# Find the woman/man of your dreams!

You're on a business trip when you meet Li Xiang, the woman/man of your dreams in the hotel bar. The two of you spend an amazing night together chatting about puzzles, trigonometry, and polyrhythms. (You both love the rhythmic pattern 3 against 5.) But when you wake up the next morning, Li is gone without a trace!

Dejected, you begin to pack your bags...but what's this? Inside your suitcase, you find a mysterious note scribbled on hotel paper (with immaculate handwriting):

What is Li's message?

Hint 1:

Wolfram Alpha can be quite helpful for this problem.

Hint 2:

the x and y formulae each include exactly two trigonometric functions: sine and cosine.

Hint 3:

Sine and cosine are nested.

Hint 4:

The following operations are used exactly once in each formula: sin, cos, +, *

Hint 5:

Being a rhythm, 3 and 5 correspond to frequency, not amplitude.

Hint 6:

y=cos(n+sin(3n)), x=sin(n+cos(5n))

• It's worth noting that n = 0 to 2π is a full circle in radians. Nov 28 '17 at 17:50
• Polyrhythm - 3 against 5 & 5 against 3 Nov 28 '17 at 21:18
• A note for future puzzles - this puzzle basically comes down to "guess the function I used here" which from the information provided in the original puzzle would be a needle-in-a-haystack search. Puzzles should incorporate clues to how they should be solved - as part of the puzzle itself, not grafted on in spoiler-tagged "hints". If the hint is effectively required for anyone not inside your head to solve the puzzle then it's not a "hint", it's an essential part of the puzzle--and in many cases is the only thing preventing your puzzle from being "guess what I was thinking".
– Rubio
Dec 14 '17 at 2:31

The message is:

Meet me at taoyuan airport.

She/He has probably left by now though :P

It is obtained by:

Superimposing the graph of $y=cos(n+sin(3n)), x=sin(n+cos(5n)), n=0to2\pi$ onto the image with the letters and start from the black square beside m in clockwise direction.

• Aw darn, I had something like that, but I dismissed "taoyua" as gibberish... Dec 12 '17 at 9:23
• @Lolgast I feel ashamed reaping up your glory! Sorry about that. I was about to give up myself, but I said why don't I google this at least before leaving it. As I started typing taoy, google autocomplete it (all of it, even the airport part) and I was like booyah!!. That's why I included the link, so people won't think it's just gibberish. After all, it makes perfect sense giving the name in the question is Li Xiang. :D Dec 12 '17 at 15:32
• Ah well, it happens... At least I can live with the knowledge that I found the method and the first part of the message :) Dec 12 '17 at 16:03

From the hints and the "rhytmic pattern 3 against 5", I got the idea that we might have a formula in the form of
$x = 3*sin(n)+5*cos(n)$
$y = 3*cos(n)+5*sin(n)$
i.e. an ellipse. I tried then overlaying it on the letters, both with the square near m on the origin as on $(3,5)$ (n=0) and with different sizes, hoping to see the ellipse move through different letters which would give a sensible answer. Unfortunately, I couldn't get anything out of it. Perhaps the formulae/shape is a bit more complex, or maybe I just didn't try hard enough.