How Ikura Puzzles Work | Addition Cross Sum Rules

Summary: This page takes you from the opening move on an easy Ikura board to the pair technique that cracks a hard one where nothing falls out for free. Ikura is a 3×3 addition cross sum: nine circles, the digits 1 to 9 used once each, and six totals to satisfy.

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. A few circles are already filled in. Your job is to fill the rest so all six totals come out right.

It is an addition cross sum: every total on the board is a sum. There are no +, − or × signs to read, so if you see a number beside a line, add.

The black discs down the left list the digits still to place — the ones not yet printed on the board. Cross them off as you go. 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.

Every published puzzle has exactly one solution. You will never have to choose between two valid answers, and you will never need to guess.

Where to Start

First, check the totals. Add up the three row totals: they always come to 45. So do the three column totals. The nine digits 1 to 9 sum to 45, and the three 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. If your totals do not, you have misread the grid — find out now, not four deductions later.

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

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

Write it in and cross 5 off your list. On an easy or standard board there is usually a second single-blank line waiting once you do.

Work in Pairs, Not in Cells

Sooner or later every line has two blanks. The useful question is no longer “what goes here?” but “which pair of remaining digits could go here?”

Continue the example. After placing the 5, the digits still unplaced are 3, 4, 6 and 9.

Step one — narrow a line. The right-hand column reads blank, 1, blank and totals 16, so its two blanks must add to 15. Of the digits left, only 6 and 9 make 15.
Step two — write down what that tells you. You do not yet know which of the two is on top, and you do not need to. What you have learned is that 3 and 4 are excluded from that column.
Step three — use the exclusion. The middle column is already full (7, 5, 8). So 3 and 4 have nowhere to go but the left column. Check it: the left column totals 9 and already holds the 2, so its two blanks must total 7 — and 3 + 4 = 7. Consistent.
Step four — break the tie. The top row totals 19 and already holds the 7, so its two blanks must total 12. Of the digits left, only 3 and 9 make 12. The top-left circle is in the left column, which takes only 3 or 4 — so the top-left is 3, and the top-right is 9.
Step five — the rest falls out. The left column’s blanks were 3 and 4, and the top took the 3, so the bottom-left is 4. That leaves 6 for the bottom-right. Check the bottom row: 4 + 8 + 6 = 18. Nine digits placed.

The habit worth building: “these two circles must total 15” is a far stronger statement than “this circle might be a 6”.

Pick the right line to work

Which line you attack matters, because a required sum is not always narrow. Late in a puzzle, with three or four digits left, a target usually has one candidate pair. Early on, from the full pool, a target of 10 has four — 1+9, 2+8, 3+7 and 4+6.

So do not scan the lines in order. Go to the line whose total is extreme, very high or very low, because those are the sums with the fewest ways to make them. In the worked example the 15 was reachable one way only, which is exactly why it opened the board. A line asking for 10 would have told you almost nothing.

Opening a Hard Board

Here is that idea on a puzzle where nothing is free — two givens, seven digits to place.

A harder Ikura puzzle with only two given digits

No line has a single blank, and four of the six lines have three. So the fewest-blanks move is unavailable and you go straight to pairs, hunting the tightest total.

The left column is the place to look: it reads blank, 9, blank and totals 22, so its two blanks add to 13. That is a high target, which is what makes it useful. 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, on a board that looked closed. From here the method is the one you already used: you know the left column holds 5 and 8 in some order, so those two digits are now excluded from the middle and right columns — and every line crossing the left column has just lost most of its candidates. Work the crossing rows next, and break the 5/8 tie the way step four did, from whichever row pins one of them.

Check yourself when you finish. Every digit 1 to 9 should appear exactly once, and both sets of three totals should still add to 45. If they do, your answer is the answer — there is no second one to have found instead.

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 to open with.
Four or three givens — Medium. Four is the standard printed puzzle. Expect a free move or two, then pair reasoning.
Two or fewer — Hard. Nothing opens by subtraction, so you start by hunting the tightest pair sum, as above.

The fewer digits you are given, the more of the solve rests on pairs and exclusion — which is the skill the puzzle is really teaching.

FAQ

What is an Ikura puzzle?

A 3×3 addition cross sum: nine circles filled with the digits 1 to 9, each used once, where every row and every column must add up to the total printed beside it.

Can an Ikura puzzle have more than one solution?

No. Every published puzzle is checked before it goes out, 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 — check the columns if you have been working across the rows.

Do you need to be good at maths to solve Ikura?

The arithmetic itself is simple: adding three single digits, and small subtractions from a two-digit total. What actually takes practice is the logic — narrowing a total to the one pair of digits that can make it, then excluding the rest from that line. That combination suits solvers from about age 10 up; the easiest boards, at five givens, come down a good deal younger than that.

What if every line has three blanks?

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

What is the difference between Ikura and a magic square?

A magic square wants every row, column and diagonal to reach the same total. Ikura prints six totals that are set independently and ignores the diagonals, which makes it the friendlier puzzle: the totals are handed to you rather than deduced.

Try One

The hard board above is still unsolved, and everything you need to finish it is on this page. Start with the left column, take the 5 and 8, and work the crossing rows.

Every puzzle here was made with Puzzle Maker Pro, which generates Ikura puzzles at any difficulty in your own styling. Easy boards at five givens make a good warm-up or filler page; hard boards at two givens or fewer stand on their own as a proper challenge. The free demo generates real puzzles, one at a time, so you can see whether they suit your readers before deciding anything.