How Constraint Puzzles Actually Work — The Six Ideas Behind Every Logic Grid

Most people learn sudoku by scanning. You look at a row, mutter the digits you can see, and hope one cell announces itself. That works for a while. Then you meet a grid that refuses to announce anything, and you stall.

The fix is not more scanning. It is a different question. Solvers who never stall are not faster at looking; they ask something narrower, and they have a handful of named tools for answering it. Those tools are not sudoku tools. They work on letter grids, logic grids, ordering puzzles, graph puzzles — anything where a fixed set of values must be distributed under rules.

Here are the six, in the order you should learn them, each with an example you can verify on paper.

1. Candidates, not answers

The most useful habit in constraint puzzles is to stop asking what goes here and start asking what cannot go here.

For every empty cell, keep the list of values still permitted by the rules. That list is the cell's candidates. In a classic sudoku, a cell in row 4, column 7 starts with all nine digits. Its row already contains 1, 2, 5 and 8. Its column contains 3, 4 and 9. Its box contains 6. Cross those off and one candidate survives: 7.

Notice the shape of that move. You did not find the 7. You destroyed the other eight. Every technique below destroys candidates, and the answers are what stands when you run out of things to destroy.

This is why the skill transfers. In Terraxis, countries are arranged so each row increases left to right by its own rule — area, population, capital latitude, land neighbours, the length of the English name. A cell's candidates are the countries that could still sit there. If a row increases by population, its leftmost cell cannot hold the most populous country in the pool. Different puzzle, identical move.

Write your candidates down. Holding them in your head is where accuracy goes to die.

2. The naked single and the hidden single

These are the two ways a cell can become forced, and they are opposites.

A naked single is a cell with one candidate left — the example above. Call this casewise elimination: you narrowed a container until only one occupant fit.

A hidden single is a value with one home left. Take the top-left box and hunt for 7. Say a 7 already sits in row 1 to the right of the box, and another in row 3 to the right. Rows 1 and 3 are closed to 7 inside the box, so its 7 lives in row 2: r2c1, r2c2 or r2c3. Now say column 1 holds a 7 lower down, and column 2 holds one too. That kills r2c1 and r2c2. Only r2c3 survives, so r2c3 is 7.

Here is the part that trips people up: r2c3 may still have three or four candidates written in it. It does not matter. No other cell in the box can host the 7, and the box needs one. Call this valuewise elimination: you narrowed an occupant's options until only one container fit.

Alternate between the two. They find different things, and each placement feeds the other. A grid that feels stuck usually means you have been doing one of them for ten minutes.

3. Naked pairs and triples

Now we stop placing values and start deleting them in bulk.

In row 5, two cells both hold exactly the candidates {4, 6}. You cannot tell which is which, and you do not need to. Between them they consume row 5's 4 and its 6. So no other cell in row 5 can be 4 or 6. A third cell reading {4, 6, 9} is now a naked single: 9.

That is a naked pair: two cells, two candidates, one unit.

The naked triple is the same argument with three cells, and it carries a detail almost everyone gets wrong at first. The cells do not each need all three candidates. What matters is that the union of their candidates has exactly three values. So {2,7}, {7,9} and {2,9} form a triple on {2,7,9}, though no cell shows all three. So do {2,7}, {2,7,9} and {7,9}. Three cells, three values: strike 2, 7 and 9 from the rest of the unit.

4. Hidden pairs and triples

This is the inverse, and the one most solvers never look for.

Scan a unit value by value and ask where each value could go. In column 4, the 3 is possible in only two cells, and the 8 is possible in those same two cells. Two values, two cells. They must take the 3 and the 8 between them, in some order.

So everything else written in those cells is wrong. If they read {1,3,5,8,9} and {3,4,8}, both collapse to {3,8}. You deleted five candidates without placing a value — and created a naked pair, which now clears 3 and 8 from the rest of the column.

Hidden pairs hide well because the cells look busy. A cell crowded with five candidates does not feel constrained, so the eye skips it. Sweep by value, not only by cell. Hidden triples work identically: three values appearing in only three cells of a unit own those cells outright.

5. Intersections

The first technique that uses two units at once.

Take the centre box and the value 2. Every cell in that box that could hold a 2 lies in the box's middle row. You do not know which one it is. But the box needs a 2, and every candidate sits on that row, so the row's 2 is inside this box. The rest of that row — six cells — cannot be 2. That is block-line reduction, often called pointing.

Run it backwards for line-block reduction. If every cell in a row that could hold a 2 falls inside one box, the box's 2 must lie on that row, so strike 2 from the box's other six cells.

One idea, twice: when two units overlap, a value confined to the overlap is banned from both remainders.

6. The rectangle (X-Wing)

Everything so far has been local. The X-Wing is the first technique that reaches across the grid.

Pick a value, say 4. In row 2 the only candidates for 4 sit in columns 3 and 7. In row 8 the only candidates for 4 sit in columns 3 and 7 as well. Four cells, one rectangle.

Row 2 puts a 4 in column 3 or column 7. Row 8 does the same. They cannot pick the same column, because two 4s cannot share one. So one takes column 3 and the other takes column 7, and you never need to know which way round. Either way, both columns already have their 4. Delete 4 from every other cell in columns 3 and 7.

It generalises at once. Three rows in which a value is confined to the same three columns is a swordfish; four is a jellyfish. The argument never changes: n units, n positions, every position spoken for.

The absence of a mark is a mark

Here is the idea worth more than the other six combined.

In Alfaelo, a 9×9 letter sudoku, a dot between two adjacent cells means the letters are consecutive in the alphabet. Beginners treat the dots as the puzzle and the blank edges as background. That is backwards. Dots are rare. Blank edges are everywhere, and each is a constraint: these two letters are not consecutive.

Place an E, and every undotted neighbour loses D and F at once. One placement, up to eight eliminations, and the grid drew you nothing. The same holds in Confinia, where every drawn link stands for a real shared land border. A link says two countries touch. A gap says the countries in those slots do not — often the sharper fact.

Positive clues narrow one cell. Negative clues narrow the whole grid at once, quietly, and only for the solver who bothers to read them.

Logic grids make this explicit: you spend the puzzle writing crosses, not ticks, and the ticks appear on their own when a row of crosses leaves one gap. Same skeleton as every technique above.

The order to apply them

Sweep in this order, restarting from the top after every placement.

• Naked singles. Free, instant, always first.
• Hidden singles, unit by unit, value by value.
• Naked pairs and triples.
• Hidden pairs and triples.
• Intersections.
• Rectangles, only when the cheaper passes are exhausted.

The order follows cost. Each step costs more to look for than the last, and each placement hands the earlier steps new work. Restarting after every move feels wasteful and is the fastest route through a hard grid.

One rule holds it together: if you are guessing, you have stopped solving. A guess that turns out right teaches you nothing and hides the elimination you missed. Stuck never means you have run out of logic. It means you have run out of the kind of logic you were using. Switch from cells to values, or from one unit to two.

That skeleton — list what is possible, destroy what is impossible, continue until one thing stands — is the whole of it. It drives every constraint puzzle here, including Meridiano, built from country relations, border walks of an exact length and map jigsaws, where there is not a number in sight and the method does not care.