We first need to find the encoding and decoding functions used by Boris and Natasha. We know that these two must be linear functions with 1 as the coefficient of x. Then the encoding function must have the form:
[tex]f(x)=x+b[/tex]Where x is the number associated with the letter and b is a constant that we don't know. The decoding function is its inverse:
[tex]f^{-1}(x)=x-b[/tex]Now let's take a look at the table that associates the letters with numbers. The minimum number is 1 associated with A and the maximum is 27 associated with Blank. Now let's write the encoded version of these two:
[tex]\begin{gathered} f(1)=1+b \\ f(27)=27+b \end{gathered}[/tex]And let's find the difference between their encoded values:
[tex]f(27)-f(1)=(27+b)-(1+b)=27-1+b-b=27-1=26[/tex]So the difference between their encoded values is the same as the difference between their decoded values. Since 1 and 27 are the minimum and maximum decoded values their difference is the greatest of all the difference between two decoded values. Then there's no other pair of decoded values with a difference equal to 26 and since the difference between two encoded values is the same as the difference between two decoded values we can assure that 26 is the maximum difference between two encoded values and it corresponds to the pair A - Blank.
This implies that if the difference between the minimum and maximum value in the message sent by Boris and Natasha is 26 we can assure that this pair of values is the one corresponding to A and Blank.
The minimum and maximum values in the message are 15 and 41 and their difference is 41 - 15 = 26. This means that 15 is the encoded value of A and 41 is that of Blank. Then we can construct two equations using the encoding function:
[tex]\begin{gathered} f(1)=1+b=15 \\ f(27)=27+b=41 \end{gathered}[/tex]By substracting 1 from both sides of the first equation and 27 from both sides of the second equation we obtain b:
[tex]\begin{gathered} 1+b-1=15-1\Rightarrow b=14 \\ 27+b-27=41-27\Rightarrow b=14 \end{gathered}[/tex]So b=14 and the encoding function is f(x)=x+14.
Then the decoding function is f⁻¹(x) = x - 14.
Now we need to decode the message. We simply need to evaluate the decoding function at all the numbers in the encoded message:
[tex]\begin{gathered} f^{-1}(25)=25-14=11 \\ f^{-1}\left(19\right)=19-14=5 \\ f^{-1}(30)=30-14=16 \\ f^{-1}(41)=41-14=27 \\ f^{-1}(17)=17-14=3 \\ f^{-1}(15)=15-14=1 \\ f^{-1}(26)=26-14=12 \\ f^{-1}(27)=27-14=13 \\ f^{-1}(28)=28-14=14 \\ f^{-1}(18)=18-14=4 \\ f^{-1}(29)=29-14=15 \\ f^{-1}(34)=34-14=20 \\ f^{-1}(22)=22-14=8 \end{gathered}[/tex]Then we replace each encoded value by its respective decoded value so the message in numbers is:
11 5 5 16 27 3 1 12 13 27 1 14 4 27 4 15 27 20 8 5 27 13 1 20 8
Using the table associating numbers and letters we obtain the final message:
KEEP CALM AND DO THE MATH