# A Mathematical Cipher

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I'm just learning to use $$\mathrm{\LaTeX}$$, and I need practice, so I made this cipher puzzle.

Code:

$$\dfrac{1}{16}\sqrt{s(18}.00\sqrt{-1.5}\ \ \ \ \dfrac{1}{10}\sqrt{16}\dfrac{1}{2}\dfrac{1}{10}\ \ \ \ \dfrac{1}{15750}\sqrt{xy18}\dfrac{1}{2}\ \ \ \ \dfrac{1}{10}\begin{matrix}n\\k\end{matrix}\Box2)\!\!\overline{\ \ 21}\dfrac{1}{2}\ \ \ \$$ $$\overline{)}\dfrac{1}{18}4)\!\!\overline{\ \ 12}2)\!\!\overline{\ \ 21}\sqrt{5}\overline{\Large{3}}\overline{.5}\overline{\Large{3}}\overline{.5}\Box\ \ \ \ \dfrac{1}{16}\overset{\uparrow}{\bigodot}\dfrac{1}{2}\ \ \ \ \overline{\Large{3}}\dfrac{1}{100}\ \ \ \ \overline{\Large{3}}\dfrac{1}{10}\ \ \ \ \overline{)(}2)\!\!\overline{\ \ 21\!500}\dfrac{1}{10}\overline{s-b}\Box$$

Question: Solve this cipher and find the hidden three-word phrase.

Sorry for my low-quality cryptic-clue...

Hint 1:

Diacritics should be ignored after you cracked the cipher.

Hint 2:

You may have noticed the tag, and you may be asking for the enumeration.
Well, here it is!
(k,i,j)

Hint 3:

What? An enumeration with letters instead of numbers?
Just simply encode the letters (not the punctuation) using the same key
and remove all diacritics.

Hint 4:

The hidden three-word phrase may often appear in some school exercise books.

Hint 5:

B minus A equals \$. (Encode the characters in code fences)

Hint 6:

The area of a triangle with edges length a, b, and c can be expressed as ÅÆÇÈÇÉÊ where "s" is half the triangle's perimeter.

• I seem to have found a loophole in the rules. Aug 18, 2020 at 8:54
• I fear your '[closed]' joke in the title may well be putting a lot of people off even clicking on your puzzle to read it, let alone trying to solve it! ;-)
– Stiv
Aug 18, 2020 at 12:12
• I would suggest removing your "[closed]" and putting something better to make people want to solve it more Jul 5, 2021 at 18:01
• You've put two very inviting word patterns in plain sight, but I'm still coming up empty because it is hard to find the edges of the letters in the longer words. The first hint suggests it's a substitution (text encoding?) that I already know, but I don't recognize it. Mar 18 at 15:12
• I'm sorry. I got as far as rot13(fnlf irel fzhtyl rkcynvavat jul vg vf pybfvat) (which I now see is a little bit off after today's hint) by solving it as a homophonic substitution, but the first word escapes me. I see your newest hint and I recognize you-know-who's formula and can even pattern match it with the given text, but I still don't see the system to the substitution. Mar 23 at 21:31

Since I can't seem to locate a key document, the best I can do is a partial answer.

Plaintext of cryptic clue:

EXPLAINING WHY IT IS CLOSED

Solve path:

I assumed this cipher is a monoalphabetic substitution, and that the word spaces are genuine, but the boundaries between letters are unclear. The first hint was not useful when taken literally since I have no diacritics to look at, but if diacritics are to appear later on, it means that there must be several math symbols that substitute for each vowel (or rather, for letters which usually take diacritics). So I began to solve it as a homophonic substitution where only vowels (and perhaps C and N) are allowed to have homophones.

I started with the shortest words, assuming IT, IS, and SAYS. This gave me a few letters in the longer words. The word that would be EXPLAINING was a perfect candidate for pattern matching due to the I?I? near the end, but it still took several trials because I didn't know exactly how long the word was.

At the same time, I assembled a substitution key for my emerging solution, which let me spot a few patterns like the lexically ascending fractions for the letters ST.V.XY which made me more confident in the words already assumed, and led me to the W and the word WHY. At this point I had several symbols for A and a couple for E, but I had also assigned both of DG to the same symbol. Hint #6 was also posted around this time, which gave me an E and C and confirmed an A.

A $$\sqrt{s(}$$
A $$\sqrt{5}$$
A $$\sqrt{16}$$
A $$\begin{matrix}n\\k\end{matrix}$$
B
C $$\overline{)(}$$
D $$\Box$$
E $$\overline{)}$$
E $$\sqrt{xy}$$
E $$\sqrt{-1}$$
F
G $$\Box$$
H $$\overset{\uparrow}{\bigodot}$$
I $$\overline{\Large{3}}$$
J
K
L $$2)\!\!\overline{\ \ 21}$$
M
N $$\overline{.5}$$
O $$\overline{500}$$
P $$4)\!\!\overline{\ \ 12}$$
Q
R $$\overline{18}$$
S $$\dfrac{1}{10}$$
T $$\dfrac{1}{100}$$
U
V $$\dfrac{1}{15750}$$
W $$\dfrac{1}{16}$$
X $$\dfrac{1}{18}$$
Y $$\dfrac{1}{2}$$
Z

Hint #6 refers to Heron's formula. By writing the formula above the given plaintext (which the formula "deciphers" to), I learned a few symbols for the letters ACE. Better yet, I noticed that the given plaintext letters are all adjacent in the Unicode numbering system, and wrote their code points.

The way the code points are assigned sequentially to parts of Heron's formula is suggestive of a book cipher. The ascending fractions for ST.VWXY. also seem to support this idea. I suspect a LaTeX document (one containing said formula) was used as a substitution key for this cipher. The math symbols in the document were divided up into small chunks and then each unique chunk of symbols was assigned a number in order of appearance. Those numbers are then used as Unicode code points. For example, to encipher the letter G, find its Unicode code point (71 in decimal), find the 71st chunk of symbols in the document, and use that chunk of symbols as the ciphertext for that letter. Or it could be an index into a symbol cache used by a LaTeX processor... I don't know exactly how it works.

I had no success locating any such document. Hints #2, #3, and #5 are asking me to encipher letters whose symbols I don't know. Without the document (or whatever it is that the key is derived from), I can only make weak guesses about their encipherments using the letters I've already recovered. At best, I can say the second value in the clue's enumeration is 3, but that might not even be right because I don't know whether the recovered I has a diacritic.

One problem with my current decipherment is that two letters are assigned to $$\Box$$. Another is that I can't find any options for the first word that would form a proper cryptic clue, except maybe a person's name which is an anagram of ANSWER.

• I am amazed at how many long standing ciphers you've been able to solve or shed light on in such a short time.
– Amoz
Apr 27 at 15:53