2868. Hexagonal Pasture Network

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.