How these sheets are made
A Killer Sudoku gives you nothing. That is the point.
A classic Sudoku hands you about a third of the answer and asks for the rest. A Killer Sudoku hands you nothing at all: no digit is printed anywhere on the grid. What replaces the givens is arithmetic, and the whole puzzle turns on which totals can only be made one way.
The totals that can only be made one way are where every solve starts
A two-cell cage totalling 3 has to be 1 and 2. There is no other pair of different digits that makes 3, so those two squares are pinned before you have looked at anything else. A two-cell cage totalling 10 could be 1 and 9, 2 and 8, 3 and 7, or 4 and 6, and tells you almost nothing on its own. Every Killer Sudoku is solved by finding the first kind and working outwards.
- One pair only, a foothold
- Two pairs
- Three or more
The one that catches people out
A cage total is not a Sudoku constraint on its own. Two squares in the same cage totalling 10 could be 4 and 6, but if those squares also share a row, that is fine; if they share a row and a 6 already sits in it, that pair is gone. The cage rule and the grid rules have to be read together, and the sheet is only hard until you start doing both at once.
Difficulty here is the deduction you are forced to find
Every level on this page is named after a technique, and a sheet is only offered at that level when the generator has confirmed it cannot be finished without it. That is a stronger promise than a clue count, and it is the same definition the Futoshiki page uses.
| Level | The deduction it forces | What that means at the table |
|---|---|---|
| Easy | Rule of 45 on one house | Every row, column, and box holds 1 to 9, so it totals 45. Add the cages inside one house and the difference names the cell left over. |
| Medium | Naked and hidden subsets | Two cells in a house that can only hold the same two digits take those digits away from every other cell in that house, and the same for three. |
| Hard | Innies and outies across houses | Apply the same 45 arithmetic across two or three houses at once, so a cage poking out of a band names a cell several boxes away. |
| Expert | Following a candidate to a contradiction | Take a cell down to two possibilities, pencil one in, and follow it until the grid breaks. If it breaks, the other one was right. This is proof, not guessing: the wrong branch is ruled out rather than abandoned. |
Read from the engine at build time: each row names the technique the sample sheet for that level actually required.
Fewer and larger cages is harder. That is a result rather than a setting: nobody chose it, and the generator was measured before it was believed.
The structure of a sheet follows its difficulty rather than causing it, and the direction is worth knowing before you print. A harder level does not mean more cages to read. It means fewer of them, each covering more squares, each therefore saying less.
| Level | Cages on the sheet | Mean cage size | Largest cage | Cages with only one possible set |
|---|---|---|---|---|
| Easy | 34.0 | 2.38 | 4.3 | 5.0 |
| Medium | 34.7 | 2.34 | 3.7 | 2.7 |
| Hard | 29.7 | 2.74 | 4.7 | 2.3 |
| Expert | 28.0 | 2.90 | 5.0 | 2.7 |
3 generated sheets per level, measured at build time. The last column is the count of footholds: cages whose total admits exactly one set of digits.
The same puzzle at its gentlest and its hardest
Both sheets are real output from the builder above.
The rule of 45, which is the technique worth learning first
Every row, column, and box holds the digits 1 to 9 exactly once, so every one of them totals 45. Add up the cages that sit wholly inside a row, and whatever is missing from 45 belongs to the squares that stick out. When exactly one square sticks out, you have just been handed its value.
This is not an advanced trick on this puzzle, it is the basic one, and it is why every level on this page assumes it. A grid with no givens cannot be started any other way.
Printing choices that matter here more than elsewhere
- One puzzle per page is the default for a reason: a 9x9 grid with a small total in every cage corner needs the room, and two on a sheet halves both.
- Colour cages make the boundaries obvious and cost a lot of ink across a class set. Black and white uses the dashed outline alone and prints just as clearly.
- Ink saving thins the grid rules but leaves the cage outlines and totals at full strength, because those are the puzzle rather than the furniture.
- Answer keys print two or four to a page. A pack of six with keys at four per page is nine sheets rather than twelve.
What the generator refuses to print, and why that took the longest
The first version of this engine grew cages at random and checked whether the result had a single answer. Measured over 30 layouts, 26 of them had more than one solution. Only four were unique. A cage total is a far weaker constraint than it looks, and a random cage layout is not a puzzle: it is a grid with some arithmetic written on it.
What fixed it was repairing layouts rather than throwing them away. When the solver stalls, the cage covering the square it cannot resolve is split in two, which replaces one loose total with two tighter ones. Repeat until the puzzle can be finished by reasoning. Because the solver only ever eliminates digits it has proved impossible, a sheet it can finish is also a sheet with exactly one answer, so the uniqueness check and the no-guess guarantee are the same check.
Then the opposite move, which is what makes the levels mean anything
Repairing alone stops the moment a puzzle becomes solvable, so it lands on whatever difficulty that happens to be. With repair alone, only 13 per cent of Hard sheets actually needed the technique Hard is named after. So the generator then merges cages back together for as long as the level's own techniques still reach the answer, which removes information rather than adding it. Every cage boundary that survives is one the solve genuinely needs. That took Hard from 13 per cent to 88 per cent, and made generation faster rather than slower.
One consequence is printed under the builder rather than hidden: when a sheet in your pack comes out easier than the level you asked for, the preview says so and says how many. Every sheet is still checked to have one answer reachable without guessing; it is only the named technique that was not forced.
How this page is made and reviewed
Print Puzzles is part of the Hamilton Digital Media Limited puzzle network. This page is written and reviewed by Brian Hamilton and Karan Hamilton. The engine was checked with a build-time harness that generated puzzles across all four levels and confirmed, for every one, that the solution is a valid Sudoku, that the cages cover all 81 squares exactly once, that every printed total matches the answer, that no digit repeats inside a cage, that every cage is a connected shape, that the logical solver reaches the answer key exactly, and that the puzzle is not solvable one rung below its own level. Uniqueness was confirmed separately against an exhaustive search using no solving techniques at all. Read more about how we make puzzle packs, or tell us about a sheet that misbehaves through the contact page.
Nothing here costs anything and there is no account. Puzzles are generated in your own browser rather than fetched from a server, which is why a harder pack takes a few seconds and why the worksheet title you type never leaves the page. The terms of use let teachers, tutors, clubs, and families print and copy these sheets and their keys freely.