Wolf's Tricks

author's note: this task caused me to fail a final when i was 1st year in geodesy.
if yall wanna see what it looks like, hit me up on discord.

Things progressed quickly after that. The ladies started discussing things amongst themselves, chatting both with Adanita and with the Viscountess. Seeing how the challenge was issued in her house, Lady Ansatz was declared the judge and witness along with two of her closest friends. She sent her servant to fetch two lawyers, and they began discussing the rules for the duel.

Adanita and Viscountess Kuplung both agreed that the time limit would be tomorrow at three in the afternoon, when the ladies will have to return to her house and turn in their tasks. Both ladies were going to give the other one a single task which can be reasonably solved in that time. They had to know how to solve a task they were giving out, of course. Viscountess asked for the winner to be allowed to take a single item in the loser's possession, to which Adanita added a rule that the item must fit in her pocket. The last thing she was going to do was let that woman take her ship. The pair of lawyers the Lady Ansatz found were clearly used to the similar antics of the nobility, scribbling down the specifics of the duel. Lady Ansatz took Viscountess aside to discuss her task, while a few more ladies chatted with Adanita.

"She's an incredible mathematician, you know? You've got quite a challenge on your hands."
"I will be sure to take that into account," she replied.
"You think you can stand on equal grounds with her? Where did you get your education?"
"I've no formal education."
"Oh, goodness. You will surely lose, then."
"Thank you for the vote of confidence."
"She means it, though, Captain. It's not shameful to lose to your betters, although I suppose it is embarrassing to challenge them at all. She is a highly educated and intelligent woman. Why, I've heard she checks her husband's work every day."
"Oh? And what does the Viscount do?"
"He has been put in charge of the Land Books for the colony."
"Is that so?"
"Have you ever done any mathematics related to surveying?"
"No, I have not," she said honestly.
"Oh, goodness."

"Ladies, please excuse me, I would like a word with the Captain," Lady Ansatz said, shooing them away. "Captain Vervain. Have you considered the task you'd like to give her?"
"Aye," she said quietly. "Great circle navigation. Let her calculate the distance between two cities. Hipparcos and Bretonian Capital, for example."
"Gods," she said, running her fingers through her hair. "Neither of you plans to take this lightly, I see. Fine, then. Everybody, please listen up and gather around the table. I'll need the atlas, Geordie. Viscountess, is your drawing complete?"
"Yes, it is," she said, approaching the table with a few papers.
"Right, then." Lady Ansatz took a deep breath.

"After hearing out the tasks the ladies have planned to present, I've come to the conclusion that they are of equal difficulty. Considering that neither party has encountered the task they will give each other today, we will need to adjust them slightly for the sake of simplicity. Viscountess, let me have a look..." she said, taking the papers from her. She took a quill and started scribbling on the papers. "I will remove seconds from angle measurements and round the coordinates to meters. Captain, regarding your task... Let's say Hipparcos is on 3°41' northern latitude, 55°25' western longitude from the Bretonian meridian, and the Capital would then be 51°13' northern latitude, 0°27' eastern longitude from the Bretonian meridian," she said, writing it down. "Do you agree with this, Captain?"
"Aye."
"I am willing to allow Captain Vervain two meters of grace, an error I will accept in her calculations," Viscountess said.
"Fair. I'll allow an error of a hundred kilometers, then. It is your first time doing great circle navigation, after all."
"Very gracious of you, Captain."
"Please explain the task to each other, then," Lady Ansatz said.

"Mine's simple. I want you to calculate the shortest distance between Hipparcos and the Capital," Adanita said. "Take these coordinates and approximate the radius of the world to 6400 kilometers."
"Sounds deceptively simple."
"It is nothing for a navigator. In a real situation, you'd also need to calculate the correct direction towards the city. I am giving you some leeway considering your lack of experience in this field."
"Hmpf." Viscountess hid her mouth with her fan, clearly annoyed. Those were fighting words, after all.

She began explaining her task, a situation they had on the field some month ago. Adanita closely listened, looking at her sketch on the paper.

There were three points of known coordinates, Viscountess named them D2, C5, and E1. C5 and E1, she explained, are high points on terrain, in this case towers, which cannot be reached by normal means. It meant the instrument they measured with couldn't be set on them. Adanita was meant to calculate coordinates of points 74, 75, and 76, which were lined up in a row between D2 and E1. There was one additional measurement of point M, which created two triangles on the sketch: D2-M-C5, and 76-M-E1. She got measurements of angles and distances, and the coordinate system was set on a simple plane.

So, if she gave Viscountess a task that was deceptively simple, then this had to be deceptively complex.

As if.

Where do you even start here? And there seemed to be an issue with the measurements, too, Viscountess graciously let her know they all had some amount of error in them. Hopefully she managed to keep a proper poker face, but damn it all, how do you do this? She got distracted when Lady Ansatz spoke up again.

"Now that we're all set, these gentlemen will accompany our competing ladies home. Neither of you is allowed to look up similar tasks, as the problems are highly well-known within their respective fields. You will have to work out the solution on your own, and our lawyers should be there with you at all times to prevent accusations of cheating. Of course, considering the nature of the tasks, I will lend each lady a copy of trigonometric tables. If both ladies agree, please sign the documents. We will gather here again tomorrow in the afternoon," Lady Ansatz declared. "If you are not here by three or earlier, you will lose by default."

Adanita and Viscountess Kuplung signed the contracts, followed by the witnesses of this duel. The entire ordeal took them so long that Adanita, accompanied by the lawyer, got home several hours later than she expected.

Rosamund was worried, but she was feeling excited. Viscountess clearly wanted to show off her skills and embarrass Adanita, but it wasn't as if Adanita went easy on her either. It wasn't going to end in death, but it was certainly a real duel she had on her hands. She tried scribbling out some possibilities in the carriage, but she didn't get far and her writing was ugly thanks to the rickety roads. She reached the library and spread out the papers on a table, trying to make sense of the task one more time.

"Captain Vervain, I implore you to tell me what happened," Rosamund said. He was using his strict voice, the one he used when other people were around and they pretended to be on bad terms.
"Viscountess Kuplung challenged me to a mathematical duel. I have until tomorrow to solve this task," she said.
He tried to peek at the problem, but the little lawyer that came with her stopped him. The man was far more courageous than he looked, and he fearlessly spoke up. "Lord Governor, I am well aware of your position and power, but I must ask you to refrain from looking at this or helping Captain Vervain. This is a matter of honor, and I cannot-"
"Enough. I know better than to get involved in another person's duel," Rosamund cut him off. "Do you plan to stay awake for the whole night?"
"Yes, my lord. Worry not, this is something I am accustomed to. I will keep watch over Captain Vervain. Please understand, she wasn't found untrustworthy by the ladies, nobody doubts she'd conduct honorably in this duel. I am here to ensure that nobody can accuse her of cheating of any kind, in case somebody decides to doubt her later."
"Fine. I'll have my servants send you both some coffee and light up the library properly."
"Thank you, Lord Governor," she said.

Now, let's see. From each point, there was a measurement of an angle towards the neighbouring two points. Those angles seemed to be random numbers, so she assumed that the instrument they used must've had the "zero" pointing at random directions at every point. The angle between the neighbouring points would therefore be calculated as a difference between the measured angles. So far, so good. However, what she needed was an angle between the ordinate axis and the line towards the next point. If she had that and the distance between points, she could figure out how much she moved on both axes to get to the point's coordinates. It was just a simple right angle triangle, after all.

Rosamund returned, along with a servant who carried coffee for them both. He gave her a small case containing her new reading glasses and wished her luck.

Gods, he was so considerate ♡

Right, then. How to get the angle of direction? What did it have to do with the known three points? Why was that extra point M there, what did these two triangles mean?

Alright, time to think logically. This whole set of measurements started with D2, measuring towards C5 and onwards to 74. So it would make sense if- ooooh. Right, then it had to end with E1 measuring to C5, because it had to start and end with two points of known coordinates. Then, we can calculate the direction from coordinates, and sort out the value of directions for every next point from the previous. Makes sense! So, how to get the angle from E1 to C5? Well, from coordinates. But she also needed the angle that would have been measured if they could've put the instrument on E1, right? Does she need it...? Well, she has it with the first two points. Surely she needs it. But what for...

Well... she got stuck here.

But not for long! She can calculate the value of an angle D2-C5-E1. Then, from measurements, figure out the angle between D2-C5-M. The difference between them was the first angle in the triangle M-E1-C5. Second angle she knew because there was measurements from M to both C5 and E1, so she could just calculate it. Then, she'd know the third angle in this triangle, and the angle she was looking for all along would have been the difference from 360 and that. HAH!

Then it took her two more hours to figure out how to get angles of direction and correct the errors of angle measurements. It turned out to be simple trigonometry, hooray for calculating on a plane! And she was right, she needed the last angle to determine whether the sum of all angles between lines towards the points would add up to the direction from E1 to C5, and then correct them depending on their values. How clever she was! The little lawyer sat on the couch behind her in absolute silence, sipping coffee and letting her work uninterrupted. Bless him.

There also had to be an error in measuring the coordinates themselves, right? When she checked how they all came along by adding them together, she didn't end up in the exact same coordinate as the end point E1. Figuring out how to correct those was hell in its own right.

Gods, she really underestimated that woman. She gave her a task that a professional would solve within maybe two hours, but Adanita was running in circles around it throughout the night. She was missing the knowledge that woman had, that was it. Well, she gave her a task of similar difficulty, at least. This was exciting, this was fun. She hoped Viscountess was suffering just as much as she was, staying awake by the candlelight, looking at a task she's never before seen in her life and trying to sort out how a person from the field would solve it. The coffee held her awake, but her tempo was slowing down as she was approaching the solution.

It was already dawning when she finally couldn't take it anymore.

Adanita fell asleep on her papers, resting her head on the twice-circled end result.