Rules
Calcudoku, also called Mathdoku, is a Latin square with a little arithmetic. There are no boxes. On an n×n grid every row and every column holds 1 to n once.
- One of each digit. Every row and every column holds 1 to n once. A 4×4 uses 1 to 4, a 6×6 uses 1 to 6, and a 7×7 uses 1 to 7. There are no boxes.
- Cages hit a target. Heavy borders mark the cages. The corner label is a target and an operator, or a bare digit when the cage is one cell. A + cage adds. A × cage multiplies. A − cage and a ÷ cage have two cells, and you always use the larger digit first.
- Repeats only on a diagonal. A digit may repeat inside a cage when the two cells share neither a row nor a column. A straight cage in one row cannot repeat.
- A one-cell cage is filled. It starts filled. You can select it. You cannot change it. The grid is finished when every row and column has each digit once and every cage hits its target.
The 3− is 1 and 4. The 24× is 4, 2, and 3. The 3÷ is 3 and 1. The 6+ along the top adds. The two givens stay bold ink, and their corner labels are hidden because the digit is the clue. A finished grid turns the other digits and the cage labels teal. The frame stays ink.
- Cage target
- Cage that hits its target
Tap a cell. Its row and column are tinted, and any cell with the same digit is a little darker. Type a digit or use the keypad. Erase, Backspace, Delete, or 0 clears it. Arrow keys move. Notes (the Notes button, orN) are pencil marks. Shift with a digit toggles one note without turning Notes on. When you place a digit, that digit leaves the notes in the rest of its row and column. It stays in the rest of the cage. A repeat there can be legal.
The line under the keypad names the selected cage and how many cells are still open: “11+ in three cells”, then “· 2 to fill”, “· done”, or “· check”. A given reads “Given 3”. It does not list the sets. On a 6×6, 3− is 1 and 4, 2 and 5, or 3 and 6, in either order.
A row or a column adds up to 10 on a 4×4, 15 on a 5×5, 21 on a 6×6, and 28 on a 7×7. You will not need that on an easy grid. Section 3 is the rule the button uses: one + cage, in one straight line, leaving one cell or two. The how-to on the play page subtracts every + cage in a nearly full row. That reading is broader. This article follows the button.
The same one-digit-per-line idea is what a Sudoku row and column ask for. This grid has cages instead of boxes. A Killer Sudoku cage is a sum only, digits do not repeat inside it, and the grid has boxes. Kakuro puts a target on a run of cells. The steps below are this puzzle’s.
First moves
Start with a cage that has only one set of digits, then a cell with one digit left, then a digit with one home in its row or column.
One way to fill a cage
Start with a cage that can be made in only one way. The order is the second question. Hint writes one cell at a time.
| Size | One way |
|---|---|
| 4×4 pairs | 3+ is 1+2, 4+ is 1+3, 7+ is 3+4, 3− is 1 and 4, 3÷ is 1 and 3, 4÷ is 1 and 4 |
| 4×4 products | 2×, 3×, 4×, 6×, 8×, and 12× are the pairs 1×2, 1×3, 1×4, 2×3, 2×4, and 3×4 |
| 6×6 pairs | 3+ is 1 and 2, 11+ is 5 and 6, 30× is 5 and 6, and 5− is 1 and 6. 3− and 3÷ are not one way |
A 4× in two cells is 1 and 4, not 2 and 2, because those two cells share a line. 1− is not one way: it is 1 and 2, 2 and 3, or 3 and 4. 2÷ is 1 and 2, or 2 and 4. On a 4×4 a straight three-cell cage is one way too: 6+ is 1+2+3, 7+ is 1+2+4, 8+ is 1+3+4, 9+ is 2+3+4, and 6×, 8×, 12×, and 24× are 1×2×3, 1×2×4, 1×3×4, and 2×3×4. A bent 6+ is not one way: 1+1+4 is legal when the two 1s are diagonal. A bent 12× can be 1×3×4 or 2×2×3. On a 6×6, a × target with a factor of 5 needs a 5. A ÷ of 3 is 1 and 3, or 2 and 6.
When the cage has one unordered set and nothing is written inside it, the toast says the shape fixes the order. That is true on the first figure. It is not true on the third. When one digit of the set is already inside the cage, the toast names that digit and writes the next cell.
The 11+ cage in three cells can only be 3, 4 and 4. Its shape fixes the order, so this cell is 4. The two 4s share neither row nor column. The middle cell shares a column with the first 4 and a row with the second, so it is the 3. A straight cage in one row could not hold two 4s.
The 11+ cage in three cells can only be 3, 4 and 4. With 4 already placed, this cell is 3. The 4 stays violet. The new 3 takes the ring.
The 4+ cage in two cells can only be 1 and 3. The toast says the shape fixes the order, so this cell is 1. Both orders, 1 then 3 and 3 then 1, fit the shape. Column 3 already has a 3, so the top cell cannot be 3. The 3 at the bottom of that column is what decides it, not the cage shape. The partner is 3, and Hint writes it on the next press. This picture shows only the 1.
A cage with more than one set does not get this sentence until one cell is left. Then Hint does the arithmetic. The sum version is the worked picture in section 5, after two digits are on the board. The minus version is the second hint of section 3. A 5×5 can open on the same shape lesson, a 2× that can only be 1, 1, and 2. The only home in the line continues that grid, and the 1 is already there.
An easy grid finishes from cages with one ordered filling. A naked single and a hidden single are easy sentences too, and the packed easy grids do not ask for them. Harder grids ask you to compare the fillings with the row and the column.
Only one digit left
Once the rows and columns have used every other digit, the last digit is forced. This is a Sudoku naked single. There are no boxes, and a cage can still have several fillings, so the row is what writes the digit.
The same sentence names a column when the column is the full one, and it names the row and the column together when each is only part of the strike. The packed easy walks never say this. The first time it shows up is on a medium or a hard grid. The sentence is still the easy one.
Row 2 already has 1, 2 and 4, so this cell is 3. The 1 comes from the cell to its right, the 2 from the cell to its left, and the 4 from the end of the row. The 1− cage under the new 3 is not one way, and it is not the reason. The given in the corner stays bold ink.
The only home in the line
A digit can be forced even when the cell still has several candidates. The row, or the column, has only one cell left that can hold it. This is a Sudoku hidden single. On this site it is an easy Calcudoku sentence. On the Sudoku guide it is a medium step. The packed easy grids do not ask for it.
Row 4 needs a 2, and this is the only cell in it where a 2 still fits. That cell is not down to one digit. The cell to its left sees the 2 above it, the next cell sees the 2 below it, and the last cell sees the 2 at the end of row 1. The first cell of the row sits in an 8+ that can only be 3 and 5. The new 2 is in the 40×. This grid opened on a 2× that can only be 1, 1 and 2, and the 1 at the top left is that digit.
Column 4 needs a 1, and this is the only cell in it where a 1 still fits. The cell is still not a single: its 4− can only be 1 and 5, so 1 is one of the candidates. The top cell is in a 60× of 3, 4, and 5. The cell under that is in a 40× of 2, 4, and 5, and the 2 to its left is already written, so those two are 4 and 5. The 2 just written blocks the next cell. The bottom cell sees the 1 to its left. That leaves the ringed cell.
Line and fillings
A line has a fixed sum. Use it when one cage fills the rest of the line, and when every filling of a cage agrees on one cell.
The rest of the line
A row of a 4×4 adds to 10, a row of a 5×5 adds to 15, a row of a 6×6 adds to 21, and a row of a 7×7 adds to 28. Columns use the same totals. When one + cage fills the rest of that line, the missing cell is the total minus the target.
When the cage leaves two cells, those two add up to the difference, and a digit you already have splits them. The cage is a + cage, it sits in one straight line, and it does not contain the cell you write. Hint does not add several cage targets together. This is a medium rule. It is common on 4×4 medium. It did not appear on the packed 5×5 medium walks, or on any packed 6×6 or 7×7 walk. The two first hints below happen to be hard grids. A medium grid can still say it later. Do not expect every medium grid to start here.
The 8+ cage fills the rest of row 3. Every row adds up to 10, so this cell is 10 − 8 = 2. The 8+ does not contain the empty cell. The given 3 is not in this row, and it is not why the 2 is forced.
The 1− cage already has a 2, and 1 is the only digit that makes 1− with it. 1− is not one way. One cell is left, so the sentence is the partner, not the shape. The 2 stays violet.
The 9+ cage covers all of row 1 but two cells, so those two add up to 15 − 9 = 6. Here that makes this cell 5. The toast does not say the other cell is 1. It is the given at the end of the row, so 6 − 1 = 5. The 9+ does not contain the new 5. That cell’s own cage is the 11+ that continues down the column.
Every way to fill the cage
A cage can have several fillings and still force one cell, when every filling puts the same digit there. You do not need the order. Count the fillings that still fit the rows and columns. If they share a digit in one cell, write it.
On a 5×5, 6×6, or 7×7 medium grid this is the usual place the button starts, and it usually starts as a chain: “No single digit is forced yet.” Section 5 is that chain, on a smaller board. The figure here is the direct sentence, where one look really does write the digit. Many medium grids never show you this wording. They show you the chain.
The 17+ cage can still be filled 4 ways, and every one of them puts a 5 in this cell. The unordered set is 3, 4, 5, and 5. The three cells in row 2 share a row, so only one of them can be 5. The other 5 has to be the corner. The four orders are 5, 3, 4, 5; 5, 3, 5, 4; 5, 4, 3, 5; and 5, 4, 5, 3. The two givens do not remove any of those fillings.
A medium grid adds the line leftover or a cell every filling shares. The button may open on the chain instead of on either sentence, and on a 4×4 it may open on the next section even though the grid still grades medium.
Hard moves
Digits a cage always places on one line come off the rest of that line. Two cells can also share the same two digits.
Digits the cage must put on the line
Look at every filling of a cage, and at the digits those fillings always use in one row or one column. Those digits cannot sit anywhere else in that line. When that leaves one digit in an outside cell, write it. The cage does not contain the cell you write.
This is the step that makes many grids hard. It is also a one-step first hint on some medium grids. The grid below is one. Its from-scratch pass is still medium, because the line leftover, repeated with the one-way cages, finishes the blank grid. The button does not do that repeat. It writes the 4 first. A medium grid can use this sentence.
Every way to fill the 6× cage puts 1, 2 and 3 in column 1, so no other cell in column 1 can be any of those. That leaves 4 here. The six fillings are the six orders of 1, 2, and 3. The 7+ above the cage is 3 and 4, so 4 is a digit that cage can take. The toast lists 1, 2, and 3, which are the struck digits, not the 4.
The next hint on this board is the 7+, now that the 4 is written: “The 7+ cage in two cells can only be 3 and 4. With 4 already placed, this cell is 3.” That is section 2’s sentence.
The 6+ cage covers all of column 2 but two cells, so those two add up to 10 − 6 = 4. Here that makes this cell 1. The toast does not mention the 3. The other complement cell is already 3, so 4 − 3 = 1. The 6+ does not contain the new 1.
Two cells, the same two digits
Sometimes two cells in a row, or in a column, can only be the same two digits, and no other cell in that line can be either. Sudoku calls that a naked pair. On a packed Calcudoku the button never says that sentence. The pair is real. The toast is the longer chain, because the pair alone does not write the digit in one look.
Some hard grids never use a pair. The digits a cage must place on a line are the common hard step. The pair is the extra one.
No single digit is forced yet. Ruling candidates out around the 6+ cage comes first, and that puts 5 here. The first two cells of the row can only be 1 or 5. The next two can only be 3 or 4. Both pairs are on the board. The button names the 6+, which is the cage of the cell it writes. It does not say “two cells”. The 1 and 5 stay penciled in the partner. The next hint fills that partner, and this picture stops.
A hard grid adds the digits a cage must place on a line, and on some grids a pair of cells that share two digits. The pair shows up as the longer chain. Nothing in the ladder asks you to guess.
What if
There is no trial step. You do not write a digit to see whether the grid breaks, and Hint will not ask you to. When one look only crosses a candidate out, the button still returns one digit, and the sentence says that ruling candidates out comes first.
No single digit is forced yet. Ruling candidates out around the 6+ cage comes first, and that puts 1 here. After the given 2 is struck from the row, the 6+ can be 1, 3, 2 or 3, 1, 2 or 4, 1, 1. The ringed cell is 1 or 3. The 4 is gone. That look does not remove the 3, and it does not write a digit. The chain is what removes the 3. The button still writes the 1.
No single digit is forced yet. Ruling candidates out around the 6+ cage comes first, and that puts 3 here. The 1 stays violet. The player hears the chain again, about the same cage.
The 6+ cage already adds up to 4, so the last cell is 6 − 4 = 2. The cage is not one way. One cell is left, so Hint does the subtraction. A × cage would divide. A − or ÷ cage would use the partner sentence from section 3.
The 8× is only 2 and 4, and the solution order is 4 then 2. A lone 2 still has a legal partner, so nothing turns red. The line under the keypad says “8× in two cells · 1 to fill”. Check says keep going. Hint says nothing is forced, and it does not erase the 2. A 2 that is the solution digit does not by itself make Hint refuse. The refusal is a placed digit that is not the solution digit. The mistakes below are the failures Check does catch. This one is the quiet failure.
Common mistakes
A repeated digit, a digit no filling can use, and a full cage that misses its target are marked red while error highlight is on, and only on the cells the check cites. A duplicate does not turn the cage label red. The other two kinds do. Empty cells are not red. A quiet wrong digit stays violet.
Two 4s in column 1, in two different cages. The 8× can be 2 and 4. The 24× can be 2, 3, and 4. The repeat is the column. Both cage labels stay ink.
A 3 in the 8×, which is only 2 and 4. The partner stays empty and is not washed. The label turns coral. The keypad line does not list the sets. The red cell is the signal.
Two 3s on the 11+ diagonal. They share neither a row nor a column, so they are not a line duplicate. The only filling is 4, 3, 4, which section 2 already drew. The empty middle cell is not washed. The label turns coral.
1 and 2 in the 8×. The product is 2, not 8. Both cells turn red, and the label turns coral. The line under the keypad says “8× in two cells · check”.
2 then 4 is still a product of 8, and the two digits do not repeat in the row. The cage turns teal, label included. The digits stay violet. Teal digits are for a solved grid. Check says keep going. Hint says nothing is forced, and it does not swap the digits.
What to play next
Easy finishes from one-way cages. Medium adds a line leftover or a cell every filling shares, and the button may open on the chain. Hard adds the digits a cage must place on a line, and on some grids a pair.
- EasyOne-way cagesThe daily size. Start with the one-way cages: on a 6×6, 3+ is 1 and 2, 11+ is 5 and 6, 30× is 5 and 6, and 5− is 1 and 6. A 4×4 easy grid is the same lesson on a smaller board, and it is where the 11+ repeat in this guide came from.Play Easy →
- MediumFillings, or a chainWhen a cage has more than one filling, see if every filling agrees on one cell. The button often says that nothing is forced yet, and then writes the digit the chain finds. A line leftover is the 4×4 version of the same idea. A 7×7 medium grid is the larger step.Play Medium →
- HardA line, then a pairWhen those stop, look for the digits a cage must place in one row or one column, and take them off the rest of that line. On some grids two cells share the same two digits. Hint describes that as the longer chain. No guess.Play Hard →
4×4, 5×5, and 6×6 come in easy, medium, and hard. 7×7 comes in easy and medium. Try 4×4 easy and 7×7 medium. The daily puzzle is a 6×6 easy grid: today’s Calcudoku opens the same size and setting as 6×6 easy. Those grids finish with one-way cages.
Print a Calcudoku pack. A 4×4 or 5×5 sheet holds four puzzles. A 6×6 or 7×7 sheet holds two. Large print is one. 7×7 comes in easy and medium. The answer key starts on its own page, with the cages kept and the digits filled in.
- GuideSudokuRows, columns and boxes, from singles to triples.
- GuideKiller SudokuCage sums and house totals do the work.
- GuideKakuroOne-way sums first, then short runs and longer lists.
- GuideFutoshikiSigns first, then chains, and one home in the line.
- GuideHashiCount bridges, never cross, connect every island.
- GuideBinairoSpot pairs, fill the gaps, balance every line.
- GuideStar BattleOne or two stars in every row, column and region.
- GuideSlitherlinkDraw one loop. Start from the 0s and the 3s.
- GuideNonogramsOverlap long clues, cross the gaps, and read the crossing line.
- GuideHitoriShade repeats, never side by side, keep white connected.
- GuideNurikabeGrow each island to its number and keep one sea.
- GuideLight UpLight every cell. Start beside the 4s and the 0s.
- GuideShikakuEach number is an area. Start with the ones that fit one way.
- GuideMasyuTurn on black pearls, go straight through white ones.