Problems

Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?

Multiplication of numbers. Restore the following example of the multiplication of natural numbers if it is known that the sum of the digits of both factors is the same.

Restore the example of the multiplication.

There is a 12 litre barrel filled with water, and two empty barrels (one is 5 litres in volume and the other is 8 litres). Using these barrels, try to

a) divide the water into 2 parts with volumes: 3 litres and 9 litres;

b) divide the water into 2 parts with equal volume.

Michael thinks of a number no less than \(1\) and no greater than \(1000\). Victoria is only allowed to ask questions to which Michael can answer “yes” or “no” (Michael always tells the truth). Can Victoria figure out which number Michael thought of by asking \(10\) questions?

There are 6 locked suitcases and 6 keys to them. At the same time, it is not known to which suitcase each key fits. What is the smallest number of attempts you need to make in order to open all the suitcases for sure? And how many attempts will it take there are not 6 but 10 keys and suitcases?

Decipher the following rebus (see the figure). Despite the fact that only two figures are known here, and all others are replaced by asterisks, the example can be restored.

The parliament of a certain country has two houses with an equal number of members. In order to make a decision on an important issue all the members voted and there were no abstentions. When the chairman announced that the decision had been taken with a 23-vote advantage, the opposition leader declared that the results had been rigged. How did he know it?

Three people A, B, C counted a bunch of balls of four colors (see table).

Each of them correctly distinguished some two colors, and confused the numbers of the other two colours: one mixed up the red and orange, another – orange and yellow, and the third – yellow and green. The results of their calculations are given in the table.

How many balls of each colour actually were there?

A family went to the bridge at night. The dad can cross over it in 1 minute, the mom can cross it in 2, the child takes 5 minutes, and grandmother in 10 minutes. They have one flashlight. The bridge can only withstands two people at a time. How can they all cross the bridge in 17 minutes? (If two people pass, then they go at the lower of their speeds.) You can not move along a bridge without a flashlight. You can not shine it from a distance.