Problems

Welcome back! We hope you all had a great summer and now you are ready for the new school year full of fun problems in mathematics. We decided to start with warm-up topic called dissections, so today we will cut various shapes into more elaborate geometric figures in order to reassemble them into a different shape.

Cut the following shape into four equal figures.

Cut the following shape into two equal parts.

Today we will practice to encrypt and decipher information using some of the most common codes. Majority of the codes in use can be alphabetic and numeric, namely one may want to encode a word, a phrase, or a number, or just any string of symbols using either letters, or numbers, or both. Some of the codes, however may use various other symbols to encrypt the information. To solve some of the problems you will need the correspondence between alphabet letters and numbers

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:
$$\begin{array}{ccccccccccc}Y& q& o& l& u& e& c& d& a& i& n\\ w& a& r& l& a& w& e& a& t& y& k\\ s& n& t& c& a& e& k& c& e& a& m\\ t& o& d& r& w& e& a& t& a& h& r\\ a& c& n& t& n& e& o& d& t& r& h\\ n& i& d& n& l& g& m& e& x& s& z\end{array}$$

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.

After mastering the Caesar shift cypher one may wonder how to generalize it. One possible way is to use Affine cypher. The difference between these two methods can be described as follows:

• In case of Caesar cypher we took a letter with position $n$ from $1$ to $26$ and added to its position a number $d$ obtaining the number $n+d$, then we compute its residue modulo $26$.

• In case of affine cypher we take a letter with position $n$ and consider a number $nx+d$ modulo $26$.

To decipher such code we need to know values $x$ and $d$, then if we have a letter in the code with position $m$, we can find $n$ as $n=(m-d)x^{-1}$ modulo $26$. Here we have to explain what is $x^{-1}$: for a number $x<26$ we are looking for such a number $y$, that $26$ divides $xy-1$.

• Does there always exist a number $x^{-1}$ modulo $26$ for any $x$?

• Using data $x=3$, $d=8$ encrypt the word "SOLUTION".