Eight points P1, P2, …, P8 stand in a straight row, equally spaced, in this order. Each point has an altitude (a real number):
| point | P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 |
|---|---|---|---|---|---|---|---|---|
| altitude | 8 | 11 | 9 | 11 | 9 | 9 | 1 | 11 |
altitude
11 | o o o
10 |
9 | o o o
8 | o
... |
1 | o
+--------------------------------------------------
P1 P2 P3 P4 P5 P6 P7 P8A knight arrives at P1 at time 0 and wants to reach P8.
Jumps. Think of each point Pk as the point (k, altitude of Pk) in a vertical plane. A jump must be an ordinary chess-knight move in that plane. From Pk the knight may jump to Pj only if one of these holds at the moment of the jump:
- |j − k| = 1 (an adjacent point, left or right) and the two altitudes differ by exactly 2 (up or down), or
- |j − k| = 2 (skipping one point, left or right) and the two altitudes differ by exactly 1 (up or down).
The altitudes compared are the current ones, which may be fractional or negative; a jump onto a point of any altitude, including a negative or fractional one, is allowed as long as the difference is exact. Jumps take no time.
Sinking. Before each jump the knight may wait at its current point for any length of time t ≥ 0 (t may be fractional). Suppose the knight is on a point of altitude A when the wait begins. The moving class is the set of all points whose altitude equals A at that moment. It includes the knight's own point, and it includes points already visited as well as unvisited ones. Let n be the number of points in the moving class. While the knight waits:
- every point in the moving class sinks at a rate of 1/n units per minute (so a wait of t minutes lowers each of them by t/n);
- the point opposite the knight rises at a rate of 1/n units per minute (Pk and P(9−k) are opposite, so the pairs are P1/P8, P2/P7, P3/P6, P4/P5), whether or not it has been visited;
- exception: if the opposite point also has altitude A when the wait begins, it neither sinks nor rises, and it is not counted in n;
- no other point moves.
The set of moving points and the value of n are fixed when the wait begins; they do not change during the wait, even if other points reach altitude A meanwhile. Altitudes may become fractional or negative, and all changes persist for the rest of the tour.
Other rules.
- Every point other than P8 may be visited at most once. The start at P1 counts as a visit.
- The tour ends the moment the knight first arrives at P8.
- Every wait must be necessary. Precisely: if a move consists of waiting t > 0 minutes and then jumping to P, there must exist at least one value t′ ≥ 0, t′ ≠ t, such that replacing this wait by t′ (keeping the same target P and all later moves unchanged) would make this jump illegal or would make some later jump in the tour illegal. It is enough that one such t′ exists; the condition does not require that every other t′ fails, nor that t′ = 0 fails. A wait of t = 0 is always allowed.
A tour is written as a list of moves (t, P): wait t minutes, then jump to P.
Question. What is the longest possible total time of a tour from P1 to P8?
Answer format: a single number of minutes.