Hexagonal Pasture Network hexagon.X
Farmer John recently acquired some new land to expand his farm. His
cows have taken a liking to the hexagonal structure of bee honeycombs,
and, ever willing to please his herd, Farmer John has set up a new
system of pastures and cowpaths in this format.
The full plot of pastures and cowpaths forms a hexagon with side
length K (2 <= K <= 50). Pastures are conveniently numbered
1..3*K*(K-1)+1 starting in the bottom left and ending in the upper
right using the pattern generalized from this illustration where K
= 3:
Each pasture is connected to all of its immediate neighbors. This
means that if a pasture is on the inside of the hexagon, it is
adjacent to exactly six other pastures. For example, in the diagram
above, pasture #10 is adjacent to pastures #5, #6, #11, #15, #14,
and #9. Pastures on the edge (but not on a corner) of the structure
are adjacent to exactly four other pastures (for example, pasture
#4 is adjacent to #1, #5, #9, and #8) while pastures at a corner
are adjacent to only three pastures (e.g., pasture #1 is connected
to pastures #2, #5, and #4). The length of any cowpath connecting
two pastures is 1 and the distance between two pastures is defined
to be the length of the shortest possible route between them.
Farmer John's Holstein cows have been munching on the grass in
pasture H (1 <= H <= 3*K*(K-1)+1) for several days now and have
grown fat and lazy. To force his cows to get some exercise, Farmer
John has laid down tasty cow treats in pastures exactly distance
of L (1 <= L <= 2*K-2) away from the cows. He guarantees the cows
that he has placed at least one treat, but he doesn't tell the cows
the pastures in which he's placed them.
Please help the cows avoid any unnecessary exercise by printing the
number of possible pastures which might hold the treats and a list
of those possible pastures in ascending order.
By way of example, suppose K = 3, the Holsteins are in pasture #1, and
Farmer John says he's placed some treats in pastures a distance of
2 away. The possible locations of the treats are pastures #3, #6,
#10, #9, and #8, as shown below.