We prove by mathematical induction that all horses in the world are of the same colour.
Base case: There is a single horse. It has some coat colour. Because there are no other horses, all the horses have the same coat colour.
Induction step: We have \(n\) horses. We assume all of them have the same coat colour. Now we add an additional \((n+1)\)st horse. We don’t know what colour it has, but if we for now get rid of one horse from the group we had before, we suddenly have a group of \(n\) horses which includes the new one. Since we have our claim proven for \(n\), all of these horses have the same coat colour and therefore the new horse has the same coat colour as all the other ones. So every group of \(n+1\) horses has the same colour.
The third step: due to mathematical induction rule, all the horses in the world have the same coat colour. THUS WE HAVE PROVED THAT ALL HORSES IN THE WORLD ARE OF THE SAME COLOUR!
Alice finally decided to do some arithmetic. She took four different integer numbers, calculated their pairwise sums and products, and the results ( the pairwise sums and products) wrote down in her wonderful book. What could be the smallest number of different numbers Alice wrote in her book?
Alice wants to write down the numbers from 1 to 16 in such a way that the sum of two neighbouring numbers will be a square number. The Hatter tells Alice that he can write down the numbers with this property in a line, but he believes that it is absolutely impossible to write the numbers with this property in a circle. Show that he is right.
If you are on a boat and toss a suitcase overboard, will the water level rise or fall?
You have 26 constants, labeled \(A\) through \(Z\). Let \(A\) equal 1. The other constants have values equal to the letter’s position in the alphabet, raised to the power of the previous constant. That means that \(B\) (the second letter) = \(2^A=2^1= 2\), \(C = 3^B=3^2= 9\), and so on. Find the exact numerical value for this expression: \[(X-A)(X-B)(X-C)\dots (X-Y)(X-Z).\]
Ten little circles are drawn on a squared board \(4\times4\).
Cut the board into identical parts in such a way that each part contains 1, 2, 3, and 4 drawn circles correspondingly.
Philip and Denis cut a watermelon into four parts. When they finished eating watermelon (they ate the whole thing), they discovered that there were five watermelon rinds left. How is it possible, if no rind was cut after the initial cutting?
Cut a square into a heptagon (7 sides) and an octagon (8 sides) in such a way, that for every side of an octagon there exists an equal side belonging to the heptagon.
Cut a rectangle into two identical pentagons.
(a) Cut the rectangle into two identical quadrilaterals.
(b) Cut the rectangle into two identical hexagons.
(c) Cut the rectangle into two identical heptagons.