Kakuro solving techniques

1. Forced combinations: the keys that turn first

Some clues can be made exactly one way and no other. Two squares adding to 17 have to be 8 and 9, because no other pair of different digits gets there. Clues like that are free information, so when a board hands you one, that is where the solve starts.

These are the ones worth keeping in your head:

The shape of it: for runs of 2 to 7 squares, the two lowest and the two highest clues are always forced. Long runs are kinder still. In an 8-square run every clue is forced, because eight different digits leave exactly one of the nine sitting out, and a 9-square run is always the whole 1 through 9 in some order. The full list lives in the combinations chart.

2. Crossing runs: where certainty comes from

A forced clue tells you which digits a run uses. It says nothing about where they sit. Position comes from the crossings: every white square belongs to one across run and one down run, and whatever digit ends up there has to be on both of their lists.

The classic case is a two-square 17 crossing a two-square 16. The 17 can only be 8 and 9. The 16 can only be 7 and 9. The square they share has to take a digit from both lists, and only the 9 is on both, so in it goes. That single move then settles the 8 next to it and the 7 below it, which is the nice thing about crossings: they rarely arrive alone.

17 across is 8 and 9, and 16 down is 7 and 9. The only digit both runs allow in the square they share is the 9, and everything else on this little board follows from it: 8 beside it, 7 beneath it, and a 1 left over in the corner.

This is the engine of the whole puzzle. Every time you pencil a run's combination in, go straight back along its squares and ask what the crossing run will allow in each one. Almost every elimination you ever make comes from that intersection rather than from any new adding up.

3. High and low limits: pruning without a chart

Plenty of clues have several combinations, and they are still not silent. The extremes of a clue rule digits out all by themselves:

• A small clue puts a ceiling on its digits. In a two-square 6, nothing can be 6 or more, because its partner would have to be a 0. And nothing can be a 3, because its partner would have to be another 3. That is 1, 2, 4 and 5 left, and you have not looked at a single crossing yet.
• A large clue puts a floor under them. In a two-square 15, nothing can be 5 or less, because its partner would need to be 10 or more. Only 6, 7, 8 and 9 survive, and sure enough the two combinations are 6 and 9, or 7 and 8.

Extreme always means extreme for that run's length. A 24 is the absolute top of a three-square run and thoroughly ordinary in a five-square one. So the instinct worth growing is to weigh every clue against the range its own length allows, and to go hunting near the ends of that range, because that is where the board is most talkative.

When you would rather be exact than instinctive, the chart has a combinations column that is the complete candidate list for any clue, and an "always uses" column marking the digits that are certain even while several combinations are still in play. Those certain digits are the ones to go crossing with first.

4. The 45 rule: the long-run shortcut

Add 1 through 9 together and you get 45, every time. So a run of all nine digits always totals 45, and rather more usefully, a run of eight squares is that whole set with exactly one digit missing:

The same idea stretches to whole regions once you have the knack. Pick a group of runs, add their clues, then add the clues of the runs crossing them. Any square covered by both sides cancels out, so what the subtraction really weighs is the leftover squares on one side against the leftover squares on the other.

The lopsided regions are the ones worth hunting. If every leftover square sits on one side, the difference is their combined value, and if that side's leftover is a single square, you have just read its digit straight off the board. That is the famous version of the trick. With leftovers on both sides you only learn the gap between two groups, which is occasionally still useful and usually a hint to redraw the region somewhere better. Finding the right region takes practice; the arithmetic never gets harder than adding a few clues and taking one number off another.

5. Pencilmark discipline: make deductions visible

None of the four patterns above are worth much if what they tell you evaporates the moment you look at another corner of the grid. Pencilmarks are how a deduction gets kept:

• Only write candidates down once a clue has narrowed them to two or three digits. Marking all nine in a square teaches you nothing and buries the marks that mean something.
• The second a digit lands anywhere in a run, strike it out of every other square in that run. Stale marks cause more wrong answers than bad arithmetic ever will.
• Make your colors mean something and keep them meaning it. Plenty of solvers keep one color for "digits this square could still be" and another for "digits this combination would need", so the two sorts of maybe never get confused for each other.

Then just trust the loop. Tightest clue, pencil the combination, walk every crossing, ink in whatever is certain, go round again. When you stall, work back down the list in order. Nine times in ten some run's surviving combinations share a digit you have not cashed in yet, and the chart will show you which. When they do not, the move you are missing is almost always a crossing you have not looked at since a digit landed in it, or a region with something to say about the 45.

And when none of that works, put the board down. Kakuro is remarkably good at solving itself while you are making a cup of tea, and a cat will tell you, at considerable length, that the correct response to a hard problem is a nap and a fresh look after.

Practice the patterns

Catkuro puts the combinations for whichever square you have selected right under the board, and trims the list as digits land. Pencilmarks in five colors, strikeouts that stay struck, and hints that talk you through the next forced deduction when there is one, or point you at the most constrained corner of the board when there is not. No hint ever writes a digit for you. All 2,000+ puzzles are free, and no ad ever interrupts a solve.