A certain bathroom has N + 2 stalls in a single row; the stalls on the
left and right ends are permanently occupied by the bathroom guards. The
other N stalls are for users.
Whenever someone enters the bathroom, they try to choose a stall that is as far
from other people as possible. To avoid confusion, they follow deterministic
rules: For each empty stall S, they compute two
values LS and RS, each of which is the number of empty
stalls between S and the closest occupied stall to the left or right,
respectively. Then they consider the set of stalls with the farthest closest
neighbor, that is, those S for which min(LS, RS) is
maximal. If there is only one such stall, they choose it; otherwise, they choose
the one among those where max(LS, RS) is maximal. If there
are still multiple tied stalls, they choose the leftmost stall among those.
K people are about to enter the bathroom; each one will choose their
stall before the next arrives. Nobody will ever leave.
When the last person chooses their stall S, what will the values of
max(LS, RS) and min(LS, RS)
The first line of the input gives the number of test cases, T.
T lines follow. Each line describes a test case with two integers
N and K, as described above.
For each test case, output one line containing
Case #x: y z,
x is the test case number (starting from 1),
y is max(LS, RS), and
is min(LS, RS) as calculated by the last person to
enter the bathroom for their chosen stall S.
1 ≤ T ≤ 100.
1 ≤ K ≤ N.
Small dataset 1
1 ≤ N ≤ 1000.
Small dataset 2
1 ≤ N ≤ 106.
1 ≤ N ≤ 1018.
Case #1: 1 0
Case #2: 1 0
Case #3: 1 1
Case #4: 0 0
Case #5: 500 499
In Case #1, the first person occupies the leftmost of the middle two stalls,
leaving the following configuration (
O stands for an occupied
. for an empty one):
O.O..O. Then, the
second and last person occupies the stall immediately to the right, leaving 1
empty stall on one side and none on the other.
In Case #2, the first person occupies the middle stall, getting to
O..O..O. Then, the second and last person occupies the leftmost
In Case #3, the first person occupies the leftmost of the two middle stalls,
O..O...O. The second person then occupies the middle of
the three consecutive empty stalls.
In Case #4, every stall is occupied at the end, no matter what the stall
In Case #5, the first and only person chooses the leftmost middle stall.