數和 Kakuro

Kakuro: rules and solving techniques

Fill unique digits to match every across and down sum. This guide reflects the controls and win conditions in the current browser build. It is reviewed by the PocketJoy game-testing team and maintained by Long-Term New Media Limited.

The rules

A Kakuro grid looks like a crossword but the cells hold digits. The board is made of black and white cells, and each black cell carries clue numbers split by a diagonal line: the number in the upper right is the sum of the run of white cells extending to the right, and the number in the lower left is the sum of the run extending downward. You fill every white cell with a digit from 1 to 9 so that each run adds up exactly to its clue, and no digit repeats within a single run. Note that the no-repeat rule applies only within a run — separate runs are independent of one another, which is a genuine difference from sudoku's row, column and box constraints. Three difficulty levels set the grid size and the distribution of run lengths. When you have filled the grid you can press check, and the game tells you either that the puzzle is complete or that some digits still do not match the clues. Completion time is recorded, with a fastest time stored separately for each difficulty on your device.

Advanced solving

The real lever in Kakuro is the extreme sum. Because every run uses distinct digits from 1 to 9, a run of a given length has a firm minimum and maximum. A run of two ranges from 1+2=3 up to 8+9=17; a run of three ranges from 1+2+3=6 up to 7+8+9=24. When a clue sits exactly at one of those limits, the combination is unique: a two-cell run summing to 3 must be 1 and 2, a two-cell run summing to 17 must be 8 and 9, a three-cell run summing to 6 must be 1, 2 and 3, and one summing to 24 must be 7, 8 and 9. These forced runs are your way into the grid and should all be found and filled first. Then look for the next tier: a two-cell run of 4 can only be 1+3, one of 16 only 7+9; a three-cell run of 7 only 1+2+4, and one of 23 only 6+8+9. The genuine reasoning happens at crossings, where one cell belongs to both a horizontal and a vertical run. Intersect the candidate sets of the two runs and often only one or two possibilities survive. If a horizontal run of 3 (candidates {1,2}) appears to cross a vertical run of 17 (candidates {8,9}), those runs cannot intersect at all — a contradiction like that means an earlier entry was wrong. Another practical technique is counting the shortfall: in a four-cell run summing to 30 where you have already placed 9 and 8, the remaining two cells must make 13 without repeating and without using 9 or 8, which leaves 6 and 7 as the only option.

Common blind spots

The most common mistake is importing sudoku's rules and assuming that whole rows and columns cannot repeat. In Kakuro the no-repeat constraint applies only within a single clued run; cross a black cell and you are in a new run where the same digits may appear again. Misunderstanding this eliminates a great many legal placements and drives you into a dead end. The second mistake is skipping the extreme-sum sweep and starting from the top-left corner on instinct. Kakuro grids are usually designed with a few forced runs near the edges or corners as an entry point, and passing over them means giving up the easiest part of the puzzle. The third is looking at horizontal or vertical runs in isolation. Nearly all of Kakuro's information lives at the crossings: a three-cell run of 20 has several possible combinations on its own, but paired with the run crossing it, the possibilities often collapse to one. The fourth is failing to re-examine the grid after an error. Because the solution is unique, a single wrong digit produces a contradiction somewhere far away, often after a dozen more entries, and at that point the check button is a better tool than trying to remember where you might have slipped.

Who it suits best

Kakuro suits people who enjoy sudoku but want more arithmetic in the mix. It exercises two skills at once: enumerating combinations, meaning which sets of digits can make this sum, and cross-referencing, meaning which candidates survive at a shared cell. That is one layer more than pure positional deduction. The entry barrier is slightly higher than sudoku's, because you need to become familiar with a handful of extreme sums first, and the first two or three puzzles can feel heavy going. Once the unique combinations for runs of two and three are committed to memory, however, solving speed improves dramatically, and that transition from stuck to fluent is where the satisfaction lies. There is no luck involved: every puzzle has a unique solution and every cell is deducible, so guessing is never required. Solving times vary with difficulty, from roughly five to ten minutes on the easy setting to half an hour or more on the hard one. Nothing counts down against you — the timer is a reference for comparing with yourself, and you can stop and think at any point.

Feature highlights

Three difficulties set the grid size and the distribution of run lengths, with a fastest time stored per difficulty in the records panel so they never overwrite one another and each can be challenged in turn. The sidebar timer shows both your elapsed time on the current puzzle and your best for that difficulty. Two assist buttons have clearly separated jobs. Check validates whether the current grid satisfies every clue, and when it does not it says specifically that some digits do not match rather than reporting a vague failure — which tells you the problem is a wrong entry rather than an incomplete one. Hint settles a single cell outright, a way to reopen a position where you are stuck. Clue numbers for the across and down runs are separated by a diagonal and placed in the upper-right and lower-left of the black cell, so which number governs which run is immediately obvious and needs no working out. Completion shows your total time and offers a next puzzle directly.

Controls and record keeping

Control is a two-step action: tap the white cell you want to fill, which highlights it, then tap a digit from the number pad at the bottom of the screen. This means phone players never need the system keyboard and can complete a whole puzzle one-handed, with desktop and touch behaving identically. To correct an entry, select the cell again and enter a new digit, which overwrites without needing to clear first. Black cells cannot be selected and are ignored automatically, so mis-taps are not a concern. Fastest times per difficulty are kept in the browser's local storage on your own device — nothing is uploaded to any server and no account is required. Records do not follow you to another device, and clearing browser data removes them. The language button switches between Traditional Chinese and English instantly, updating every prompt, panel label and result string without a page reload. Once the page has finished loading, the game itself needs no further network connection, which suits commutes and places with poor reception.

Rules Q&A

The unique combinations are tabulated in Kakuro Magic Combinations and Reference Tables.

Reviewed by the PocketJoy game-testing team · Operated by Long-Term New Media Limited · hklongtermad@gmail.com