Pirate Blackbeard's Lucky Flag

Pirate Blackbeard is preparing his ship for a perilous voyage in search of hidden treasure. According to ancient buccaneering superstition, the ship's flag must be a right-angled triangle with whole-number side lengths, and its area must be exactly equal to its perimeter, for good fortune at sea.

What is the largest possible area of such a flag that Blackbeard can make?

The Sea Monster of Dead-Gale Passage

Since the dawn of time, a gigantic sea monster has lurked beneath the Dead-Gale Passage. On any given day:

If it is asleep, it will still be asleep the next day with probability 1/2.

If it is awake, it will still be awake the next day with probability 11/12.

Blackbeard and his crew will cross these treacherous waters tomorrow.

What is the probability that the crew will have to contend with the monster on the day they cross?

The Monster's Wrath

As expected, the sea monster attacked Blackbeard and his crew who narrowly managed to escape, but their ship's mainsail was not so fortunate. Before the attack, the sail was a proud triangular canvas with integer side lengths and a perimeter of 156 feet.

During the fierce battle, the sea monster's razor-sharp claws tore straight through the sail, ripping it into two right-angled integer triangles.

What was the original area of the mainsail, in square feet?

Blackbeard's Shadow in the Cabin

There is a leak in one of the dark lower cabins.

Blackbeard who is 2m tall turns on the 5m high lantern in the corner of the cabin.

He runs away from the lantern towards the hole in the hull at 6m/s.

At what rate is his shadow lengthening?