Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the line segments between them represent intercity highways.
A salesperson needs to make a trip. She needs to start from a city, visit each of the remaining cities exactly once, and finally return to the same city from which she started.

Which one of the following options is then true?

Question

In the 2020 summer Olympics' Javelin throw finals, Neeraj Chopra exhibited a spectacular performance to win the gold medal. The silver medal was won by Jakub Vadlejch and the bronze medal was won by Vitezlav Vesely. There were six rounds of throws with each athlete having one throw per round. The best of all the throws of each athlete is considered for the medal. Following were the observations about the throws:
(i) The first and second rounds were dominated by Neeraj Chopra with a gold medal performance in his second throw, while the other two athletes did not have any medal winning throws in these rounds.
(ii) The throws in the last round by both Jakub Vadlejch and Vitezlav Vesely were fouls and were not considered for scoring.
(iii) After four rounds, Vitezlav Vesely was in the second position and could not improve upon his best throw in the succeeding rounds.
(iv) In the fourth round, the throw by Jakub Vadlejch was the best in that round.
In which round did Vitezlav Vesely have his best throw?

Answer 1

This question tests whether a closed tour touching every city exactly once, a Hamiltonian circuit, can be drawn on each of the two road maps.

Option A, trip possible for (i) but not (ii): In the 4 by 4 grid, start at the top left city and go straight down the first column, cross to the second column, weave up and down through columns 2 to 4, and come back along the top row to the start. Because the grid has an even count of cities, $4 \times 4 = 16$, this weave closes into a single loop covering every city once. In the 5-city map, three cities sit on just two roads each, and both of those roads run to the same pair of hub cities. Making every one of those roads part of the loop would put three roads on one hub city, but a loop can carry only two roads per city, so the loop cannot be completed. This matches option A exactly.
Option B, trip possible for (ii) but not (i): Wrong, since the argument above shows (i) does allow a full round trip.
Option C, trip possible for both: Wrong, since (ii) fails the round trip test.
Option D, trip possible for neither: Wrong, since (i) does allow a full round trip.

So the correct choice is option A: the round trip works for (i) but breaks down for (ii).

Answer 2

This question tests whether a closed tour touching every city exactly once, a Hamiltonian circuit, can be drawn on each of the two road maps.

Option A, trip possible for (i) but not (ii): In the 4 by 4 grid, start at the top left city and go straight down the first column, cross to the second column, weave up and down through columns 2 to 4, and come back along the top row to the start. Because the grid has an even count of cities, $4 \times 4 = 16$, this weave closes into a single loop covering every city once. In the 5-city map, three cities sit on just two roads each, and both of those roads run to the same pair of hub cities. Making every one of those roads part of the loop would put three roads on one hub city, but a loop can carry only two roads per city, so the loop cannot be completed. This matches option A exactly.
Option B, trip possible for (ii) but not (i): Wrong, since the argument above shows (i) does allow a full round trip.
Option C, trip possible for both: Wrong, since (ii) fails the round trip test.
Option D, trip possible for neither: Wrong, since (i) does allow a full round trip.

So the correct choice is option A: the round trip works for (i) but breaks down for (ii).

Show Hint

The Correct Option is A

Solution and Explanation

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