Technique

Four deduction techniques that work in any puzzle

These four tools solve the vast majority of logic puzzles. The first three are the ones to master; the fourth is the one worth holding back.

People who have been solving these puzzles for a while do not think in terms of individual clues, but in terms of patterns. They recognise the shape of the deduction before they read the detail. These are the four main shapes, from most to least frequent.

1. Cross elimination

The most basic one, and the one used most often. If an element occupies a position, no other element of its own category can occupy that position; and that element cannot be in any other.

It sounds obvious, but it gets applied less than it should. Every time you confirm something, you have to sweep the whole row and the whole column, crossing out what has just become impossible. Plenty of people confirm and carry on reading clues without closing that sweep, and then get stuck on information they already had.

Cross elimination also has a version that gives more than it looks: if an element has only one possible position left, it is solved, even if no clue says so. And the other way round: if a position has room for only one element, the same applies. Those are two different checks and it is worth doing both.

2. Contagion between linked elements

When a clue ties several elements together — "the mill, the cow and the lettuce are in the same column" — those elements stop being independent: they start behaving as a block.

The practical consequence is powerful: any elimination on one of them copies across to all the others. If you work out that the lettuce cannot be in the first position, then automatically neither can the mill nor the cow. No additional clue is needed to say so.

It is the technique with the best return on effort, and also the one most often forgotten, because it requires remembering that those three elements are still tied together long after you read the clue that tied them.

3. Reasoning by ranges

This one shows up with ordering clues: "A is to the left of B", "A comes before B". The trap is that these clues look like they say nothing until one of the two is placed. That is not true.

From the very first moment, an ordering clue hands you two free eliminations: A can never be in the last position — there would be no room to its right — and B can never be in the first. That is free information you can mark before reading anything else.

And it keeps paying out on its own: when A only has, say, positions 2 and 3 left, then B cannot be in 1 or in 2, because in the best case A is in 2 and B has to come after. You do not need to know exactly where A is; it is enough to know the earliest position it can still occupy.

The same thing works mirrored from the other end, and with "between" clues: if the middle element cannot sit on the edges, the two outer elements are a fixed distance apart, and any elimination on one translates into an elimination on the other.

4. Proof by contradiction, in moderation

This means assuming an element sits in a particular position, following the consequences, and seeing whether you reach an absurdity. If it blows up, that position is eliminated.

It always works, which is why it is tempting. But it has two problems. The first is that it is slow and error-prone: if you slip halfway down the chain, you eliminate something that was valid and the puzzle becomes unsolvable without your knowing why. The second is that, in a well-built puzzle, you almost never need it: if you find yourself reaching for it, the odds are there is a direct deduction you have not spotted.

My advice: use it only on short chains you can verify at a glance, of the "if the cow were in the first column, the carrot would have to be in column zero, which does not exist" kind. That is not guessing, it is a one-step check. Save the long chains for when you are certain there is nothing left to squeeze.

The working order I recommend

With all four tools on the table, the routine that produces the fewest dead ends is this:

1. Collect what is free. Go through the ordering and "between" clues and mark the eliminations they hand you without needing anything else.
2. Place whatever is absolute. The clues that pin down a specific position, and the immediate consequences.
3. Sweep after every confirmation. Row and column, no exceptions.
4. Go back to the linked clues. Every new elimination can spread.
5. Repeat until nothing changes. Only then consider a short hypothesis.

The step most people skip is the fifth. They take one pass, see nothing obvious and jump straight to guessing. There was almost always something there.