How to play Kakuro

Kakuro, also known as Cross Sums, is a number crossword. Instead of words, you write digits from 1 to 9 into white cells, and instead of definitions, the clues are sums. Every horizontal or vertical line of white cells - a run - must add up to the number printed in the dark cell just before it, and no digit may appear twice in the same run. Every white cell belongs to one run across and one run down, so each digit has to satisfy two sums at once.

Every puzzle here is made fresh by the game when you start it, the shape of the grid included, so no two boards look alike. Each has exactly one solution and can be solved by reasoning alone, with no guessing. The four levels differ in the size of the grid and in how deep the combination logic goes, from short runs with obvious sums on Easy to area sums on a large board on Crazy.

How to play on skill.games

When you open the game, the settings popup appears. Choose a level under DIFFICULTY, choose how mistakes are treated under MISTAKES, and press PLAY. A new puzzle is generated in a moment.

• The board fills most of the screen. Dark cells hold the clues: a dark cell split by a diagonal line shows the sum of the run to its right in its upper-right half, and the sum of the run below it in its lower-left half. The white cells are yours to fill; the digits you enter appear in blue.
• Choosing a cell: tap any white cell. Both of its runs are shaded and the clues of those two runs light up, so you always see which sums you are working with. Every cell holding the same digit is highlighted too.
• Entering a digit: with a cell chosen, tap a number on the number pad (under the board, or beside it when your screen is wider than it is tall).
• NOTES: tap NOTES to switch notes on. Numbers you tap are now pencilled into the cell in small print as candidates, and tapping one again removes it. Tap NOTES again to go back to entering digits. When you enter a digit, it is removed automatically from the notes of every cell in its two runs.
• ERASE clears the chosen cell (its digit or its notes).
• UNDO takes back your last action - a digit, a note or an erase - all the way back to the start. Undo is free and unlimited.
• Highlighting a digit: with no cell chosen, tap a number on the pad to highlight every cell and note holding that digit. Tap it again to clear the highlight.

On a computer you can also use the keyboard: the arrow keys (or W, A, S, D) move between white cells, skipping the clue cells, 1 to 9 enter a digit, Shift and a digit pencil a note, Backspace, Delete or 0 erase, N switches notes on and off, and Z undoes.

The top line shows the clock, the level and your mistakes. The clock runs only while you play: it stops when the menu or a popup is open. The menu button in the top-left corner pauses the game, starts a new puzzle or switches the sound on or off. The game is saved after every move, so if you close the page you can press CONTINUE later and carry on with the same grid, digits, notes, clock and undo history.

When the last cell is filled correctly, the board ripples and the results screen shows your time, your best time at that level (a star marks a new record), your mistakes and the puzzle's code, together with your wins, losses, win rate and streaks.

Rules

Runs. A run is an unbroken line of two or more white cells, across or down. The clue for an across run sits in the dark cell to its left; the clue for a down run sits in the dark cell above it. The digits of a run must add up to its clue.

No repeats in a run. Only the digits 1 to 9 are used, and a digit may appear only once in each run. The same digit can appear many times in the grid, as long as it never repeats within a run.

No givens. The grid starts empty at every level; the clues alone fix the solution, and every puzzle has exactly one.

Mistakes - Unlimited (the default). A digit that is already in one of its runs is marked in red and the mistakes counter goes up by one. So is a digit that breaks a sum: a full run that doesn't add up to its clue, or a run whose digits already leave no possible total for its empty cells. Nothing else is checked: a digit that breaks no rule you can see is not marked, even if it turns out to be wrong later - just as on paper. Because the solution is unique, a full grid with nothing marked is always the right one.

Mistakes - 3 lives. Every digit you enter is checked against the solution at once. A wrong digit is marked in red and costs a life; the third mistake ends the game and counts as a loss. Undo can take a wrong digit back, but not the life it cost.

Abandoned puzzles. Starting a new puzzle before finishing the current one counts as a loss in your statistics for that level.

Solving techniques

These are the methods the game's own solver uses to grade every puzzle, from the simplest up.

• Run combinations: many sums can only be made one way. In two cells, 3 is always 1 + 2, 4 is 1 + 3, 16 is 7 + 9 and 17 is 8 + 9. In three cells, 6 is 1 + 2 + 3, 7 is 1 + 2 + 4, 23 is 6 + 8 + 9 and 24 is 7 + 8 + 9. In four cells, 10 is 1 + 2 + 3 + 4 and 30 is 6 + 7 + 8 + 9. For other sums, a digit that appears in none of the possible combinations can be crossed out.
• Crossing runs: each cell must suit both of its runs, so its candidates are only the digits both runs allow. Where a two-cell run of 4 (1 and 3) crosses a two-cell run of 3 (1 and 2), the crossing cell must be 1 - the only digit the two have in common.
• Naked singles: a cell where only one digit is still possible.
• Short-run permutations: in a run of two or three cells, check that a combination can really be placed - in a two-cell run of 17, if only one cell can take a 9, that cell is 9 and the other 8.
• Long-run permutations: the same reasoning in runs of four or more empty cells, where the combinations and their orders take more care.
• Area sums: take a band of whole rows. The across runs in it are all inside, so the band's digits add up to their clues. The down runs crossing the band's edge leave some cells inside it; subtract the down runs lying wholly inside, and what is left is the total of those cells. When that leaves one cell, it is fixed; when it leaves two to four cells, only the digits that can make their total survive. The same works for bands of columns.
• Advanced: XY-Wings and XY-Chains - linked cells with two candidates each, where one of two ends must hold a digit - all still without guessing.

Strategy tips

• Learn the unique sums. The two-, three- and four-cell sums that can only be made one way are the keys to every Kakuro. Start with the runs that use them.
• Look at the crossings. A cell where a run of 4 (1 and 3) crosses a run of 3 (1 and 2) must be 1. Short runs crossing each other often settle each other.
• Mind the big and small digits. A run of 2 adding up to 10 can't hold a 5; a run of 3 adding up to 7 can't hold anything above 4. Cross out the impossible digits first.
• Use notes. Pencil in the candidates for a run whose combinations you have worked out, then compare them with the crossing run.
• Count the corners. In a corner where two short runs meet, the area sums often fix a cell the runs alone can't.
• Don't guess. Every puzzle here can be solved by logic. If you feel stuck, look again at the runs crossing the cells with the fewest candidates.

Difficulty levels

The game generates every puzzle on the spot: it draws a symmetric grid shape, fills it, and reworks the digits and the shape until its solver - which works the way a person does, always taking the easiest step available - can finish the empty grid. A puzzle is given to you only if its hardest required step matches the level you chose.

• Easy: a 6x6 playing area with runs of up to four cells; run combinations and crossings are all you need.
• Normal: 7x7, runs of up to five; at least one step needs checking the orders of a combination in a short run.
• Hard: 8x8, runs of up to six; at least one step needs the same in a long run, or a one-cell area sum.
• Crazy: 9x9, runs of up to seven; at least one step needs an area sum over several cells or a chain.

Statistics and best times are kept separately for each level. If you are new to Kakuro, start on Easy and keep the table of unique sums in mind before moving up.