Alternating liars, limited truths, particular days and conditional behaviors. Each game is followed immediately by its solution and explanation.
1. The alternator
Marco makes three consecutive statements. You know that he always alternates truth and lies, but you do not know which he starts with. He says: 1) “2+2=4.” 2) “Rome is in France.” 3) “5 is odd.” What kind of statement did he start with?
Solution
He started with a truth.
Explanation
The three statements are true, false, true: the pattern is V-F-V, where V denotes true and F false.
2. The liar’s day
Paolo tells the truth on Tuesday, Thursday and Saturday; he lies on the other days. Today he says: “Tomorrow I will lie.” What day could it be?
Solution
Any day from Monday to Saturday, but not Sunday.
Explanation
The statement is true exactly when tomorrow is a lying day. It matches today’s behavior when today and tomorrow have opposite roles. This happens on Monday, Tuesday, Wednesday, Thursday, Friday and Saturday. On Sunday he lies, but the next day, Monday, is also a lying day: the statement would be true and is therefore incompatible with Sunday’s behavior.
3. Three people, three behaviors
Anna always tells the truth, Bruno always lies, and Carlo can say either true or false things. One of them says: “I am Carlo.” Who could have said it?
Solution
Bruno or Carlo, but not Anna.
Explanation
Anna could not say it because it would be false. Bruno can say it precisely because he lies. Carlo can say it because, being Carlo, the statement is true and he is also allowed to tell the truth.
4. Only one truth
Luca makes three statements and you know that only one is true: 1) “I have 20 euros.” 2) “I have at least 10 euros.” 3) “I do not have 20 euros.” How much money could he have?
Solution
Less than 10 euros.
Explanation
With 20 euros, 1 and 2 would be true. With an amount between 10 and 19 euros, 2 and 3 would be true. With less than 10 euros, only 3 is true.
5. Two truths out of three
Giulia thinks of an integer from 1 to 10. Exactly two statements are true: 1) “The number is greater than 5.” 2) “The number is even.” 3) “The number is less than 8.” Which numbers are possible?
Solution
2, 4, 7, 8 and 10.
Explanation
For 2 and 4, statements 2 and 3 are true. For 7, statements 1 and 3 are true. For 8 and 10, statements 1 and 2 are true. The number 6 makes all three true; 1, 3, 5 and 9 make only one true. The possible numbers are therefore exactly those listed.
6. The inconsistent witness
A witness makes three statements and you know that exactly two are true: 1) “I was at home.” 2) “I was alone.” 3) “I was not at home.” What can you say about the second statement?
Solution
“I was alone” must be true.
Explanation
The first and third statements negate each other: one is true and one false. To have two truths in total, the second must be true.
7. The liar on alternate days
Franco alternates one truthful day and one lying day. Today he says: “Yesterday I lied.” Is he telling the truth or lying today?
Solution
It cannot be determined.
Explanation
If he tells the truth today, he lied yesterday: everything is consistent. If he lies today, he told the truth yesterday, so “Yesterday I lied” is false: this is also consistent. Another piece of information is needed.
8. The two brothers
There are two brothers, A and B. Each day, one always tells the truth and the other always lies; the following day they swap roles. Today A says: “B is lying.” Can you determine which brother is truthful today?
Solution
No.
Explanation
If A is truthful, B lies and the statement is true. If A lies, B is truthful and the statement is false. Both situations are possible.
9. One lie every three statements
A man always follows the pattern: two truths, then one lie. He says: “7 is prime,” “9 is odd,” “10 is less than 3.” Where is he in the cycle?
Solution
He has just completed a full V-V-F cycle.
Explanation
The first two statements are true and the third is false: they match his pattern exactly.
10. The suspect
Three suspects each make a statement. You know that exactly one lies. A: “It was not me.” B: “It was C.” C: “B is lying.” Who is guilty?
Solution
B or C; the information does not distinguish them.
Explanation
B’s and C’s statements negate each other, so exactly one of them is false. Since there must be only one lie in total, A’s statement must be true: A is innocent. If B is guilty, the statements are true, false, true. If C is guilty, they are true, true, false. Both possibilities satisfy the condition.
11. The secret number
An integer is between 1 and 10. Exactly one statement is false: 1) “It is greater than 3.” 2) “It is less than 8.” 3) “It is even.” Which numbers are possible?
Solution
2, 5, 7, 8 and 10.
Explanation
For 2, only statement 1 is false. For 5 and 7, only statement 3 is false. For 8 and 10, only statement 2 is false. The numbers 4 and 6 make all three true; 1, 3 and 9 make two false.
12. Truth about others
Anna always lies when speaking about herself, but always tells the truth when speaking about others. She says: “Bruno is truthful.” What do you know about Bruno?
Solution
Bruno is truthful.
Explanation
Anna is speaking about another person, so in this case she must tell the truth.
13. Lies about others
Carlo tells the truth when speaking about himself and lies when speaking about others. He says: “I have a sister.” Then he says: “Marco has a sister.” What can you conclude?
Solution
Carlo has a sister; Marco does not have a sister.
Explanation
The first statement concerns Carlo himself and is true. The second concerns Marco and is false.
14. The lying yes
A person always tells the truth when answering “no” and always lies when answering “yes.” Asked “Are you over 30?”, the person answers “yes.” What do you know?
Solution
The person is not over 30.
Explanation
“Yes” answers are always lies. Therefore the statement “I am over 30” is false.
15. The lying no
A person always tells the truth when answering “yes” and always lies when answering “no.” Asked “Is the number you are thinking of even?”, the person answers “no.” What do you know?
Solution
The number is even.
Explanation
Since the answer was “no,” that answer must be false. The number is therefore not odd: it is even.
16. Exactly two lies
Four people say: A: “B is lying.” B: “C is lying.” C: “D is lying.” D: “A is lying.” Can exactly two statements be false?
Solution
Yes.
Explanation
One possible configuration is: A true, B false, C true, D false. A correctly says that B lies; B lies by saying that C lies; C correctly says that D lies; D lies by saying that A lies.
17. The thief among three
A, B and C know who stole. You know that exactly one tells the truth. A: “It was B.” B: “It was not me.” C: “It was B.” Who could be the thief?
Solution
A or C.
Explanation
If B were the thief, A and C would both tell the truth. If it is A or C, only B tells the truth. The possibilities are therefore A and C.
18. The calendar
Marta lies on Monday and Tuesday but tells the truth on the other days. Today she says: “Yesterday I lied.” What day is it?
Solution
Monday or Wednesday.
Explanation
On Monday, she lies: yesterday was Sunday, when she told the truth, so her statement is false. On Wednesday, she tells the truth: yesterday was Tuesday, when she lied, so her statement is true. On Tuesday the statement would be true during a lying day; on the remaining days it would be false during truthful days. Only Monday and Wednesday work.
19. Alternation between two people
A and B speak in turn. Each new statement must have the opposite truth value to the previous one. A says “2+2=4.” B says “5 is even.” A says “Rome is in Italy.” Does the sequence follow the rule?
Solution
Yes.
Explanation
The three statements are true, false, true. Each statement has the opposite value to the previous one.
20. The detective
A detective knows that the suspect tells the truth in exactly two answers out of three. The suspect answers “Yes” to “Did you enter through the window?”, “No” to “Did you take the money?”, and “Yes” to “Were you alone?” Which answer is certainly true?
Solution
No particular answer.
Explanation
We only know that two answers are true and one false, but without another constraint we cannot identify the lie.
21. The number and three properties
A number is between 1 and 9. Exactly two statements are true: “It is odd.” “It is greater than 5.” “It is less than 8.” Which numbers are possible?
Solution
1, 3, 5, 6 and 9.
Explanation
The numbers 1, 3 and 5 are odd and less than 8, but not greater than 5. The number 6 is greater than 5 and less than 8, but not odd. The number 9 is odd and greater than 5, but not less than 8. The number 7 makes all three true; 2, 4 and 8 make only one true.
22. The selective liar
A man always lies when speaking about even numbers and always tells the truth when speaking about odd numbers. Asked “Is your number 6?”, he answers “no.” What can you deduce?
Solution
His number is 6.
Explanation
The question concerns the even number 6, so the man must lie. His “no” is false: the number is 6.
23. Three friends
Three friends A, B and C: one always tells the truth, one always lies, and one alternates truth and lies. A says: “B always lies.” B says: “C is the one who alternates.” C says: “A always tells the truth.” Can you determine the three roles with certainty from this single round?
Solution
Yes. A always lies, B always tells the truth, and C alternates; in this round C lies.
Explanation
If A always told the truth, B would be the habitual liar and C the alternator. But B’s statement would then be true, which is impossible for the habitual liar. If C always told the truth, A would also always tell the truth, contrary to the distinct roles. The habitual truth-teller must therefore be B. B’s true statement identifies C as the alternator, leaving A as the habitual liar. A’s statement is false because B is truthful; C’s statement is false because A is the liar. This configuration satisfies all the conditions.
24. The same question twice
A person alternates truth and lies with every answer. You ask the same question twice in succession: “Is the number greater than 10?” The person first answers “yes,” then “no.” What can you deduce about the number?
Solution
Nothing.
Explanation
The answers must be opposite precisely because one is true and the other false. This happens both if the number is greater than 10 and if it is not; it depends only on which answer is the true one.
25. The stationmaster
A stationmaster tells the truth on even-numbered days of the month and lies on odd-numbered days. Today he says: “Tomorrow I will tell the truth.” Is this possible?
Solution
Yes, on the last day of a month with 31 days, or on February 29 in a leap year.
Explanation
Within a month, today and tomorrow have opposite parity. On an even day, the statement would be false although he should tell the truth; on an odd day, it would be true although he should lie. At the end of a month with 31 days, or on February 29, however, an odd-numbered day is followed by day 1, which is also odd. He lies today and will lie tomorrow: his prediction that he will tell the truth is false, as required. At the end of a month with 30 days, or on February 28, parity changes and no solution arises.