Kakuro (Cross Sums)

Kakuro is a cross-sums number puzzle: fill every white cell with a digit from 1 to 9 so each horizontal and vertical run of white cells adds up to its clue, and no digit repeats inside a single run. It is played on a grid of white and black cells, and your job is to make the numbers add up correctly in two directions at once. Each horizontal block of white cells — a run — must total the clue printed to its left, shown after a → ; each vertical run must total the clue printed above it, shown after a ↓ . The black cells carry the clues and break the grid into these runs.

What is the no-repeat rule in Kakuro?

There is one more rule, and it is the heart of the puzzle: a digit may not repeat within a single run. So a run of three cells summing to 6 cannot be 2+2+2 — it has to be 1+2+3. The same digit is perfectly free to appear again in a different run, including the run that crosses it; the no-repeat rule applies only inside one straight line.

Why every Kakuro has one logical solution

Because the clue fixes a total and the no-repeat rule limits which digits can share a run, every cross sums puzzle can be reasoned out step by step. Each white cell sits at the meeting point of one across run and one down run, so it must satisfy both clues at the same time — and that double constraint is usually enough to pin down a single digit. Every board on this page has exactly one solution reachable by logic alone, with no guessing required.

To solve Kakuro, work the shortest, most constrained runs first, learn the magic combinations, and cross-reference where across and down runs meet so each cell satisfies both clues. Good kakuro solving is a few dependable techniques applied in the right order — from the runs that give the most away to the ones that give the least. These are the core kakuro tips:

• Start with the shortest, tightest runs. A two-cell run has far fewer possibilities than a five-cell one, and extreme totals are even stingier. A two-cell run summing to 3 can only be 1+2; one summing to 17 can only be 8+9. Place those first.
• Learn the magic combinations. Many length-and-total pairs allow only one possible set of digits (more on these below). When a run is one of them, you already know exactly which digits live there — you just need to work out the order.
• Cross-reference where across and down runs meet. A cell must obey both its row clue and its column clue. List the digits each run allows, keep only the digits that appear in both lists, and often a single candidate survives.
• Watch the box two runs share. When a forced across combination and a forced down combination overlap in one cell, the digit that belongs to both sets is locked — and fixing it usually removes a candidate from every other cell in both runs.

Make forced moves first, and never guess

Build the habit of making every forced move first — the cells that can only be one digit — and re-checking the crossing runs after each one, because every digit you write narrows the runs around it. And never guess: on a proper Kakuro there is always a cell you can prove next.

A magic combination is a run length and total that has only one possible set of digits — such as 17 in two cells (8+9) or 24 in three cells (7+8+9) — so you instantly know which digits are present even before you know their order. This is the single most useful idea in how to solve kakuro, and it is often enough to crack a corner of the grid. The most valuable ones are the extremes, where the digits are squeezed hardest:

• A two-cell run totalling 3 must be 1+2.
• A two-cell run totalling 4 must be 1+3.
• A two-cell run totalling 16 must be 7+9.
• A two-cell run totalling 17 must be 8+9.
• A three-cell run totalling 6 must be 1+2+3.
• A three-cell run totalling 24 must be 7+8+9.
• A four-cell run totalling 10 must be 1+2+3+4.

Kakuro magic combination table

This quick-reference table lists the clues that lock to a single digit set. When you see one of these sums on a run of the given length, the digits are fixed — only the order is left to work out:

What happens when two magic combinations cross?

Spotting these on sight is what separates fast solvers from slow ones. When two magic combinations cross, the puzzle almost solves itself: you simply keep the digit their two sets have in common. Printed Kakuro books often include a full combinations table, but for the 4×4 and 5×5 boards here, the handful above will carry you a long way.

When a two-cell 17 → run (8+9) crosses a two-cell 16 ↓ run (7+9), the shared cell must be the digit common to both sets — 9 — which then forces the other cells to 8 and 7. Picture two short runs that cross. A two-cell across run is clued 17 →, and a two-cell down run clued 16 ↓ shares the across run’s left-hand cell. Both are magic combinations: 17 in two cells can only be 8+9, and 16 in two cells can only be 7+9.

Now look at the shared cell — the one counted by both runs. It must be a digit from the across set {8, 9} and from the down set {7, 9}. The only value in both sets is 9, so that corner cell is forced to 9 — no guessing required.

Once the 9 is placed, the rest follows at once: the other across cell must be 8 to reach 17, and the other down cell must be 7 to reach 16. A single crossing of two magic combinations has settled three cells. Each digit you fix trims the candidates in its neighbours, and these little forced crossings are exactly how a Kakuro unravels — patient elimination, never guesswork.

Kakuro here comes in two sizes: a 4×4 board with short runs and many magic combinations for learning, and a harder 5×5 board with longer runs, more crossings, and more candidates per cell. The 4×4 is the gentle introduction — short runs, plenty of two-cell entries, and magic combinations everywhere you look, so it is the ideal place to learn the cross-sums idea. A 5×5 board adds longer runs and more crossings, which means more candidates per cell and deductions that chain further before they resolve.

Difficulty in Kakuro is really about how much the runs constrain each other. Longer runs and middling totals (which allow many digit sets) are harder than short runs and extreme totals (which allow only one). A bigger grid simply contains more of the harder kind, so it asks you to hold more possibilities in mind at once — but, like every board here, it is always solvable by pure logic.

Tap New for a fresh puzzle at either size whenever you like, or play the shared Daily to get exactly the same board as everyone else that day — a nice way to compare solving times with a friend.

Kakuro is also known as Cross Sums (its original name) and is sometimes spelled Kakkuro — all the same puzzle: fill the white cells so each across and down run hits its clue, with no repeats inside a run. In its country of origin it was — and often still is — called Cross Sums, and you will also see it written Kakkuro.

Kakuro belongs to the number-placement family, so if you enjoy it you will likely take to its relatives. Sudoku shares the “digits 1–9, no repeats” instinct but never adds them up — the numbers are pure symbols there. Futoshiki keeps the no-repeat rule and replaces sums with greater-than and less-than signs between cells. Learn the cross-referencing habit in any one of the three and the others come noticeably faster.

The key Kakuro terms are run (a straight block of white cells totalling one clue), clue (the number in a black cell), black cell, crossing cell, magic combination, and candidate. A handful of words turn up in every Kakuro guide:

• Run (or entry) — a straight block of white cells, across or down, that must add up to one clue with no repeated digit.
• Clue — the number in a black cell giving a run’s total: the figure after → is the across total, after ↓ the down total.
• Black cell — a filled cell that holds clues and separates the runs; you never write digits in it.
• Crossing cell — a white cell shared by one across run and one down run, so it must satisfy both clues.
• Magic combination — a run length and total with only one possible set of digits, such as 16 in two cells (7+9).
• Candidate — a digit that could still legally go in a cell, given both of its runs.

You do not need the jargon to play — tap a cell and pick a number — but the names make strategy guides far easier to follow.

The most common Kakuro mistakes are repeating a digit in a run, treating the no-repeat rule as per-grid instead of per-run, solving one direction in isolation, ignoring the magic combinations, and guessing. Most Kakuro frustration comes from a few avoidable habits:

• Repeating a digit in a run. The no-repeat rule is easy to forget on longer runs — a three-cell run summing to 6 is 1+2+3, never 2+2+2.
• Forgetting the rule is per run, not per grid. The same digit can appear in a crossing run; only the single straight line must stay free of repeats.
• Solving one direction in isolation. An across run might allow several combinations on its own; it is the down clue crossing it that usually decides the cell. Always check both.
• Ignoring the magic combinations. Skipping past a two-cell 17 (8+9) or a two-cell 3 (1+2) means doing by trial what you could have read off instantly.
• Guessing. One unproven digit can quietly break several runs before you notice. There is always a logical next move — find it instead.

Kakuro is worth playing because it blends deep logic with only light single-digit arithmetic, every board has one solution reachable by reason alone, and it makes a calming daily ritual. The sums never get harder than adding a few single digits, yet the deductions can run deep, so it rewards clear thinking far more than maths ability. Every board has one answer reachable by reason alone, which means progress always feels earned — there is no luck to blame and nothing to unlock.

It also makes a calming daily ritual. A 4×4 grid is a five-minute reset; a 5×5 is a satisfying stretch of focus, and the shared Daily gives you the same board as everyone else and a small streak to keep. The cross-referencing habit it builds carries straight across to the rest of this collection — try Sudoku next, or Futoshiki for the same no-repeat logic without the sums.

Kakuro was born in the United States in 1966 as “Cross Sums” in Dell Magazines, devised by Jacob E. Funk; the Japanese publisher Nikoli later renamed it Kakuro, short for kasan kurosu (“addition cross”). Although the name is Japanese, the puzzle itself is American: Funk’s original was a cross-shaped grid of runs that each had to add up to a clue, with no digit repeated in a run.

It found a second home in Japan. Nikoli introduced the puzzle there and gave it the name Kakuro, a contraction of kasan kurosu — “addition cross”. (You will also see it spelled Kakkuro.) Under that name it became one of the great Japanese grid puzzles, sitting alongside Sudoku as a fixture of newspapers and puzzle books, and it is by the Japanese name that most of the world now knows the old American cross sums puzzle.