mental arithmetic. A square grid of N×N cells has to be filled with the numbers 1 to N, so that each number appears exactly once in every row and every column. The grid is also split into outlined groups of cells called cages. Each cage shows a target and an operation in its top-left corner; the numbers in the cage must produce that target with that operation.
A cage marked 11+ adds up to 11. A cage marked 24× multiplies to 24. Subtraction and division only appear in cages of exactly two cells: 3− means the larger number minus the smaller one is 3, and 2÷ means the larger number divided by the smaller one is 2, in either order. A cage of one cell holds just the number shown, and it is filled in for you. A bent cage may contain the same number twice, as long as the two copies are not in the same row or column.
Pick a grid size from 4×4 to 9×9 and a difficulty, then press Start. The clock stops while the game is paused or the tab is hidden, and your progress is saved in this browser.
Controls
- Mouse: click a cell, then type a number or click it on the number pad.
- Touch: tap a cell, then tap a number on the pad under the grid. The page does not scroll while your finger is on the grid.
- Keyboard: arrow keys move the selection, number keys fill the cell, Backspace or Delete clears it. N switches notes on and off, and Shift plus a number writes a note directly.
- Notes are pencil marks for the numbers a cell could still hold. A placed number removes the same note from its row and column.
- Undo and Redo (U and Y, or Ctrl+Z and Ctrl+Y) step through every change.
- Check (C) marks every wrong number with a slash. Hint (H) fills in one correct number, in the selected cell unless that one is already right. P pauses and hides the grid, F switches to full screen.
- Cage combinations (K, or the bar under the buttons) opens a panel with every set of numbers that makes the target of the selected cage. A set is struck out once it no longer fits the numbers already in the cage, its rows and its columns. On a narrow screen the sets sit in one row that you swipe sideways, with the possible ones first.
A number that repeats in its row or column is shown red and underlined, and a full cage that misses its target gets its label crossed out.
How it works
Without the cages, the grid is a Latin square: an N×N array in which every symbol occurs once per row and once per column. There are 576 different Latin squares of size 4×4 and about 5.5 × 1027 of size 9×9, so the rows and columns alone are far from enough. The cages add arithmetic clues until only one filling is left.
The basic tool is listing the combinations a cage allows. In a 6×6 grid, a two-cell cage 11+ can only be 5 and 6. A three-cell cage in one row marked 6+ must hold 1, 2 and 3, so none of those numbers can appear elsewhere in that row. For a product cage, factor the target:
The Cage combinations panel does this listing for the selected cage. A number can appear twice in a set only when the cage bends, so that the two copies sit in different rows and columns: a three-cell 12× cage shaped like an L in a 6×6 grid allows {2, 2, 3} as well as {1, 2, 6} and {1, 3, 4}, a straight one only {1, 2, 6} and {1, 3, 4}.
Then the usual grid logic applies. If a number can go in only one cell of a row, it goes there. If two cells of a column can only hold the same two numbers, no other cell of that column can use them. Hard puzzles need these pair and row-and-cage deductions, while Easy puzzles fall to cage combinations and single candidates.
All puzzles were generated in advance and checked by two separate solvers: one proves that each has exactly one solution, the other that it can be solved by deduction alone, without guessing. Easy grids use cages of at most three cells and a few givens. Medium grids use cages of up to four cells. Hard grids have no givens (a 4×4 may have one) and cages of up to four cells, or five on 7×7 and larger. Each puzzle can also appear rotated or mirrored, which keeps its single solution.
Where to start
- Start with single-cell cages and cages that allow only one combination, such as 3+ in two cells (1 and 2).
- Look for a large product with few cells: 60× in three cells of a 6×6 grid must be {3, 4, 5} or {2, 5, 6}, so it always contains a 5.
- Use the row sum. Every row of an N×N grid adds up to N(N+1)/2, which is 21 for 6×6. If cages cover a row except one cell, that cell is 21 minus their totals.
- Play a 4×4 Easy first to learn the cage symbols, then step up one size at a time.
- Solve a puzzle without Check or Hint and try to beat the best time on the start card.
Frequently asked questions
Can a number repeat inside a cage?
Yes, if the cage bends so that the two cells sit in different rows and different columns. Rows and columns never repeat a number.
Which way round do subtraction and division work?
Always larger first. A 2÷ cage can hold 3 and 6 in either order, and a 1− cage any two neighbouring numbers.