Instructions [29 - 33 ]

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.

What was Jano's total spend in the two countries he visited, in Aurels?

Correct Answer: The correct answer is 1500.

Let us solve the puzzle step-by-step to find Jano's total spend in the two countries he visited, converted to Aurels.

Step 1: Identifying the Visited Countries
We have four countries: Aurevia (A), Brelosia (B), Cyrenia (C), and Zerathania (Z, the starting point).
Three travelers (Jano, Kira, and Lian) each visit exactly two countries from {A, B, C}, and each country is visited by exactly two travelers.
From the citizen observations:
• Aurevia citizen: Jano and Kira visited Aurevia.
• Brelosia citizen: Kira and Lian visited Brelosia.
• Cyrenia citizen: Lian visited Cyrenia.
Since each country is visited by exactly two travelers, Cyrenia must also have been visited by Jano. Thus:
• Jano visited Aurevia (A) and Cyrenia (C).
• Kira visited Aurevia (A) and Brelosia (B).
• Lian visited Brelosia (B) and Cyrenia (C).

Step 2: Defining Spends and Exchange Rates
Let the exchange rates (value in Zentars, Z) be:
• 1 Crown (Cyrenia) = 0.5 Z (given).
• 1 Aurel (Aurevia) = $e_A$ Z.
• 1 Brin (Brelosia) = $e_B$ Z.
Across the 6 visits, the local currency spends must be from the set {1000, 2000, 3000}. There are exactly two spends of 1000, one spend of 3000, and therefore three spends of 2000.
From the Brelosia citizen:
• Kira spent 2000 Brins in Brelosia.
• Lian spent 3000 Brins in Brelosia.
Since the single spend of 3000 is taken by Lian, no other traveler can spend 3000 in any country.
Each traveler has different spends in their two visited countries:
• Kira visited A and B. Kira's spend in B is 2000, so Kira's spend in A cannot be 2000. Since it cannot be 3000, Kira's spend in A must be 1000 Aurels.
• The remaining spends to distribute among the other visits (Lian in C, Jano in A, Jano in C) are {1000, 2000, 2000}.
• Jano visited A and C and must have different spends. Thus, Jano's spends must be one 1000 and one 2000 in some order.
• This leaves Lian's spend in Cyrenia to be 2000 Crowns.

Step 3: Calculating Exchange Rates using Travel Costs
The flight costs of Kira and Lian are 5000 Z and 6000 Z, in some order. Jano's flight cost is 4000 Z.
Let's analyze Kira's Travel Cost:
From Aurevia's citizen, Kira's Travel Cost = 8000 Aurels. In Zentars:
Travel Cost (K) = $8000\times e_A$ Z.
From Brelosia's citizen, Kira's Travel Cost = 4000 Brins. In Zentars:
Travel Cost (K) = $4000\times e_B$ Z.
Equating the two:
$8000\times e_A=4000\times e_B\Rightarrow e_B=2\times e_A$.
Kira's Travel Cost in Zentars is also the sum of Flight Cost and Spends:
$\text{Travel Cost (K)}=\text{Flight(K)}+(1000\times e_A)+(2000\times e_B)$
Substituting $e_B=2\times e_A$:
$8000\times e_A=\text{Flight(K)}+1000\times e_A+4000\times e_A$
$\text{Flight(K)}=3000\times e_A$.

Now let's analyze Lian's Travel Cost:
From Cyrenia's citizen, Lian's Travel Cost = 36000 Crowns. In Zentars:
Travel Cost (L) = $36000\times 0.5=18000$ Z.
Lian's Travel Cost in Zentars is:
$18000=\text{Flight(L)}+(3000\times e_B)+(2000\times 0.5)$
$18000=\text{Flight(L)}+3000\times e_B+1000$
$17000=\text{Flight(L)}+3000\times e_B$.

Since Flight(L) can be 5000 Z or 6000 Z:
If Flight(L) = 5000 Z, then:
$17000=5000+3000\times e_B\Rightarrow 3000\times e_B=12000\Rightarrow e_B=4$.
This gives $e_A=e_B/2=2$, and Flight(K) = $3000\times 2=6000$ Z. This perfectly matches the set of flight costs {5000, 6000}.
(If we set Flight(L) = 6000 Z, we get a non-integer value for Flight(K) which does not equal 5000 Z).
Therefore, the exchange rates are:
• $e_A=2$ (1 Aurel = 2 Zentars)
• $e_B=4$ (1 Brin = 4 Zentars)

Step 4: Jano's Spends and Total Spend
From Aurevia's citizen, Jano's Travel Cost = 3500 Aurels. In Zentars:
Travel Cost (J) = $3500\times 2=7000$ Z.
We know Jano's Flight Cost is 4000 Z:
$7000=4000+\text{Spend(J in A in Z)}+\text{Spend(J in C in Z)}$
$\text{Spend(J in A in Z)}+\text{Spend(J in C in Z)}=3000$ Z.
Let Jano's local spends be $S_{JA}$ (in Aurels) and $S_{JC}$ (in Crowns), which must be {1000, 2000} in some order:
$(S_{JA}\times 2)+(S_{JC}\times 0.5)=3000$.
Testing the combinations:
• If $S_{JA}=1000$ and $S_{JC}=2000$:
$(1000\times 2)+(2000\times 0.5)=2000+1000=3000$ Z (this is correct).
• If $S_{JA}=2000$ and $S_{JC}=1000$:
$(2000\times 2)+(1000\times 0.5)=4000+500=4500$ Z (incorrect).
So, Jano spent 1000 Aurels in Aurevia and 2000 Crowns in Cyrenia.

Step 5: Converting Total Spend to Aurels
• Spend in Aurevia = 1000 Aurels.
• Spend in Cyrenia = 2000 Crowns.
Since 1 Crown = 0.5 Zentars, 2000 Crowns = 1000 Zentars.
Since 1 Aurel = 2 Zentars, 1000 Zentars = $1000/2=500$ Aurels.
Therefore, Jano's total spend in Aurels is:
$1000+500=1500$ Aurels.