Pencil and Paper Puzzles: Four Kinds and How to Solve Them

Pencil and paper puzzles fall into four main groups: placement puzzles, where you put pieces on a board so that none of them clash; tour puzzles, where you trace one route through every cell or line; grid deduction puzzles, where you fill or shade a grid from a few clues; and liar tables, where you sort speakers into truth-tellers and liars. All you need is a pencil, a sheet of paper and the rules, because every step can be written down, checked and erased.

This guide explains each group, gives a fully solved example for each one, and ends with a table that compares them and a few habits that work for all of them.

What counts as a pencil and paper puzzle?

A pencil and paper puzzle is any puzzle you solve by marking a drawing rather than by moving parts. The drawing might be a chessboard, a grid of empty cells, a figure made of lines or a short list of statements.

The pencil is the real tool. It lets you record what you have ruled out, so you never have to hold the whole puzzle in your head. Good solvers write down small marks constantly: a dot where a piece cannot go, a light letter for a guess, an X for a dead end. When a guess fails, the eraser takes you straight back.

What are placement puzzles?

A placement puzzle asks you to put pieces on a board so that they obey a rule, usually that no piece may share a line with another or touch it. The classic is the queens puzzle: place queens on a chessboard with no two in the same row, column or diagonal.

Worked example. On a four-by-four board, number the rows 1 to 4 from the top and the columns 1 to 4 from the left. Put the queens in column 2 of row 1, column 4 of row 2, column 1 of row 3 and column 3 of row 4:

No two queens share a row, a column or a diagonal. This board has exactly two solutions, and the other one is this grid seen in a mirror.

A newer twist lets you place pawns, and a pawn blocks any queen's line that runs through it. A paper published on 8 December 2016 notes that one pawn lets you fit one queen more than the board's width on any board at least six squares wide ^[1]. With as many pawns as you like, the limit on an even board is a quarter of its squares, because the board splits into two-by-two blocks and two queens in one block would always sit side by side ^[1].

Pieces with long moves make the board feel small. A lone rook or queen on an empty board can reach any square in at most two moves, and a 2021 paper, published on 7 December, studied how many pawns it takes to make some trips longer ^[2].

How to start: fill the board one row at a time. Lightly mark every square the new piece covers, and the moment a row has no safe square left, go back and move the previous piece.

What are tour puzzles?

A tour puzzle asks for a single route that visits every square, or uses every line, exactly once. You trace it with the pencil, so a wrong turn is easy to see and easy to undo.

Line-drawing puzzles are tours of a figure: draw the whole figure in one stroke, never lifting the pencil and never going over a line twice. A good first step is to count the lines that meet at each point.

Worked example. Draw a square with corners A at bottom left, B at bottom right, C at top right and D at top left. Add both diagonals, crossing at E, and a roof peak F joined to D and C. Count the lines at each point: A and B have three each, and every other point has an even number. Start at A, one of the odd points. The route A, B, C, D, A, E, C, F, D, E, B uses all ten lines once and ends at B, the other odd point.

The knight's tour is the board version: a chess knight must land on each square exactly one time. If the last square is one knight's move from the first, the tour is closed ^[3].

Worked example. Here is an open tour of the five-by-five board. Follow the numbers from 1 to 25; each step is a knight's move:

It cannot be closed, and the reason is a neat colouring argument. Colour the board like a chessboard: every knight's move lands on the other colour, so a closed tour must alternate colours all the way round and needs an even number of squares. This board has 25. The same kind of two-colour argument shows that a cube of cells with an odd side length has no closed tour either ^[3]. Smaller boards are harsher still: the four-by-four board has no tour at all, open or closed ^[3]. A 1991 result, restated in a paper published on 9 May 2007, lists exactly which rectangles allow a closed tour ^[3].

How to start: in a knight's tour, write each step's number in light pencil as you land, so a dead end shows exactly how far to back up. In a line drawing, count the lines at every point first and pick your start before the pencil touches the page.

What are grid deduction puzzles?

Grid deduction puzzles give you an empty or partly filled grid and a few clues, and ask you to complete it so that every rule holds. The group is large, and plain descriptive names suit it best:

• Number-place puzzles: fill the grid so that every row, column and box holds each symbol once.
• Shading puzzles: shade some cells so that the clue numbers count the shaded cells around them.
• Loop puzzles: draw one closed loop through the grid, guided by numbers that say how many of a cell's sides it uses.
• Region-splitting puzzles: cut the grid into areas whose sizes match the numbers inside them.
• Path-joining puzzles: link matching pairs of symbols with lines that never cross or share a cell.

A well-made grid has one answer that you can reach without guessing. Each clue removes a few possibilities, and every placed mark removes more.

Worked example. Here is a small number-place puzzle with letters. Each row, each column and each of the four two-by-two boxes must hold A, B, C and D once:

Row 1 still needs B and C. Column 2 already has a C, so row 1 reads A, B, C, D. Row 4 still needs A and D, and column 1 already has an A, so row 4 reads D, C, B, A. The top-left box now needs C and D, and column 1 already has a D, so row 2 starts C, D. The top-right box needs A and B, and column 4 already has an A, so row 2 ends A, B. Every cell of row 3 now has one letter left in its column, giving B, A, D, C. The finished grid is unique.

Here is a grid riddle where one tap has to light every tile:

How to start: look for the row, column or box with the most filled cells, and ask which symbols it still needs and where each of them can go.

What are liar tables?

A liar table is a puzzle about speakers. Each one is either a truth-teller, whose every statement is true, or a liar, whose every statement is false, and your job is to label each one. The table is your working sheet: one column per assumption, one row per speaker.

These puzzles have a surprising limit. When the only speakers are truth-tellers and liars, trading each truth-teller for a liar and each liar for a truth-teller in a valid answer always gives a second valid answer, so the statements alone can never single out one answer ^[4]. A paper published on 4 February 2020 added a third kind of speaker, who can only make statements that contradict themselves, such as saying that they are a liar, and showed that puzzles with a single answer then become possible ^[4].

Worked example. A says that B is a truth-teller. B says that B is a liar. C says that A is a liar. With only two kinds of speaker this puzzle has no answer at all, because neither a truth-teller nor a liar can call themselves a liar. Allow the third kind and it has exactly one:

The chest riddle below runs on the same skill: exactly one label tells the truth, and the rest do not.

How to start: pick one speaker, assume they tell the truth and follow the consequences. Should that lead to a clash, that speaker must be lying. Then run the opposite assumption through the same test.

How do the four groups compare?

Which habits work for every pencil and paper puzzle?

Mark what is impossible, not only what is certain. Most progress comes from ruling things out. A dot in a cell that cannot hold a queen is as useful as the queen itself.

Change one thing at a time. After every new mark, check what it forces elsewhere before you make the next one. Chains of forced moves are how these puzzles are meant to be solved.

Guess lightly and record the guess. When you are stuck, write a guess faintly and follow it. If it leads to a clash, erase everything after it and you have learned something certain.

Check the finished answer against every rule. A full grid or a complete tour is not proof. Read each rule once more and test it against what is on the page.

Where can you try one now?

The lock riddle below is a pure pencil and paper puzzle: write down the possible codes, then let each clue strike some out.