How many possible trade routes contest. 15M gold prize!

[Captain dungeness]

Here is an interesting contest for you all:

Announcing the Winner:

Haron with his calculation of 28,350!

Congratulations on your 15M gold and well done with your maths skills

How many total unique trade routes are there in Avonmora? Post your guess or calculation and the closest post after 1 week wins 15M gold from the Bank of Captain Dungeness!

This is a somewhat complicated problem because some trade routes are duplicates of other trade routes (such as 1-2-3-4 and 2-3-4-1) (also 1-2 and 1-2-1-2 and 2-1-2 and 1-2-1).
[Edit: also ports that sell the same trade goods are not valid to trade with eachother: so Kanoni-Aiora is not valid. Neither is Kanoni-Aiora-Tortuga etc.]

I'm considering reverse direction trade routes to be the same as the other direction (so Pania-Goroum-Hannes = Hannes-Goroum-Pania) (1-2-3 = 3-2-1). Also, the options like "store in warehouse" and "never buy from market" and gold bars don't count as different trade routes.
[EDIT: Haron corrected me: 1-2-3 IS NOT EQUAL TO 3-2-1 because or port prices]

Shaydo and I are currently compiling a list of all possible trade routes and eliminating the duplicates. Our estimate will be used as the "correct" answer. Once we post our answer, if someone can prove our estimate is wrong then I will give them 30M gold.

Everyone can change their answer as many times as they want. I will just use the last number you posted.

Good Luck!
Shaydo (#22355) and Captain Dungeness (#3397)

[Bmw]

im out

[ChaIbaud]

I'm considering reverse direction trade routes to be the same as the other direction (so Pania-Goroum-Hannes = Hannes-Goroum-Pania) (1-2-3 = 3-2-1).

840 trade routes if you consider tzogos to aiora one trade route and aiora to tzogos another (but like that to all of them)

He counts them as the same!

[Bmw]

yes Chaulbad that is why I said that.

[Slindur]

2,593,920 trade routes.

Slindur

[Brazzler]

176,820

[Dejanira]

Well, it's all here https://www.mathsisfun.com/combinatorics/combinations-permutations.html


There are 21 ports to choose from, and we choose 4 of them (with repeat)
so the number must be 21!/[4!*(21-4)!]=5985

[Dejanira]

must be 21!/[4!*(21-4)!]=5985

IF you consider 1-2-3 = 3-2-1 and also = 2-3 -1
in other words if Aiora-Tortuga-Tzogos it's the same as Aiora-Tzogos-Tortuga, 143640 routes otherwise.

[Not me]

210 for when there is 2 ports in a trade route, 1330 when it's 3, and 5985 with 4. so 7525 different combinations I believe.

[Haron]

There are 21 ports. Trade routes can consist of 2, 3 or 4 ports. Considering all permutations (NOT removing duplicates), we get:

2 ports: 21x20 = 420
3 ports: 21x20x19 = 7980
4 ports: 21x20x19x18 = 143640

Now, what are different trade routes? Clearly, A-B and B-A are the same. A-B-C, B-C-A and C-A-B are also the same. However, A-B-C and A-C-B are NOT the same trade routes. Considering these trade routes as identical is WRONG. I'll therefore present both the RIGHT and the WRONG answer.

2 port routes have 2 of each route, so we must divide by 2. 3 ports route have 3 of each route (you can start at A, B or C and go the same route), so we should divide by 3. Doing this WRONG, you'll want to divide by 6. 4 port routes should be divided by 4. Doing this WRONG, wou'll divide by 24. so we get:

RIGHT: 420/2 + 7980/3 + 143640/4 = 38780
WRONG: 420/2 + 7980/6 + 143640/24 = 7525

Since this is a proof, I'll also accept the 30M reward if you should somehow end up with a different result

If I were rich enough to offer a reward, I'd propose another contest - a reward for the player finding the shortest route through ALL of Avonmoras ports.