3. In the Magic and Wizardry club, all first and second-year students wear red robes, third-year students wear blue, and fourth-year students wear black.

Last year, at the general assembly of students, there were 15 red, 7 blue, and several black robes, while this year - blue and black robes are equal in number, and red robes are twice as many as blue ones.

(a) How many black robes will there be at the general assembly next year? (1 point)

(b) How many first-year students are there this year? (3 points)

(c) In what minimum number of years will the number of blue and black robes be equal again? (6 points)

(d) Suppose it is additionally known that each year the number of first-year students is 1 less than the number of fourth-year students. In how many years will the number of red robes first be three times the number of blue ones? (10 points)
# Solution.

(a) 7 mantles. Since 7 blue mantles from last year will turn black this year and there are an equal number of blue and black mantles this year, there will also be 7 blue mantles this year, which will turn black next year.

(b) 6 students. This year, there are 7 blue, 7 black, and $14=2 \times 7$ red mantles. Therefore, last year there were $8=15-7$ first-year students who will become second-year students this year. Thus, this year there are $14-8=6$ first-year students.

(c) In three years. Last year, 8 new students joined the club, and this year 6 did. If 6 new students join next year, then in three years there will be 6 blue and 6 black mantles:

$$
(6,8,7,7) \rightarrow(6,6,8,7) \rightarrow(?, 6,6,8) \rightarrow(?, ?, 6,6)
$$

If a different number of new students join next year, the number of blue and black mantles will differ for the next three years. Therefore, the smallest number of years after which the number of blue and black mantles can be equal is three.

(d) In 17 years. The composition of students for the next year is uniquely determined by the composition of students for the current year: $(a, b, c, d) \rightarrow(c-1, a, b, c)$. If we sequentially list the composition of students for each year:

$$
(6,8,7,7) \rightarrow(6,6,8,7) \rightarrow(7,6,6,8) \rightarrow(5,7,6,6) \rightarrow \ldots
$$

then in 17 years, we will first obtain a composition of students that satisfies the condition. This will be the composition $(2,1,1,3)$, in which $2+1=3$ red mantles and one blue mantle.