Is it possible to cut such a hole in \(10\times 10 \,\,cm^2\) piece of paper, though which you can step?
A square \(4 \times 4\) is called magic if all the numbers from 1 to 16 can be written into its cells in such a way that the sums of numbers in columns, rows and two diagonals are equal to each other. Sixth-grader Edwin began to make a magic square and written the number 1 in certain cell. His younger brother Theo decided to help him and put the numbers \(2\) and \(3\) in the cells adjacent to the number \(1\). Is it possible for Edwin to finish the magic square after such help?
Cut a square into \(3\) parts which you can use to construct a triangle with angles less than \(90^{\circ}\) and three different sides.
Suppose that a rectangle can be divided into \(13\) equal smaller squares. What could be the side lengths of this rectangle?
Cut a square into two equal:
1. Triangles.
2. Pentagons
3. Hexagons.
Using the representation of Latin alphabet as sequences of \(0\)s and \(1\)s five symbols long, encrypt your first and last name.
Decipher the quote from Philip Pullmans "His Dark Materials":
Erh csy wlepp orsa xli xvyxl, erh xli xvyxl wlepp qeoi csy jvii.
The same letters correspond to the same in the phrase, different letters
correspond to different. We know that no original letters stayed in
place, meaning that in places of e,r,h there was surely something
else.
Decipher the quote from "Alice in Wonderland" from the following
matrix:
\[\begin{array}{@{}*{26}{c}@{}}
Y&q&o&l&u&e&c&d&a&i&n \\
w&a&r&l&a&w&e&a&t&y&k \\
s&n&t&c&a&e&k&c&e&a&m \\
t&o&d&r&w&e&a&t&a&h&r \\
a&c&n&t&n&e&o&d&t&r&h \\
n&i&d&n&l&g&m&e&x&s&z
\end{array}\]
Decipher the following quote from Alice in Wonderland:
Lw zrxog eh vr qlfh li vrphwklqj pdgh vhqvh iru d fkdqjh.
The same letters correspond to the same in the phrase, different letters
correspond to different. We know that no original letters stayed in
place, meaning that in places of e,r,h there was surely something
else.
Elon is studying the Twitter server. Inside the software he found two integer variables \(a\) and \(b\) which change their values when special search queries “RED”, “GREEN”, and “BLUE” are processed. More precisely the pair \((a, b)\) changes into \((a + 18b, 18a - b)\) when processing the query “RED”, to \((17a + 6b, -6a + 17b)\) when processing “GREEN”, and to \((-10a - 15b, 15a - 10b)\) when processing “BLUE”. When any of \(a\) or \(b\) reaches a multiple of \(324\), it resets to \(0\). If \((a, b) = (0, 0)\) the server crashes. On the server startup, the variables \((a, b)\) are set to \((20, 20)\). Prove that the server will never crash with these initial values, regardless of the search queries processed.