Logical Reasoning

Math Puzzles

Hard

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.

How many Zentars did Lian spend in the two countries he visited?

Entered answer:

✅ Correct Answer: 13000

Slide 1/8

Three travelers — Jano, Kira, Lian — each visit exactly two countries, starting from Zerathania (Zentars). The three visited countries are Aurevia (Aurels), Brelosia (Brins), and Cyrenia (Crowns); each is visited by exactly two travelers.

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)	Travel Cost (Zentars)
Jano
Kira
Lian

Exchange value of one unit, in Zentars:

Aurel	Brin	Crown
		$0.5$

Each row is a traveler.

The Flight column stores the Flight Cost in Zentars.

The three country columns store the spend made in that country, in its own local currency; a dash marks a country a traveler did not visit.

The last column stores the Travel Cost (Flight Cost plus all spends), kept here in Zentars.

The small table converts one unit of each currency into Zentars; only the Crown is given, at $0.5$ .

Two facts are known at the start:

Jano's Flight Cost is $4000$ Zentars; the other two flights are $5000$ and $6000$ in some order.

The six spends form the multiset $\{1000, 1000, 2000, 2000, 2000, 3000\}$ : two of $1000$ , one of $3000$ , three of $2000$ .

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)
Jano	$4000$		—
Kira				—
Lian		—

Aurel	Brin	Crown
		$0.5$

Jano's Flight Cost $4000$ is filled.

From observation (i), Aurevia is visited by Jano and Kira.

From observation (ii), Brelosia is visited by Kira and Lian.

From observation (iii), Cyrenia is visited by Lian and one more traveler.

Kira already has two countries (Aurevia, Brelosia), and Lian already has two (Brelosia, Cyrenia).

So the second visitor of Cyrenia must be Jano, giving Jano the pair Aurevia and Cyrenia.

Dashes mark the unvisited country for each traveler: Jano–Brelosia, Kira–Cyrenia, Lian–Aurevia.

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)
Jano	$4000$		—
Kira			$2000$	—
Lian		—	$3000$

Aurel	Brin	Crown
		$0.5$

From observation (ii), in Brelosia Kira spent $2000$ Brins and Lian spent $3000$ Brins. These are filled.

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)
Jano	$4000$		—
Kira		$1000$	$2000$	—
Lian		—	$3000$

Aurel	Brin	Crown
		$0.5$

The only $3000$ in the whole set is Lian's Brelosia spend, so no other spend equals $3000$ .

Kira's Aurevia spend must differ from her Brelosia spend ( $2000$ ) and cannot be $3000$ .

The only value left is $1000$ , so Kira's Aurevia spend is $1000$ .

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)
Jano	$4000$		—
Kira		$1000$	$2000$	—
Lian		—	$3000$	$2000$

Aurel	Brin	Crown
		$0.5$

The set needs exactly two spends of $1000$ ; one is already Kira's Aurevia spend.

Jano's two spends are distinct and lie in $\{1000, 2000\}$ (no $3000$ remains), so exactly one of them is the second $1000$ .

That uses up both $1000$ s.

Hence Lian's Cyrenia spend cannot be $1000$ , and cannot be $3000$ , leaving $2000$ .

This also confirms three $2000$ s: Kira–Brelosia, Lian–Cyrenia, and Jano's single $2000$ .

Jano's pair $\{$ Aurevia, Cyrenia $\} = \{1000, 2000\}$ remains to be ordered.

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)
Jano	$4000$		—
Kira	$6000$	$1000$	$2000$	—
Lian	$5000$	—	$3000$	$2000$

Aurel	Brin	Crown
$2$	$4$	$0.5$

Let one Aurel $= a$ and one Brin $= b$ Zentars (Crown $= 0.5$ ).

Kira's Travel Cost is $8000$ Aurels (i) and $4000$ Brins (ii), the same amount:

$8000a = 4000b \Rightarrow b = 2a$ .

Kira's Travel Cost in Zentars is also Flight $_K + 1000a + 2000b = 8000a$ .

Using $b = 2a$ : Flight $_K + 5000a = 8000a \Rightarrow$ Flight $_K = 3000a$ .

Flight $_K$ is $5000$ or $6000$ .

Flight $_K = 6000 \Rightarrow a = 2,\ b = 4$ , leaving Flight $_L = 5000$ .

Flight $_K = 5000 \Rightarrow a = \tfrac{5}{3},\ b = \tfrac{10}{3}$ , leaving Flight $_L = 6000$ .

Lian's Travel Cost is $36000$ Crowns $= 36000 \times 0.5 = 18000$ Zentars, and equals Flight $_L + 3000b + 2000(0.5)$ .

So Flight $_L + 3000b = 17000$ .

Testing the first option: $5000 + 3000(4) = 17000$ . This holds.

Testing the second: $6000 + 3000(\tfrac{10}{3}) = 16000 \neq 17000$ . This fails.

So $a = 2$ , $b = 4$ , Flight $_K = 6000$ , Flight $_L = 5000$ . These are filled.

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)
Jano	$4000$	$1000$	—	$2000$
Kira	$6000$	$1000$	$2000$	—
Lian	$5000$	—	$3000$	$2000$

Aurel	Brin	Crown
$2$	$4$	$0.5$

Jano's Travel Cost is $3500$ Aurels $= 3500 \times 2 = 7000$ Zentars, and equals

$4000 + (\text{Aurevia})\times 2 + (\text{Cyrenia})\times 0.5 = 7000$ ,

so (Aurevia) $\times 2 +$ (Cyrenia) $\times 0.5 = 3000$ , with $\{$ Aurevia, Cyrenia $\} = \{1000, 2000\}$ .

Aurevia $= 1000$ , Cyrenia $= 2000$ : $2000 + 1000 = 3000$ . This holds.

Aurevia $= 2000$ , Cyrenia $= 1000$ : $4000 + 500 = 4500$ . This fails.

So Jano spent $1000$ Aurels in Aurevia and $2000$ Crowns in Cyrenia.

Traveler	Flight (Zentars)	Aurevia (Aurels)	Brelosia (Brins)	Cyrenia (Crowns)	Travel Cost (Zentars)
Jano	$4000$	$1000$	—	$2000$	$7000$
Kira	$6000$	$1000$	$2000$	—	$16000$
Lian	$5000$	—	$3000$	$2000$	$18000$

Aurel	Brin	Crown
$2$	$4$	$0.5$

Travel Costs in Zentars:

Jano $= 4000 + 1000(2) + 2000(0.5) = 7000$ .

Kira $= 6000 + 1000(2) + 2000(4) = 16000$ .

Lian $= 5000 + 3000(4) + 2000(0.5) = 18000$ .

Verification against the citizens:

Jano $7000$ Z $= 3500$ Aurels.

Kira $16000$ Z $= 8000$ Aurels $= 4000$ Brins.

Lian $18000$ Z $= 36000$ Crowns.

All match the stated observations, and every value is uniquely determined.

Lian visited Brelosia (spending $3000$ Brins) and Cyrenia (spending $2000$ Crowns).

Using $1$ Brin $= 4$ Zentars and $1$ Crown $= 0.5$ Zentars:

Brelosia $= 3000 \times 4 = 12000$ Zentars.

Cyrenia $= 2000 \times 0.5 = 1000$ Zentars.

Total $= 12000 + 1000 = 13000$ Zentars.

Question 18

What is the sum of Travel Costs for all travelers in Zentars?

Question 20

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

Question 21

One Brin is equivalent to how many Crowns?

Question 22

Which of the following statements is NOT true about money spent in the local currency?

How many Zentars did Lian spend in the two countries he visited?

Solution

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