The 4th annual offline tournament, "Best of the best".

I'm still a little confused about something – what's the deal with the Admiral's Hat? What's allowed and what isn't, according to the rules? I'd like more details.

This restriction applies to the following situations:

-- Escaping from the sea and subsequently moving with the redeemed hero, using navigation bonuses on land.
This includes the following cases:
1. Using the "Admiral's Hat" artifact or its components;
2. The hero having the "Navigation" skill at any level;
3. Having (building or taking control of) a Lighthouse(s).
This restriction is related to the possibility – in the specified cases – of having an unreasonably increased movement range on land after escaping from the sea, gaining additional movement points "out of thin air."
That's why this tactic is prohibited.
P.S. Escaping from the sea and subsequently moving with the redeemed hero in the 3rd case is allowed if the construction or capture of the Lighthouse occurs on the day of the escape.

-- Using the "Admiral's Hat" artifact, which is prohibited in the following way:
repeatedly boarding a boat and disembarking from it in approximately the same location. The use of this artifact is permitted if the main goal is to cross a body of water, which is necessary for the player to perform specific actions.
When boarding and disembarking, any artifacts the hero has that increase his movement range on land (Boots of Swiftness and Rider's Gloves) must be in the hero's backpack.
Boarding and disembarking at any point on the same land area (with the exception of a point that cannot be reached on land, either at all or without using the "Fly" or "Dimension Door" spells) does not constitute crossing a body of water.
Boarding and disembarking on the same land area without crossing a body of water for the purpose of visiting objects (collecting artifacts/resources or attacking monsters/heroes) on the sea is allowed once per day.
This restriction is related to the possibility – when manipulating artifacts that increase the hero's movement range on land and the "Admiral's Hat" artifact – of significantly increasing the movement range by repeatedly boarding and disembarking from a boat, gaining additional movement points "out of thin air."
That's why this tactic is prohibited.