Problems

Cowboy Joe was sentenced to death in an electric chair. He knows that out of two electric chairs standing in a special cell, one is defective. In addition, Joe knows that if he sits on this faulty chair, the penalty will not be repeated and he will be pardoned. He also knows that the guard guarding the chairs on every other day tells the truth to every question and on the alternate days he answers incorrectly to every question. The sentenced person is allowed to ask the guard exactly one question, after which it is necessary to choose which electric chair to sit on. What question can Joe ask the guard to find out for sure which chair is faulty?

Fred and George are twin brothers. One of them always tells the truth, and the other always lies. You can ask only one question to one of the brothers, to which he will answer “yes” or “no”. Try to find out the name of each of the twins.

It is known that “copper” coins that are worth 1, 2, 3, 5 pence weigh 1, 2, 3, 5 g respectively. Among the four “copper” coins (one for each denomination), there is one defective coin, differing in weight from the normal ones. How can the defective coin be determined using scales without weights?

How can we divide 24 kg of nails into two parts of 9 kg and 15 kg with the help of scales without weights?

The vendor has a cup weighing scales with unequal shoulders and weights. First he weighs the goods on one cup, then on the other, and takes the average weight. Does he deceive customers?

In the language of the Ancient Tribe, the alphabet consists of only two letters: M and O. Two words are synonyms, if one can be obtained by from the other by a) the deletion of the letters MO or OOMM, b) adding in any place the letter combination of OM. Are the words OMM and MOO synonyms in the language of the Ancient Tribe?

True or false? Prince Charming went to find Cinderella. He reached the crossroads and started to daydream. Suddenly he sees the Big Bad Wolf. And everyone knows that this Big Bad Wolf on one day answers every question truthfully, and a day later he lies, he proceeds in such a manner on alternate days. Prince Charming can ask the Big Bad Wolf exactly one question, after which it is necessary for him to choose which of the two roads to go on. What question can Prince Charming ask the Big Bad Wolf to find out for sure which of the roads leads to the Magic kingdom?

There are 68 coins, and it is known that any two coins differ in weight. With 100 weighings on a two-scales balance without weights, find the heaviest and lightest coin.

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse (from the node “M”) moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.

Can the cat catch the mouse in each of the cases A, B, C?

Two play a game on a chessboard \(8 \times 8\). The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?