Advanced Calcudoku Techniques for Harder Grids

In a 6 x 6 or 7 x 7 Calcudoku, the cages with sums and products of four cells have too many possible fillings to reason about one by one. These Calcudoku techniques look at a whole row or column at once. We show them on a hard LogikLand grid (6 x 6, all four operations) and every step below was checked against its solution. Beginners should first read how to solve Calcudoku.

Line totals

Every row and every column holds each digit once, so its sum and its product are fixed.

The product trick for rows and columns

Look for cages that fill two whole rows (or columns) except for one cell. If all those cages are products, multiply their clues and compare with the product of the two lines. The missing cell is the quotient. The same works with sums: if the cages of a row are all additions and leave one cell free, that cell is the line total minus the sum of the clues.

In the grid above

Rows 5 and 6 are covered by three cages, apart from the cell (5,6): the cage 120× (row 5, cells 1 to 4), the cage 90× (row 6, cells 1 to 4) and the cage 16× (cells (5,5), (6,5) and (6,6)). Together the two rows hold every digit twice, so their product is 720 × 720 = 518,400. The three cages multiply to 120 × 90 × 16 = 172,800. So the cell that is missing is 518,400 ÷ 172,800 = 3.

This cell is the whole key of the bottom right. The cage 3− covers (4,6) and (5,6). The two cells differ by 3, so the one above the 3 is either 6 or 0. A 0 does not exist, so (4,6) = 6.

Finding a block

Scan the grid row by row, and for each row (and each pair of neighbouring rows) list the cages that are entirely inside it. Ask three questions. Do these cages cover every cell except one? Are they all additions, or all products? Is the total easy to compute? For example, in a 6 x 6 grid, if the additions of one row add up to 18 and leave one cell, that cell is 21 − 18 = 3. A cage that sticks out of the rows by one cell works the same way in the other direction: the cell outside is the sum of the clues minus the total of the rows. A mix of additions and products in the same block does not work, so leave those out.

Digits locked to a line

A cage lying in a single row or column tells you which digits that line has already used. The cage 20× covers (1,6), (2,6) and (3,6), all in column 6. The only three different digits from 1 to 6 with product 20 are 1, 4 and 5. So column 6 uses 1, 4 and 5 in its top three cells, and the three cells (4,6), (5,6) and (6,6) hold 2, 3 and 6.

Now read the cage 3−, over (4,6) and (5,6): the pairs from {2, 3, 6} with a difference of 3 are only 3 and 6. So the last cell of the column, (6,6), is 2. You found it without the product trick, and the product trick then tells you which of the two is on top.

The cage 16× has cells (5,5), (6,5) and (6,6). Since (6,6) = 2, the other two multiply to 8 and are side by side in column 5, so they are 2 and 4. Row 6 already has a 2, so (6,5) = 4 and (5,5) = 2.

Prime factors: 5 and 7

A digit that is prime can only come from itself. In 6 x 6 and 7 x 7 grids the digit 5 is only a factor of 5. A product cage whose clue is a multiple of 5 contains a 5, and in a 7 x 7 grid a clue that is a multiple of 7 contains a 7. In our grid, 120× in row 5 and 90× in row 6 both contain a 5. Because each row has only one 5, the 5 of row 5 is in the 120× cage, and no other cell of that row, such as (5,5) or (5,6), is a 5.

Subtraction and division pairs

Combine this table with a locked digit set, as in the 3− example, and many two-cell cages solve at once.

What the hard levels need

The LogikLand generator keeps a Calcudoku grid only if its solver can finish it by deduction. For each cage the solver tries every way to fill it with the digits that are still possible, and it also applies the row and column rules. The product trick is not part of that solver, so no grid needs it, but it is a quick way to a cell when you find the shape. The hard level is a 6 x 6 grid with cages up to 4 cells and the four operations. The expert level is 7 x 7, with at most one single-cell cage.

Practising

Our suggestion, not a rule: when a hard grid blocks you, add up the rows and columns of cages first, then look for locked digits. To make 6 x 6 and 7 x 7 grids for a book, create a free account and use the Calcudoku assistant (1 or 2 grids per page). For gentler practice, see printable easy Calcudoku puzzles.