How Ikura Puzzles Work | Addition Cross Sum Rules

Summary: Ikura is a small addition puzzle: place the digits 1 to 9, each exactly once, into a 3×3 grid of circles so that every row and every column adds up to the total printed beside it. A few digits are given, the rest are listed alongside, and there is always exactly one correct arrangement.

Overview

An Ikura puzzle is nine circles in a 3×3 block. To the right of each row is a total; below each column is another total. Your job is to fill the empty circles so all six totals come out right.

Three things make it different from the puzzles it resembles:

Every digit from 1 to 9 is used exactly once. Not “any numbers that add up” — the nine digits are a fixed set, dealt out among the nine circles. That constraint is what makes the puzzle solvable by reasoning instead of trial and error.
There are no operator signs on the board. Every operation is addition, so printing a + twelve times would only add clutter. If you see a total, it is a sum.
It is not a magic square. A magic square wants every row, column and diagonal to hit the same total. Ikura prints six totals that are set independently, and ignores the diagonals entirely. That makes it the friendlier puzzle: the totals are handed to you rather than deduced.

The black discs down the left are the digits still to place. They are a convenience, not a clue —
they simply restate which of 1 to 9 are not already printed on the board. Some puzzles are published
without them.

The Rules

Place the digits 1 to 9, each exactly once, into the nine circles.
Each row must add up to the total printed to its right.
Each column must add up to the total printed below it.
Digits already printed on the board are fixed and cannot move.

And one promise from the other side: every published puzzle has exactly one solution. You never have to choose between two valid answers, because there is only ever one.

That promise is worth trusting, because it is not free. Addition-only grids of this kind can easily have many valid arrangements — with no digits given at all, a 3×3 board averages around ten — so the generator checks each board before publishing it, adding a given digit if the grid is not yet pinned to a single answer, and throwing the board away if it still is not.

Where to Start

Find the line with the fewest blanks. A row or column with only one empty circle is not a puzzle at all — it is subtraction.

Look at the middle row of the example: 2, a blank, 1, totalling 8. There is nothing to work out here. 2 + 1 = 3, and the total is 8, so the middle circle is 5. One digit, no branching, no guessing.

On a standard four-given puzzle that opening is always available: the given digits land on a fixed shape — the four corners or the four edge centres — and either shape leaves at least one line with a single blank. On harder settings, where fewer digits are given and they are placed more freely, usually no line has a single blank, and you open with the pair technique below instead.

Before that, one check costs you nothing. Add up the three row totals: they always come to 45. So do the three column totals. The nine digits 1 to 9 always sum to 45, and the rows between them hold all nine. In the puzzle above, the rows are 19, 8 and 18; the columns are 9, 20 and 16. Both make 45.

That is worth thirty seconds because it catches a misread grid before you commit ink to it, and because a very low row total tells you at a glance that its three digits are small — which is usually where the easiest deductions live.

When No Line Has a Single Blank

Sooner or later you will face two blanks in every line. Now the useful question is not “what goes here?” but “which pair of remaining digits could go here?”

Continuing the example: after placing the 5, the digits still unplaced are 3, 4, 6 and 9. The right-hand column reads blank, 1, blank and totals 16 — so the two blanks must add to 15. Of the digits left, only 6 and 9 make 15. You do not yet know which is which, but 3 and 4 are now excluded from that column — and since the middle column is already full (7, 5, 8), the only place left for them is the left column.

From there the grid closes. The left column needs its two blanks to total 7, and 3 + 4 = 7 — consistent. The top row needs its two blanks to total 12, and only 3 and 9 do that, so the top-left is 3 and the top-right is 9. The left column’s blanks were 3 and 4 and the top-left took the 3, so the bottom-left is 4 — which leaves 6 for the bottom-right. Check the bottom row: 4 + 8 + 6 = 18. Nine digits placed.

The habit worth building: work in pairs and sums, not in single cells. “These two circles must total 15” is a much stronger statement than “this circle might be a 6”.

One caveat, because it changes how you hunt. Late in a puzzle, with only a few digits left, a required sum usually has just one candidate pair. Early on, from the full pool, it may have three or four — a target of 10 can be made four different ways. So do not scan the lines in order: scan for the line whose total is extreme, very low or very high, because those are the sums with the fewest ways to make them.

Opening a Hard Board

Here is that technique on a puzzle where nothing falls out for free — two givens, seven digits to place:

No line has a single blank; four of the six lines have three. But the left column reads blank, 9, blank and totals 22, so its two blanks must add to 13. The digits still to place are 1, 2, 3, 4, 5, 7 and 8 — and the only pair among them making 13 is 5 and 8. Two circles narrowed to two digits, from a board that looked closed. That is your way in.

(With more than five digits still to place, the list beside the board splits into two columns so it stays beside the puzzle rather than running past it.)

Difficulty

Ikura has one difficulty lever: how many digits are given. The board is always 3×3.

Five givens — Easy. Four digits to place, and a single-blank line is always available to open with.
Four or three givens — Medium. The standard printed puzzle is four. You will get one or two free moves, then pair reasoning.
Two or fewer — Hard. Usually nothing opens by subtraction, so you start by hunting for the tightest pair sum, as above.

Asking for fewer than two givens does not generally get you fewer: a 3×3 addition board with no givens almost never has a unique answer, so the generator adds one or two before publishing. Requesting zero and requesting one produce much the same puzzles in practice.

FAQ

Can an Ikura puzzle have more than one solution?

No. Every published puzzle is checked at generation and only kept if exactly one arrangement of the digits satisfies all six totals.

Do I ever need to guess?

No. Every puzzle can be reasoned to the end. If you are stuck, the usual cause is a line you have not looked at yet — check the columns if you have been working across rows.

What if every line has three blanks?

Work the totals rather than the lines. An extreme total — very low or very high — has the fewest digit combinations that reach it, so start there; a row totalling 7 can only be 1 + 2 + 4.

Can two rows have the same total?

Yes. The six totals are whatever the arrangement produces, and nothing stops two of them matching. They only have to satisfy the 45 rule as a group.

Are the totals ever subtraction or multiplication?

No. Ikura is addition only, which is why no operator signs are printed. If you want mixed operators, negative numbers or fractions on the same style of grid, that is a different puzzle — Cross Sum.

Is the arithmetic hard?

The board only ever adds three single digits. The solving, though, is mostly small subtractions from a two-digit total — 19 minus 7, 16 minus 1 — so it suits anyone comfortable with mental arithmetic under 45.

Try One

The hard board above is unsolved, and everything you need to finish it is in this page — start with the left column.

Every puzzle here was generated by Puzzle Maker Pro, which makes Ikura puzzles at any difficulty with your own styling, small enough to use as a warm-up or filler page in a book. The free demo generates real puzzles, one at a time, so you can see whether they suit your readers before deciding anything.