One of five brothers baked a cake for their Mum. Alex said: “This was Vernon or Tom.” Vernon said: “It was not I and not Will who did it.” Tom said: “You’re both lying.” David said: “No, one of them told the truth, and the other was lying.” Will said: “No David, you’re wrong.” Mum knows that three of her sons always tell the truth. Who made the cake?
In a certain realm there are magicians, sorcerers and wizards. The following is known about them: firstly, not all magicians are sorcerers, and secondly, if the wizard is not a sorcerer, then he is not a magician. Is it true that not all magicians are wizards?
A traveller on the planet of Liars and Truth-Tellers met four residents and asked them: “Who are you?”. They received the following answers:
1st: “We are all liars.”
2nd: “Among us is exactly one liar.”
3rd: “Among us there are two liars.”
4th: “I have never lied and I’m not lying”.
The traveller quickly realised who the fourth alien was. How did they do it?
In the lower left corner of an 8 by 8 chessboard is a chip. Two in turn move it one cell up, right or right-up diagonally. The one who puts the chip in the upper right corner wins. Who will win in a regular game?
a) There are 10 coins. It is known that one of them is fake (by weight, it is heavier than the real ones). How can you determine the counterfeit coin with three weighings on scales without weights?
b) How can you determine the counterfeit coin with three weighings, if there are 27 coins?
Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.
Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).
Prove that the sum of
a) any number of even numbers is even;
b) an even number of odd numbers is even;
c) an odd number of odd numbers is odd.
Prove that the product of
a) two odd numbers is odd;
b) an even number with any integer is even.
During the election for the government of the planet of Liars and
Truth-Tellers, \(12\) candidates each
gave a short speech about themselves.
After everyone had spoken, one alien said: “So far, only one lie has
been told today.”
Then another said: “And now two have been said so far.”
The third said: “And now three lies have been told so far,” and so on —
until the twelfth alien said: “And now twelve lies have been told so
far.”
It turned out that at least one candidate had correctly counted how many
lies had been told before their own statement.
How many lies were said that day in total?