Find all solutions of the puzzle \(HE \times HE = SHE\). Different letters stand for different digits, and the same letters stand for the same digit.
Cut an arbitrary triangle into parts that can be used to build a triangle that is symmetrical to the original triangle with respect to some straight line (the pieces cannot be inverted, they can only be rotated on the plane).
The numbers from \(1\) to \(9\) are written in a row. Is it possible to write down the same numbers from \(1\) to \(9\) in a second row beneath the first row so that the sum of the two numbers in each column is an exact square?
On a Halloween night ten children with candy were standing in a row. In total, the girls and boys had equal amounts of candy. Each child gave one candy to each person on their right. After that, the girls had \(25\) more candy than they used to. How many girls are there in the row?
There are \(16\) cubes, each face of every cube is coloured yellow, black, or red (different cubes can be coloured differently). After looking at their colouring pattern, Pinoccio said that he could put all the cubes on the table in such a way that only the yellow color would be visible, on the next turn he could put the cubes in such a way that only the black color would be visible, and also he could put them in such a way that only the red color would be visible. Is there a colouring of the cubes such that he could tell the truth?
Alex writes natural numbers in a row: \(123456789101112...\) Counting from the beginning, in what places do the digits \(555\) first appear? For example, \(101\) first appears in the 10th, 11th and 12th places.
Frodo ventures into a thick fog where he is to meet his three companions: Sam, Merry, and Pippin. He can tell they are standing in a row before him – one on the left, one in the middle, and one on the right – but he cannot see who is who. To help, he asks each of them to speak.
Remember: Sam always lies, Merry always tells the truth, and Pippin sometimes lies and sometimes tells the truth.
Here is what Frodo hears:
The one on the left says: “Merry is next to me.”
The one on the right says: “The person who just spoke is Pippin.”
The one in the middle says: “On my left is Sam.”
Can you work out who is standing on Frodo’s left, in the middle, and on the right?
Using areas of squares and rectangles, show that for any positive
real numbers \(a\) and \(b\), \((a+b)^2 =
a^2+2ab+b^2\).
The identity above is true for any real numbers, not necessarily
positive, in fact in order to prove it the usual way one only needs to
remember that multiplication is commutative and the distributive
property of addition and multiplication:
\(a\times b = b\times a\);
\((a+b)\times c = a\times c + b\times c\).
Annie found a prime number \(p\) to which you can add \(4\) to make it a perfect square. What is the value of \(p\)?
Let \(a\) and \(b\) be positive real numbers. Using areas
of rectangles and squares, show that \(a^2 -
b^2 = (a-b) \times (a+b)\).
Try to prove it in two ways, one geometric and one algebraic.