Problems

Among some number of mathematicians, every seventh is a philosopher, and among some number of philosophers every ninth is a mathematician. Who are there more of: philosophers or mathematicians?

Find numbers equal to twice the sum of their digits.

Know-it-all came to visit the twin brothers Screw and Nut, knowing that one of them never speaks the truth, and asked one of them: “Are you Screw?”. “Yes,” he replied. When Know-it-all asked the second brother the same question, he received an equally clear answer and immediately determined who was who.

Who was called Screw?

Uncle Jack, the cat Whiskers, Spot and postman Pat are sitting on a bench. If Spot, sitting to the right of everyone, sits between Uncle Jack and the cat, then the cat will be at the extreme left. In what order do they sit?

Cut a square into three pieces, from which you can construct a triangle with three acute angles and three different sides.

In any group of 10 children, out of a total of 60 pupils, there will be three who are in the same class. Will it always be the case that amongst the 60 pupils there will be: 1) 15 classmates? 2) 16 classmates?

A pedestrian walked along six streets of one city, passing each street exactly twice, but could not get around them, having passed each one only once. Could this be?

When Harvey was asked to come up with a problem for the mathematical Olympiad in Sunny City, he wrote a rebus (see the drawing). Can it be solved? (Different letters must match different numbers).

In two purses lie two coins, and one purse has twice as many coins as the other. How can this be?

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?