Question 1161833: The royal court of Prince Lemon had dukes, earls, and barons. At the beginning of the Prince's reign, there were 2020 members of the court and each day, one of them killed another one in a duel. Dukes only killed earls, earls only killed barons, and barons only killed dukes. No person won a duel twice. In the end, the only member of the court left alive was Baron Orange. What was the title of the first court member who was killed?
Answer by ikleyn(52890)   (Show Source):
You can put this solution on YOUR website! .
The royal court of Prince Lemon had dukes, earls, and barons. At the beginning of the Prince's reign,
there were 2020 members of the court and each day, one of them killed another one in a duel.
Dukes only killed earls, earls only killed barons, and barons only killed dukes. No person won a duel twice.
In the end, the only member of the court left alive was Baron Orange. What was the title of the first court member who was killed?
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So, we have D (dukes), E (earls), and B (barons), and we know that B was the last and only survivor.
To solve the problem, we should to construct a chain of arguments that will lead us to the answer.
Watch my steps (!)
(1) First of all, different types of duels are possible: DD, DE, DB, ED, EE, EB, BD, BF, BB, listed by the titles of participants.
Of them, only duels DE, EB, BD are with the fatal outcome; the other are not.
Since the other outcomes do not change the population of the court, we can and we should consider only
duels with the fatal outcomes DE, EB and BD; we can omit all other duels from our consideration.
(2) On principle, there are two possibilities: the last baron won the last duel, OR the last baron
did not participate in duels, at all.
But would he did not participate in the duels, then after the last duel the other participant would be alive,
which contradicts to the problem.
Hence, the last baron Orange participated in the last duel and won it.
(3) It means that his opponent in the last duel was a duke; so, the last duel was the pair BD.
Thus the last survivor was B, and the previous survivor was D.
(4) Since the previous survivor was D, his opponent in that duel was E (nobody else could be (!)).
Thus the last survivor was B (N1 from the end); the previous survivor was D (N2 from the end); and E was N3 from the end.
(5) By moving this way BACK from the end to the beginning, we see this pattern
. . . 6 5 4 3 2 1
. . . E D B E D B
The upper line shows the number of a duel, from the LAST (N1) to the very first (in reversed order).
The bottom line is the title of the survivor in the duel.
My numeration in two previous lines is, as you understand, from the end to the beginning; from right to left.
These triples (E D B) (in this order) are repeating again and again, showing the survivor at the duels,
numerated from the end to the beginning.
(6) Now it is not difficult to determine who was the first survivor.
The number of people is 2020; the number of duels is 1 less than 2020, i.e. 2019.
When divided by 3, 2019 gives the remainder 0; so the survivor in the first duel was E, an Eagle.
His opponent, who was killed at that duel, was a baron (B).
ANSWER. The first court member who was killed at the first duel was a baron.
* * * * * Solved (!) * * * * *
It is really good, FIRST CLASS Math Olympiad and Math circle level problem.
I am very proud by myself, having solved it (!) (!)
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