Olympiad Maths Prep

Track / Stage 6 / 83 of 400 #1083 of 2000

Problem 1083

National olympiad, first round
Combinatorics Difficulty 6.1 Find the answer

11. (12 points) As shown in the figure, there are 9 small square doghouses numbered 191 \sim 9, with each doghouse housing at most 1 puppy; originally, there are 3 puppies, and the doghouses they occupy are not adjacent (adjacent small squares share a common side); when a new puppy moves in, the adjacent puppies will bark once to welcome it; now, 5 new puppies move in one after another, and each new puppy is welcomed by exactly 2 barks; it is known that the first new puppy moved into doghouse 2, the second new puppy barked 2 times. The fourth new puppy moved into doghouse 4, and it did not bark; if the doghouse numbers of the 5 new puppies are A,B,C,D,EA, B, C, D, E in order, then the five-digit number ABCDE=A B C D E= \qquad .

Official solution

【Answer】Solution: According to the problem:
(1) The first puppy is in kennel 2. The 2nd new puppy barked twice. The 4th new puppy is in kennel 4, and it didn't bark;

This means that 2 new puppies came next to the 2nd puppy. When puppies move in, exactly 2 puppies bark once, so the new 2nd puppy cannot be in the corner kennels 1, 3, 6, 7, 8, 9. It can only be in kennel 5.
(2) The 4th new puppy is in kennel 4, and it didn't bark; When puppies move in, exactly 2 puppies bark once, which means either 1 or 7 is empty. If 1 is empty, the kennels would be adjacent. Therefore, 7 must be empty.
(3) The new 2nd puppy barked twice, which means that either kennel 6 or 8 originally had 1 puppy. Therefore, 8 must be the original one, and 6 is the new one.
(4) The original three non-adjacent kennels are 1, 3, and 8. The 5th one is in kennel 9.

Therefore, the answer is: 25649

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.