# Puzzling Equations

Mathematics is a fascinating field of study, ever changing, far from finished.

Arbitrary symbols, prescribed with special meaning become extraordinary challenges for the mind.

And if you’re like me, you like a good puzzle!

Consider the following equation: 5/15 plus some other number is 5/16.

I solved an equation like this once, and my friends were impressed, but certainly I am not bragging!

It really isn’t too tricky actually. Just find a common denominator. Let’s try an easier example.

5/12 minus 3/6 is?

Clearly it is negative 1/12!

This is all fun and good, but we haven’t talked about my favorite equation yet!

9/12 plus 3/10 is equal to what then?

So lovely it is that it happens to be one.

Small hint:

The last sentence is most significant when posted anywhere but a puzzle-enthusiast webpage.

Big hint:

I left out a couple tags in an effort to be a bit more cryptic.

Hint that is more confusing than it is helpful:

The share url is solved if you’re willing to backtrack.

• In fact, $-\dfrac{1}{12}$ is a very interesting fraction... – Mr Pie Apr 8 '19 at 17:48
• A bit of a reference included ;) @user477343 – Samy Bencherif Apr 16 '19 at 20:50
• You know what? Once this bounty ends, if nobody has been awarded it, I will put on a bounty myself, and the puzzle can continue for another week :) – Mr Pie Apr 16 '19 at 20:52
• @user477343 It doesn’t look like anyone will be solving this one – Samy Bencherif Apr 20 '19 at 17:51

I think that it is

One

because

You start with "Consider the following equation: 5/15 plus some other number is 5/16." And if we use US or Asian date format then it is May 15th plus some other number is May 16th. And that some other number is One day.

Ok, ok. I know that it is wrong, because

If you subtract March 6th from May 12th you don't get January 12th of last year, and the step is only 2 days, so we cannot interpret 1/12 as "a month". But we do get International Women's Day. And if we add Computing day (256th day of the year) and a day 2 days after International Women's Day, we get a Transgender Day of Remembrance. Is it «lovely»?

• Headed towards the right ballpark with this one :) esp in that there is more than meets the eye when it comes to the math – Samy Bencherif Apr 9 '19 at 6:31

Probably not the answer but the idea had some appeal for me

The question states that
$$\frac{9}{12} + \frac{3}{10}$$ is equal to ... So lovely it is that it happens to be one.

The expression as decimal values produces $$\frac{21}{20}$$.

So I tried it using number bases $$11, 12, 13, 14, 15, 16$$ but could not get an answer of $$1$$.

However if I play around and make the divisor also the number base, because
it is at the "base" of the fraction (but swapped), then the expression becomes

$$\frac {9_{\mathit {10}}}{12_{\mathit {10}}} + \frac {3_{\mathit {12}} }{10_{\mathit {12}}}$$

which in decimal is

$$\frac{9}{12} + \frac{3}{12} = 1$$

• I like this answer. But I think I might've confused too many people with the "equation that claims to be equal 1 but isn't actually equal 1". The hint that downplays the significance of this last line is not a farce. – Samy Bencherif Apr 10 '19 at 0:03
• I've noticed a clue I overlooked. – Weather Vane Apr 10 '19 at 0:09

I will begin to start answering my own question. It is not too late to win the bounty. Once it expires I will fully explain my puzzle.

Hint 1 Explained:

The last sentence is most significant when posted anywhere but a puzzle-enthusiast webpage.

Here I am indicating that the last sentence would inform you that there is a puzzle in the first place--which is unneeded on a puzzle website. 9/12 + 3/10 is equal to 1.05 which is 5% off of what I said it was (one). The off-ness is a clue that this text is not only a wonky passage about fractions. The 5% represents the little extra there is than meets the eye. All of that aside, the point of this hint is to say that the last sentence is a bit of a red herring if you already know there is a puzzle involved.

Hint 2 Explained:

I left out tags, namely cryptography and cyphers. This is indicated by the use of the word "cryptic" and moreso by the hyperlinked period that leads to a webpage about cyphers.

Upon visiting that website you should consider that my message contains some kind of cypher. Considering my message is lines of readable text with numbers (not garbled text), many of the cyphers can be eliminated. What remains is A1Z26, ASCII, Anagram, Acrostic, Book, Substitution. It is possible that only the numbers are involved in the cypher, or perhaps both the numbers and the text together. Maybe decoding Acrostically results in some Caesarically encoded message 😈.

Hint 3 Explained:

Hint 3 truly is more confusing than it is helpful, but being the kind of person that I am, and delighted by an intriguing coincidence, I felt obligated to keep it in there. If you click "share" under the question you will be given a URL: https://puzzling.stackexchange.com/q/81535/58232.

There is a catch, but if you decrypt this url using the same cypher I used in the above message you will get the message "solved". The catch is that you must be willing to backtrack, that is, you go back to the start when you reached the end. Under this logic, if I asked you for the fifth letter of "abc" you should say "a-b-c-a-b, ah yes the 5th letter is b!"

How to solve the puzzle

Figure out what kind of cypher I used and use it to decode the passage. The first answer containing the secret message will receive the bounty.

Please feel free to ask questions if you need help completing the puzzle. If it is possible to extend bounties, I may do this.

Bonus Hint Explained:

I initially forgot about this one. In the comments for one of the answers I was talking about the significance of the last sentence and I included a little hyperlink to https://en.wikipedia.org/wiki/The_Wide_Window. This is a reference to how one of the characters of that book uses grammatical errors to hide a secret message, because only those who truly know her would recognize bad grammar as completely out of character for her.

This book was a large influence that got me interested in cyphers in the first place.

Anyway it was a good book; cypher or no cypher.

Edit: Cough Cough. Book. Cypher. Book Cypher. It's a book cypher. And it is not of the book I linked to, or any actual book in fact, As I can't guarantee everyone has access to the same editions of books. I will be posting the answer soon! :)

• (ROT13) Va lbhe ynfg fragrapr, vf gurer n ernfba lbh unir jevggra "bar" nf bccbfrq gb "1"? – Mr Pie Apr 16 '19 at 11:53
• Npghnyyl lrf! V qvq abg jvfu gb nqq n ahzore jurer vg jbhyq or pbashfvat. – Samy Bencherif Apr 16 '19 at 19:44