Five techniques, one at a time. Each step is a short lesson, a worked example, then three puzzles that genuinely need that technique. No guessing, ever: every puzzle here falls to pure logic.
Step 1 of 5
Naked Single
Sometimes a cell has nowhere left to hide.
The lesson: A cell that sees every digit but one must be that one.
Pick an empty cell and look along its row, down its column and around its box. Every digit you spot there is one this cell can’t be.
Cross those off. If only one digit survives, that’s a Naked Single: the cell has exactly one candidate left, so that’s its digit. No guessing involved.
Where to look: Busy neighborhoods. A cell whose row, column and box are already mostly filled has the fewest options.
7
4
5
1
8
8
6
4
2
3
5
9
9
4
3
8
7
1
3
9
2
8
5
1
4
8
7
9
5
3
1
2
4
Worked example. Row 2, column 2 already sees every other digit in its row, column and box, so it must be 6.
Green: the digit it proves. Purple: what the reasoning rests on. Amber: where to look.
Three for three. Counting what a cell sees is yours now. Next up: the Hidden Single.
Step 2 of 5
Hidden Single
Flip the question around: not “what goes here?” but “where does the 7 go?”
The lesson: A digit with only one possible spot in a row, column or box goes there.
Pick a box (or a row, or a column) and a digit it’s still missing. Walk its empty cells: if a cell already sees that digit in its own row or column, it’s out.
When only one cell is left standing, the digit goes there. That cell might have other candidates too. Doesn’t matter: the digit has nowhere else to go.
Where to look: Digits that are already all over the board. Each copy knocks out a whole row and column at once.
8
4
5
6
4
5
6
9
7
7
9
6
5
4
8
8
3
5
2
5
6
4
9
1
4
1
7
6
9
1
7
3
6
8
2
8
7
1
3
Worked example. In box 1, the 6 can only go in row 3, column 3. Every other empty cell there already sees a 6.
Green: the digit it proves. Purple: what the reasoning rests on. Amber: where to look.
Nicely done. Singles of both kinds will carry you through every Easy board. Next: pairs.
Step 3 of 5
Naked Pair
Two cells, two digits, and the rest of the house is locked out.
The lesson: Two cells in a house with the same two candidates claim those digits: cross them off everywhere else in the house.
When the singles dry up, fill in the candidates (the button does it for free). Now look for two cells in the same row, column or box with exactly the same two candidates, say 4 and 8.
One of them is the 4 and the other is the 8. You don’t know which yet, and you don’t need to: together they’ve claimed both digits. Every other cell in that house can cross off 4 and 8.
Those crossings-off usually leave some cell with one candidate, or some digit with one spot. That’s the payoff.
Where to look: Cells with just two candidates. Then scan their row, column and box for a twin.
23478
2
238
1
5
9
5
1
9
3
8
47
47
6
3
9
8
2
1
5
5
8
1
9
1
3
8
6
3
7
9
4
1
8
5
2
8
3
6
4
9
7
9
8
4
2
5
7
1
6
3
Worked example. Row 3, column 1 and row 3, column 2 can only be 4 or 7, so those two cells take the 4 and the 7 of box 1. No other cell in box 1 can be 4 or 7. That leaves row 1, column 2 with only one candidate: 2.
Green: the digit it proves. Purple: what the reasoning rests on. Amber: where to look. Red, struck through: candidates it crosses off.
Once you see pairs, you can’t unsee them. Next: the same idea, hidden.
Step 4 of 5
Hidden Pair
The mirror image of a Naked Pair, and much easier to miss.
The lesson: Two digits that only fit in the same two cells of a house own those cells: cross off their other candidates.
Pick a house and look at where each missing digit could go. If two digits can only go in the same two cells, those two cells belong to them.
The cells might show other candidates too. Those are decoys: cross them off, and the pair is out in the open.
Where to look: A digit with exactly two spots in a house. Then check whether another digit has the very same two spots.
4
8
3
6
1
7
2
7
8
3
6
9
3
4
8
4
1
7
236
9
5
1236
1
6
7
4
23
4
5
8
1236
3
12456
3
2
1
8
3456
Worked example. In column 9, the 4 and the 5 only fit in row 8, column 9 and row 9, column 9. Those two cells must hold them, so they can’t be anything else. That leaves the 3 only one spot in row 9: row 9, column 2.
Green: the digit it proves. Purple: what the reasoning rests on. Amber: where to look. Red, struck through: candidates it crosses off.
That’s the trickiest pair done. One more to go: the Pointing Pair.
Step 5 of 5
Pointing Pair
A box can tell a whole row what to do.
The lesson: When a digit’s spots in a box all sit in one row or column, it can’t go anywhere else in that line.
Look inside one box at one digit. If every cell that could hold it sits in the same row, then that row’s copy of the digit lives inside this box.
So the rest of that row, outside the box, can’t have it. Cross it off there. Same thing for columns, and it works with three cells in a line too.
Where to look: A digit whose candidates in a box line up. Then follow that line out of the box.
235
257
37
2567
2357
8
123467
124679
134679
9
1
6
3
4
8
4
27
8
1
9
5
8
4
9
6
3
2
3
8
2
9
9
8
3
8
5
3
8
7
2
8
Worked example. In box 1, every spot left for a 3 is in row 1. Whichever one gets it, the 3 of row 1 is inside box 1, so the rest of row 1 can’t be 3. That leaves the 3 only one spot in box 2: row 2, column 5.
Green: the digit it proves. Purple: what the reasoning rests on. Amber: where to look. Red, struck through: candidates it crosses off.
That’s the whole track. Every Hard board here is solvable with these five. Go find one.
