The villagers will initiate construction of the new hut on their own, but you need to have a villager who already has some building skill, and whose preferred skill is set to construction, before that will happen. If you don't have any builders, and you're playing v1.0, you will have to hope that more debris washes up on the beach so that you can train a builder. If you're playing v1.1, you can train new builders by having them fix the huts that they've already built.
The last 3 puzzles require Level 3 in specific Village Technologies.
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Barbara
Unicorn
Last Day of Work