Quests
NPC given quests were introduced in version 3.40 and range from collecting branches to fighting robbers. Quests are received by either chatting (c) to the quest giving NPC, or, if the quest giver believes his quest urgent, they might approach the adventurer themselves. Information about all received quests, both finished and unfinished, can be found by pressing F2. Quests and quest related areas will also marked in the adventurers world map accessed by pressing F6. There's also always a reward for completing a quest.
General tips
All quest related discussion options are under the option "Greet".
Many quests have an area for the adventurer to search for whatever it is that was requested. In these situations adventurers are reminded that climbing a tree (Alt+l) gives a good view of the surrounding area, and that one should look further than one's immediate surroundings (Ctrl+- or scrolling with mousewheel).
If one expects to find tracks in the quest related area, they can also be looked for (Alt+t) from the zoomed-out map. Finding the tracks this way is not guaranteed however.
Quests and quest specific information
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I tried just to walk to the village on the zoomed-out map, but I lost him. Now I try walking him there on the zoomed-in map (which may take a while), but that seems to work.
A wounded adventurer has crawled to a village and is resting and recuperating there. If villagers are asked "How is it going" they will tell that there's an injured adventurer in the village. This quest is received by talking to the wounded man who will tell you that he was attacked by a bear or a wolf while collecting firewood near his shelter and that his belongings, including a family heirloom which he'd like back, were left by the shelter.
The general tips work very well if one has trouble finding the shelter. The beast may still lurk around and following its tracks backwards may help locate correct area. The belongings are usually scattered within on tile of the shelter making spotting it easier. As a reward, you may get a lesson in one combat skill, or sometimes the location of a treasure.
A villager will tell you that someone's been stealing prey from his bird traps and wants you to ask an other villager about a spell to catch the thief and then cast it. You'll need to give the other villager a whole bird carcass (freshness doesn't matter) so catch a bird but don't skin or butcher it. You'll also need three feathers of a stolen catch and for that you'll need to do the most annoying part of this quest: locate the trap which is only marked by a couple of feathers lying around with an unset loop snare. The place is very easy to miss.
Once the feathers have been acquired, the spell can be cast in a village sauna or a base rock after either of these has a burnt out fire (you can check by looking with l, this will also tell if a gray ground square is base rock). The bird thief will then come to confess to the quest giver, so talk to him after a day or two to receive your reward: a lucky paw.
When asking a villager how's it going they'll tell you that a villager has been very busy lately. When asking the busy busy busy villager for a quest, they'll ask you to gather stones and take them to the sauna. If one is uncertain which building is the sauna, it'll have the sauna scoop in it. Reward: a few squirrel hide's worth of goods. These will be used when next time trading with the village.
When asking a villager how's it going they'll tell you that a villager has been very busy lately. When asking the busy busy busy villager for a quest, they'll ask you to gather branches and drop them anywhere in the village. Reward: a few squirrel hide's worth of goods. These will be used when next time trading with the village.
When asking a villager how's it going they'll tell you that the sage/shaman could use some help. When talking to the sage, he'll ask the player to collect some herbs for him. These herbs may be unknown to the player and even poisonous, but the wise old man who's been eating weird plants all his life must know what he's doing, right? If you are having trouble locating or identifying the correct herb, the plants page may be of use. For a reward, you'll receive some all-healing ailment that helps with injuries when boiled. It's uncertain whether this ailment helps with poisoning or deadly poisoning so experimentation is player's own responsibility.