Cows like to build shapes out of hay bales. Each new shape is constructed from N (1 <= N <= 25,000) 1 x 1 x 1 cubic bales.
Bale 1 lies on the floor. After placing it, each successive bale from 2 through N is attached to the shape by placing the new bale in the proper relative position to one of the bales already placed. For instance, imagine the following figure:
Bale 1 - on the floor
Bale 2 - RIGHT Bale 1 (to Bale 1's right)
Bale 3 - FRONT Bale 2 (in front of Bale 2)
Bale 4 - FRONT Bale 3
This results in an (inverted) L-shape:
+--+--+
1-> / / /| <-2
+--+--+ +
| / /|/
++--+ +
/ /|/ <-3
+--+ +
| |/ <-4
+--+
Adding another bale:
Bale 5 - OVER Bale 1
adds height to the shape:
+--+
/ /|
+--+ +--+
5-> | |/ /| <-2
+--+--+ +
1-> | / /|/
++--+ +
/ /|/ <-3
+--+ +
| |/ <-4
+--+
But adding another directive:
Bale 6 - BACK Bale 4
results in an invalid shape, since bales 3 and 6 would overlap.
Given N bales and placements described as above, output the exposed surface area of the solid figure if the shape is valid. A face is considered exposed if it is not touched by any face of any other bale and does not touch the floor (i.e., is not the bottom of a cube on the floor). Output -1 if the shape is invalid. No bale will be placed farther than 25 bale-widths from the first bale. No bale will be placed below the floor level.