Self-Referential Puzzles: How to Solve Statements About Themselves

A self-referential puzzle is a puzzle in which a statement talks about itself, about the person saying it, or about the list it belongs to. To solve one, treat the truth of each statement as an unknown, try every possible value, and keep only the choice under which every statement agrees with its own value.

That single test, which this guide calls the consistency check, handles talking labels, lists of statements that count their own true members, sentences that count their own letters and sequences built from their own earlier terms. Below you will find each kind, a step-by-step method, one full worked example and a practice puzzle with its answer.

What makes a puzzle self-referential?

In an ordinary logic puzzle, every clue points outward. It tells you something about a person, a box or a colour, and you check it against the facts. In a self-referential puzzle, at least one clue points back at itself. Its truth depends on the answer, and the answer depends on its truth.

That loop is what makes these puzzles feel slippery. You cannot read a clue, decide it is true and move on, because deciding it is true may change what it says. The way out is to stop judging clues one at a time and test whole scenarios instead.

Here are a few typical self-referring clues:

• I am a liar.
• This label is false.
• Exactly two statements in this list are true.
• This sentence has five words.
• Each new term comes from looking back at earlier terms of this same sequence.

Why can nobody say I am a liar?

Take a puzzle where every speaker is either a truth-teller, who only says true things, or a liar, who only says false things. Suppose speaker A says: I am a liar. If A is a truth-teller, the sentence is true, so A is a liar, which clashes. If A is a liar, the sentence is false, so A is a truth-teller, which clashes again. Neither type survives.

A paper published on 4 February 2020 makes the same point: no truth-teller and no liar could make that claim, and in a solvable puzzle with only those two types, nobody says I am a liar ^[1]. It also looks at statements that are neither true nor false whichever of the two types says them, calls them self-contradictory, and proves that such statements contain self-reference ^[1].

The mirror image is just as instructive. If A says I am a truth-teller, both types can say it: a truth-teller says something true, and a liar says something false. Here the trouble is the opposite one. Instead of no consistent reading, there are two, so the sentence tells you nothing about A.

What kinds of self-referential puzzles are there?

Most self-referential puzzles fall into a handful of kinds. They look different on the page, but the same consistency check solves all of them.

The first three are pure true-or-false puzzles. The self-counting sentences and self-describing numbers mix logic with careful counting. The sequences add arithmetic, but the loop is the same: the rule uses the very thing it is building.

What is the step-by-step method?

1. List the unknowns. For a list of statements, each unknown is whether a statement is true or false. For a speaker puzzle, it is each speaker's type. For a fill-in sentence, it is the missing number.
2. List the possible scenarios. A scenario fixes every unknown at once. Each extra true-or-false statement doubles the number of scenarios, so short lists stay manageable.
3. Evaluate inside the scenario. Pretend the scenario is real and ask what each statement would then say about the world, including about itself.
4. Compare. A scenario passes only when every statement marked true says something true, and every statement marked false says something false.
5. Count the survivors. One survivor is the answer. No survivors means you are facing a paradox. Two or more mean a clue is missing.
6. Prune early. You rarely need to write out every scenario. Look for a statement that clashes with itself whenever it is true, or one that forces another statement's value, and cross out whole groups at once.

The most important habit is step four. Never ask whether a statement is true in general. Ask whether it is true in this scenario, and whether that matches the label the scenario gave it.

How does the method solve a counting list?

Here is a full worked example. Five statements are written on a board:

1. Exactly one of these five statements is false.
2. Exactly two of these five statements are false.
3. Exactly three of these five statements are false.
4. Exactly four of these five statements are false.
5. All five of these statements are false.

Which statements are true?

Notice that the statements compete. Each one names a different count of false statements. Only one count can be right, so at most one statement is true. That prunes the scenarios at once: either none is true, or exactly one is.

Test the scenario where none is true. Then all five are false. But the fifth statement says exactly that, so in this scenario it would be true. The scenario contradicts itself and is crossed out.

So exactly one statement is true. That means exactly four statements are false.

Find the statement that says so. The fourth statement claims that exactly four are false. It is the true one, and the other four are false.

Check every statement in the final scenario. Four statements are false and one is true:

Answer: only the fourth statement is true. A full search of all thirty-two true-or-false combinations leaves this as the single consistent scenario.

The key move was counting the statement you are checking as one of the statements being counted. Leave it out, and the arithmetic goes wrong in every row.

How do you fill in a sentence that counts its own letters?

Self-counting sentences ask you to fill a gap with a number word so that the finished sentence describes itself. Start with a warm-up: This sentence has ___ words. Any number written as a single word gives a five-word sentence, so the only fill that works is five.

Letter counts are harder, because the number you write adds its own letters. Try this one: There are ___ letters in this sentence. Count only letters, not spaces, hyphens or the full stop.

1. Count the fixed part. The word there has five letters, the word are has three, letters has seven, in has two, this has four and sentence has eight. That is twenty-nine letters.
2. Write the condition. The missing number must equal twenty-nine plus the letters in its own spelling.
3. Narrow the range. Any whole number below one hundred is spelled with three to twelve letters, so the total must lie between thirty-two and forty-one. A number of one hundred or more cannot work, because even its longest spelling has fewer than thirty letters, which keeps the total far below one hundred.
4. Try each candidate from thirty-two to forty-one. Thirty-two has nine letters, giving thirty-eight, which is not thirty-two. Thirty-five has ten letters, giving thirty-nine, which is not thirty-five. Forty has five letters, giving thirty-four. Only thirty-nine works: it has ten letters, giving thirty-nine.

Answer: There are thirty-nine letters in this sentence. Checking every whole number from one to nine hundred and ninety-nine leaves only this fill.

Do not stop at the first fit. A small change in wording can create a sentence with two valid fills or none, so always sweep the whole plausible range before you announce an answer.

A similar puzzle asks for a number whose own spelling has to obey a rule. In the riddle below, the number, once written out as a word, must have its letters in perfect alphabetical order.

How do number sequences refer to themselves?

Some sequences are self-referential too. Instead of a fixed rule such as add two each time, each new term comes from looking up earlier terms, and the positions you look up are themselves values taken from the sequence.

A paper published on 29 April 2011 studies a class of sequences of this kind ^[2]. In one of its examples the first two terms are one and one, and the paper's table of that example begins 1, 1, 2, 3, 3, 4, 5 ^[2]. In that table each step adds zero or one, so some values repeat ^[2].

The rule behind that example works like a small look-up game:

1. Take the previous term and use it as a position. Read the term at that position, and call the result p.
2. The new term is the term at position p plus the term at the position you get by taking p away from the current position.

Try it for the third term. The previous term is one, and the term at position one is one, so p is one. The new term is the term at position one plus the term at position two, which gives two. For the fourth term, the previous term is two, the term at position two is one, so p is one again, and the new term is one plus two, which gives three.

The catch is that a look-back could point to a position you have not reached yet, and then the rule breaks down. The paper sets out conditions on the starting terms that guarantee every term is defined and that each step goes up by zero or one, a property it calls slow-growing ^[2].

It also proves a neat pattern: at positions taken from the add-the-last-two sequence, such as eight, thirteen and twenty-one, the term equals the number just before it in that sequence ^[2].

Not everything about these sequences is settled. For the version that looks up the previous term only once, rather than twice as in the example above, the paper reports an earlier result that the ratio of a term to its position has a limit of one half ^[2].

For the versions with two or more look-ups, including the example above, the paper states that whether this ratio has a limit at all is an open question ^[2].

How can you tell a puzzle from a paradox?

Once you can run the consistency check, the number of surviving scenarios tells you what kind of object you are holding.

A paradox is not a trick question with a hidden answer. If no scenario survives, the honest conclusion is that the setup cannot be consistent. A puzzle with several survivors is not broken either. It simply needs one more clue, and adding the right clue can turn it into a proper puzzle.

What mistakes trip people up?

• Judging a statement before fixing a scenario. Self-referring clues have no truth value until the scenario is fixed.
• Forgetting that the statement counts itself. In a counting list, the statement you are checking is one of the statements being counted.
• Stopping at the first fit. Fill-in sentences can have more than one valid fill, so sweep the whole range.
• Confusing false with opposite. The denial of exactly two are true is not exactly two are false. It only says the true count is something other than two.
• Skipping the final check. Read every statement against the finished scenario before you trust it.

Can you solve this practice puzzle?

Four statements, labelled A to D, are written on a sheet:

• A. Statement B is false.
• B. Exactly two statements on this sheet are true.
• C. Statements A and B are both true.
• D. Exactly three statements on this sheet are true.

Which statements are true? Use the consistency check before you read on.

Answer: only statement A is true.

C cannot be true, because it would make A and B both true, yet A says B is false. D cannot be true either: with C false, three true statements would have to be A, B and D, and again A and B cannot both be true. So C and D are false.

If B were true, exactly two statements would be true, and the only candidates left are A and B. But then A would be calling a true statement false. So B is false, which makes A true. That leaves one true statement, so B's claim of two is indeed false. A search of all sixteen true-or-false combinations leaves this single answer.

Where can you try a self-referential label puzzle?

Talking labels are a friendly place to practise. In the riddle below there are three chests, gold, silver and bronze, the treasure is in one of them, and exactly one of the three labels is true. Write down one scenario per chest, count the true labels in each, and keep the scenario where the count is one.