Problems

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

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?

On the selection to the government of the planet of liars and truth tellers \(12\) candidates gave a speech about themselves. After a while, one said: “before me only once did someone lie” Another said: “And now-twice.” “And now – thrice” – said the third, and so on until the \(12\)th, who said: “And now \(12\) times someone has lied.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly counted how many times someone had lied before him. So how many times have the candidates lied?

Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence \(1, 2, \dots , 100, 101\). After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.

A six-digit phone number is given. How many seven-digit numbers are there from which one can obtain this six-digit number by deleting one digit?

The city plan is a rectangle of \(5 \times 10\) cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?

27 coins are given, of which one is a fake, and it is known that a counterfeit coin is lighter than a real one. How can the counterfeit coin be found from 3 weighings on the scales without weights?

There are two purses and one coin. Inside the first purse is one coin, and inside the second purse is one coin. How can this be?

Many maths problems begin with the question “Is it possible…?”. In these kinds of problems, what you need to do depends on what you think is true.

• If you believe it is possible, then you must give an example that really satisfies the conditions in the problem.

• If you believe it is not possible, then you must explain clearly why it cannot be done.

When trying to build an example, it often helps to ask yourself extra questions to narrow things down: “How could it be possible?”, or “What properties must a correct example have?”.

On the other hand, if you have been trying to build an example for a while and nothing works, perhaps the answer is that it is impossible. In that case, look for a property that any example would need to have — and then show why that property cannot actually happen. Let’s see some examples!

Cut a square into two equal:
1. Triangles.
2. Pentagons
3. Hexagons.