Aurevia, Brelosia, Cyrenia and Zerathania are four countries with their currencies being Aurels, Brins, Crowns, and Zentars, respectively. The currencies have different exchange values. Crown's currency exchange rate with Zentars = 0.5, i.e., 1 Crown is worth 0.5 Zentars.

Three travelers, Jano, Kira, and Lian set out from Zerathania visiting exactly two of the countries. Each country is visited by exactly two travelers. Each traveler has a unique Flight Cost, which represents the total cost of airfare in traveling to both the countries and back to Zerathania. The Flight Cost of Jano was 4000 Zentars, while that of the other two travelers were 5000 and 6000 Zentars, not necessarily in that order. When visiting a country, a traveler spent either 1000, 2000 or 3000 in the country's local currency. Each traveler had different spends (in the country's local currency) in the two countries he/she visited. Across all the visits, there were exactly two spends of 1000 and exactly one spend of 3000 (in the country's local currency).

The total “Travel Cost” for a traveler is the sum of his/her Flight Cost and the money spent in the countries visited.

The citizens of the four countries with knowledge of these travels made a few observations, with spends measured in their respective local currencies:

i. Aurevia citizen: Jano and Kira visited our country, and their Travel Costs were 3500 and 8000, respectively.

ii. Brelosia citizen: Kira and Lian visited our country, spending 2000 and 3000, respectively. Kira's Travel Cost was 4000.

iii. Cyrenia citizen: Lian visited our country and her Travel Cost was 36000.

One Brin is equivalent to how many Crowns?

The correct answer is 8.

To determine the exchange rate between Brins and Crowns, we need to deduce the routes taken by each traveler, the amounts they spent in each country, and the exchange rate of each currency relative to Zentars. Let's break down the given information logically.

Step 1: Determine the countries visited by each traveler
The three travelers start from Zerathania and visit the other three countries: Aurevia, Brelosia, and Cyrenia. Let's refer to these countries by their currencies: Aurels (A), Brins (B), and Crowns (C). We know each country is visited exactly twice, and each traveler visits exactly two countries.
From the citizens' observations:
- The Aurevia citizen observed Jano and Kira, meaning Jano and Kira visited country A.
- The Brelosia citizen observed Kira and Lian, meaning Kira and Lian visited country B.
Since each traveler visits exactly two countries, we can determine the remaining visits. Jano has visited A and needs one more, which must be C. Lian has visited B and needs one more, which must also be C. This perfectly aligns with each country receiving exactly two visitors:
- Jano visits A and C.
- Kira visits A and B.
- Lian visits B and C.

Step 2: Determine the local currency spends for each traveler
There are 6 visits in total (3 travelers × 2 countries). Across all visits, the spends are either 1000, 2000, or 3000. We are told there are exactly two 1000s and exactly one 3000. Therefore, the complete list of the 6 spends is: 1000, 1000, 2000, 2000, 2000, 3000.
From the Brelosia citizen's observation, we know:
- Kira spent 2000 in B.
- Lian spent 3000 in B.
Because each traveler must have different spend amounts in their two visited countries, we can deduce their other spends:
- Kira spent 2000 in B, so her spend in A cannot be 2000. The only 3000 was spent by Lian, so Kira's spend in A must be 1000.
At this point, the remaining unassigned spends are {1000, 2000, 2000}, which must belong to Jano in A, Jano in C, and Lian in C.
- Since Jano must have different spends for his two countries, his spends must be 1000 and 2000.
- This leaves the final 2000 for Lian's spend in C.
To summarize the spends (in local currencies):
- Kira: 1000 in A, 2000 in B.
- Lian: 3000 in B, 2000 in C.
- Jano: 1000 and 2000 in A and C (we will determine which is which shortly).

Step 3: Establish the currency exchange rates
Let $v_A$, $v_B$, and $v_C$ represent the value of 1 Aurel, 1 Brin, and 1 Crown in Zentars, respectively.
We are given that 1 Crown = 0.5 Zentars, which gives us:

$v_C=0.5$

The Travel Cost is the sum of the Flight Cost (in Zentars) and the spends converted to Zentars. Let $F_K$ be Kira's Flight Cost. Jano's Flight Cost is 4000, leaving 5000 and 6000 for Kira and Lian.
Using the Brelosia citizen's observation, Kira's total Travel Cost is 4000 Brins. Converting everything to Zentars gives:

$4000v_B=F_K+1000v_A+2000v_B$

Subtracting $2000v_B$ from both sides gives Equation 1:

$2000v_B=F_K+1000v_A$

Using the Aurevia citizen's observation, Kira's total Travel Cost is 8000 Aurels. Converting everything to Zentars gives:

$8000v_A=F_K+1000v_A+2000v_B$

Substituting $2000v_B$ from Equation 1 into this new equation:

$8000v_A=F_K+1000v_A+(F_K+1000v_A)$

$8000v_A=2F_K+2000v_A$

$6000v_A=2F_K$

$3000v_A=F_K$

We know Kira's Flight Cost $F_K$ must be either 5000 or 6000. If $F_K$ were 5000, $v_A$ would be a fraction (5/3), which usually indicates an incorrect path in such puzzles. Let's test $F_K=6000$.
If $F_K=6000$, then $v_A=2$.
Substituting $v_A=2$ and $F_K=6000$ into Equation 1:

$2000v_B=6000+1000\left(2\right)=8000$

$v_B=4$

This gives clean integer exchange rates: 1 Aurel = 2 Zentars, and 1 Brin = 4 Zentars.
(We can quickly verify this with Jano and Lian's total costs:
- Jano's TC = 3500 A = 7000 Z. His FC is 4000. His spends in A and C are 1000 and 2000. $4000+1000\left(2\right)+2000\left(0.5\right)=4000+2000+1000=7000$. This matches perfectly!
- Lian's TC = 36000 C = 18000 Z. Her FC is 5000. Her spends are 3000 in B and 2000 in C. $5000+3000\left(4\right)+2000\left(0.5\right)=5000+12000+1000=18000$. This also matches perfectly!)

Step 4: Calculate the final conversion
We have established that:
1 Brin = 4 Zentars
1 Crown = 0.5 Zentars (so 1 Zentar = 2 Crowns)
To find how many Crowns are equivalent to 1 Brin, we multiply:

$1\text{ Brin}=4\text{ Zentars}\times 2\text{ Crowns/Zentar}=8\text{ Crowns}$

Thus, one Brin is equivalent to 8 Crowns.