Logic Grid Puzzles: How to Solve Them With X and Tick

To solve a logic grid puzzle, draw a grid that pairs every item with every other item, then use each clue to put an X where a pairing is impossible and a tick where it is certain. When only one empty cell is left in a row or column of the grid, that cell must be a tick, and every new tick carries information across to the rest of the grid.

That is the whole method. This guide shows how to set out the grid, the four moves that do most of the work, a full worked puzzle and a practice puzzle with its answer.

What is a logic grid puzzle?

A logic grid puzzle gives you a few groups of things, such as people, colours and places, and a short list of clues. Each person is matched with exactly one colour and exactly one place, and no two people share either. Your job is to find the one way of matching everything that agrees with every clue.

A well-made puzzle of this kind has a single answer. People who analyse logic puzzles formally use the same test: a puzzle is called good when it has exactly one solution ^[1]. If you ever reach two answers that both fit, either a clue was missed or the puzzle is faulty.

What does the grid look like?

The grid is a set of small square tables. For three groups, such as people, colours and places, you need three of them:

• people against colours
• people against places
• colours against places

Each small table has a row for every item in one group and a column for every item in another. With four people, four colours and four places, each table is four by four, so the whole grid has forty-eight cells.

Printed grids usually fit the three tables into an L shape: people down the side, colours and places across the top, and the colours against places table tucked underneath. When you draw your own, you can copy that layout or simply draw three separate tables. The layout does not matter. What matters is that every pair of groups has its own table.

What does mark X and tick mean?

Every cell in the grid answers one yes-or-no question, such as whether Dana carried the blue umbrella. You fill it with one of two marks:

• X means no: this pairing is impossible.
• Tick means yes: this pairing is certain.

Use a pencil, and never mark a cell on a hunch. Each mark should come from a clue or from marks already on the grid. If you want to try a guess, note it beside the grid, not inside it.

When you place a tick, finish the job at once. A tick means every other cell in its row and in its column gets an X. If Maya has the red umbrella, nobody else has red and Maya has no other colour.

How does elimination work?

Most clues do not hand you a match. They tell you what is not true, and you mark an X. Elimination is the steady build-up of X marks until only the truth is left standing.

Learn to translate each kind of clue into marks before you start:

Negative clues are easy to misread. Dana does not have the blue one tells you nothing about which colour she does have. Mark the single X and move on.

What is the only-one-left rule?

This is the move that turns X marks into answers. Look along any row or column of a small table. If every cell but one holds an X, the empty cell must be a tick, because each item has to match something.

In a four by four table, three X marks in a line are enough to place a tick. After you place it, apply the tick rule and X out the rest of its row and column. Those new X marks can leave another line with a single empty cell, and the answers start to fall one after another.

Scan for this after every new mark, not only when you feel stuck. A tick can sit waiting for several steps before you notice it.

How do you chain clues across the grid?

Chaining is the step most beginners skip, and it is the one that cracks the harder puzzles. The three small tables describe the same people, so what you know in two of them tells you something in the third.

1. Two ticks give a third. If Rami has the blue umbrella and the blue umbrella went to the park, then Rami went to the park.
2. A tick carries X marks across. If Rami has the blue umbrella and Rami did not go to the market, then the blue umbrella did not go to the market either.
3. A tick and an X give an X. If Fadi went to the library and the blue umbrella went to the park, then Fadi cannot have the blue umbrella.

Each time you place a tick, look at the other two tables and ask what it carries across. That one habit saves many rereadings of the clue list.

How does the method solve a four by four puzzle?

Here is an original puzzle to try the method on. Four friends, Rami, Dana, Fadi and Maya, went out on a rainy afternoon. Each carried an umbrella of a different colour, red, blue, green or yellow, and each spent the afternoon in a different place: the library, the park, the market or the harbour.

1. Maya carried the red umbrella.
2. Fadi spent the afternoon at the library.
3. Whoever carried the green umbrella went to the harbour.
4. Dana did not carry the blue umbrella.
5. Whoever carried the blue umbrella went to the park.

Who went where, and with which umbrella? Have a go before reading on. Every clue matters here: a computer check of every possible arrangement shows that all five clues together leave exactly one answer, and that dropping any single clue lets more than one answer through.

Step 1: the direct clues. Clue 1 gives a tick for Maya and red, so X out the rest of Maya's row and the rest of the red column. Clue 2 gives a tick for Fadi and the library, with X marks for the rest of that row and column. Clue 4 gives a single X for Dana and blue.

Step 2: chain clue 5. Blue goes with the park. Fadi is at the library, so he is not at the park, so he cannot have the blue umbrella. Mark that X.

Step 3: only one left. Look down the blue column. Maya has red, and Dana and Fadi have X marks. Only Rami's cell is empty, so Rami has blue. Tick it and X out the rest of his row. Now chain again: blue goes with the park, so Rami went to the park.

Step 4: chain clue 3. Green goes with the harbour. Maya has red and Rami has blue, so neither has green. Fadi is at the library, not the harbour, so he does not have green either. That leaves Dana, who carried green and went to the harbour.

Step 5: finish with only one left. Fadi is the only person without a colour, and yellow is the only colour left, so Fadi carried yellow. The market is the only place left, and Maya is the only person without a place, so Maya went to the market.

Check every clue. Maya has red. Fadi is at the library. Green belongs to Dana, who is at the harbour. Dana does not have blue. Blue belongs to Rami, who is at the park. All five clues hold. Always finish with this pass, because it catches a mark placed in the wrong cell.

How do grid sizes compare?

The number of possible answers grows fast. With four people there are twenty-four ways to hand out the colours and twenty-four ways to hand out the places, so 576 combinations in all. Good clues knock out whole blocks of these at once, which is why five short clues can leave just one.

Why does this method work so well?

A grid puzzle is a set of choices plus rules that forbid some combinations. A computer can work through puzzles of this kind with nobody guiding it, and several techniques for doing so have been put forward ^[1].

On paper, mathematicians treat them in one of two ways: as everyday common sense, or as statements in formal logic ^[1]. One paper on truth-teller and liar puzzles even draws each puzzle as a graph, with people as points and statements as links between them ^[1].

Computers also handle rule-out problems far larger than any grid. A well-known puzzle about fitting squares inside one large square was finally closed by a computer search that modelled it as a constraint-satisfaction problem ^[2]. An earlier article on the same problem relied on backtrack programming ^[2].

The scale is very different from a pencil grid, but the spirit is related. Each X removes an option, and when enough options are gone, what is left must be true.

What are the most common mistakes?

• Marking a guess as a fact. One wrong tick spreads through the whole grid.
• Forgetting the row and column after a tick. The X marks that follow a tick are where the next answers come from.
• Reading a negative clue as a positive one. Not blue never means some other named colour.
• Ignoring the third table. The colours against places table is where most chaining happens, so keep it up to date.
• Skipping the final check. Read every clue once more against your finished answer.

What should you do when you get stuck?

• Reread each clue that mentions an item that is still open.
• Look for clues that link two groups without naming a person, such as whoever carried green went to the harbour. They are often the key.
• Scan every row and every column of all three tables for a single empty cell.
• If you must guess, guess one cell, note it beside the grid and follow it. If it ever forces two ticks into one row, the guess was wrong, so mark that cell X.

Can you solve this practice puzzle?

Three neighbours, Nadia, Tariq and Samir, each bought one item at a market stall: a lamp, a kite or a vase. Each item was a different colour: blue, green or white.

1. Samir did not buy the lamp.
2. The kite was blue.
3. Nadia's item was green.
4. Samir's item was not blue.

Who bought what, and in which colour? Draw the three small tables and try it before reading the answer.

Answer: Nadia bought the green lamp, Tariq bought the blue kite and Samir bought the white vase. The kite is blue. Nadia's item is green and Samir's is not blue, so neither of them bought the kite, and by the only-one-left rule Tariq did. The only colour left for Samir is white. Samir did not buy the lamp, so he bought the vase, which leaves the lamp, in green, for Nadia.

Now try a live riddle that works like a small grid puzzle: three people, three fruits and two clues. Draw a three by three table and see how quickly the X marks give it away.