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How To Do Logician Puzzles
This is how I go about solving a Logician puzzle!

1. The 5 characters are Naruto, the one who uses Kunai, the one who uses Nunchaku, the one going to the Village Hidden in Mist, and the one going to the Village Hidden in Sand.
From this, we can make a list of the five distinct characters:
Naruto
Kunai
Nunchaku
Mist
Sand
For now, make a note of what options each has for the other categories. Naruto can't use Kunai or Nunchaku, and he can't be headed to Mist or Sand village, so his line looks like:
Naruto, (Senbon/Shuriken/Explosives), (Rain/Sound/Leaf)
You can do the same for each separate character, giving you:
Naruto, (Senbon/Shuriken/Explosives), (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), Kunai, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Sand
2. Shikamaru brought along Explosives.
If Shikamaru brought along Explosives, it means that Naruto did not bring Explosives, and that the person who brought Kunai or Nunchaku wasn't Shikamaru. You can eliminate those from the lines, like so:
Naruto, (Senbon/Shuriken), (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Sand
3. Hinata did not bring any Kunai.
We have Kunai located already, so for this we just eliminate Hinata from that line.
Naruto, (Senbon/Shuriken), (Rain/Sound/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Sand
4. The one who went to the Sand Village is not Lee nor the one with the Shuriken.
Likewise, we know where the Sand village is, so we eliminate Lee and Shuriken from that line.
Naruto, (Senbon/Shuriken), (Rain/Sound/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata), (Senbon/Explosives), Sand
5. Sakura went to the Village Hidden in Sound.
If Sakura went to the Sound village, that means Naruto didn't go to the Sound village, and the Sand and Mist villages weren't Sakura.
Naruto, (Senbon/Shuriken), (Rain/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Hinata), (Senbon/Explosives), Sand
6. Naruto did not bring Senbon.
If Naruto didn't bring Senbon, that leaves only Shuriken available for him. That means we can eliminate Shuriken from any other lines.
Naruto, Shuriken, (Rain/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Hinata/Lee), (Senbon/Explosives), Mist
(Shikamaru/Hinata), (Senbon/Explosives), Sand
7. The one with the Senbon went to the Mist Village.
If the Mist village is the one with Senbon, that also means that the Sand village is the one with Explosives. Referring to clue #2 shows us that Shikamaru is the one with the Explosives.
Naruto, Shuriken, (Rain/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Hinata/Lee), Senbon, Mist
Shikamaru, Explosives, Sand
8. Lee did not go to the Village Hidden in Mist nor bring any Kunai.
If Lee didn't bring Kunai, then Sakura must have, and if Lee didn't go to the Mist village, then Hinata must have. That leaves only one line that Lee could be on. We also know from clue #5 that Sakura went to the village in Sound.
Naruto, Shuriken, (Rain/Leaf)
Sakura, Kunai, Sound
Lee, Nunchaku, (Rain/Leaf)
Hinata, Senbon, Mist
Shikamaru, Explosives, Sand
9. The one who went to the Village Hidden in Rain brought the Nunchaku.
We know where the Nunchaku is, so that must be the person who went to Rain, leaving Leaf for the one remaining person.
Naruto, Shuriken, Leaf
Sakura, Kunai, Sound
Lee, Nunchaku, Rain
Hinata, Senbon, Mist
Shikamaru, Explosives, Sand

1. The 5 characters are Naruto, the one who uses Kunai, the one who uses Nunchaku, the one going to the Village Hidden in Mist, and the one going to the Village Hidden in Sand.
From this, we can make a list of the five distinct characters:
Naruto
Kunai
Nunchaku
Mist
Sand
For now, make a note of what options each has for the other categories. Naruto can't use Kunai or Nunchaku, and he can't be headed to Mist or Sand village, so his line looks like:
Naruto, (Senbon/Shuriken/Explosives), (Rain/Sound/Leaf)
You can do the same for each separate character, giving you:
Naruto, (Senbon/Shuriken/Explosives), (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), Kunai, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Sand
2. Shikamaru brought along Explosives.
If Shikamaru brought along Explosives, it means that Naruto did not bring Explosives, and that the person who brought Kunai or Nunchaku wasn't Shikamaru. You can eliminate those from the lines, like so:
Naruto, (Senbon/Shuriken), (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Sand
3. Hinata did not bring any Kunai.
We have Kunai located already, so for this we just eliminate Hinata from that line.
Naruto, (Senbon/Shuriken), (Rain/Sound/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Sand
4. The one who went to the Sand Village is not Lee nor the one with the Shuriken.
Likewise, we know where the Sand village is, so we eliminate Lee and Shuriken from that line.
Naruto, (Senbon/Shuriken), (Rain/Sound/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Sakura/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Sakura/Hinata), (Senbon/Explosives), Sand
5. Sakura went to the Village Hidden in Sound.
If Sakura went to the Sound village, that means Naruto didn't go to the Sound village, and the Sand and Mist villages weren't Sakura.
Naruto, (Senbon/Shuriken), (Rain/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Hinata/Lee), (Senbon/Shuriken/Explosives), Mist
(Shikamaru/Hinata), (Senbon/Explosives), Sand
6. Naruto did not bring Senbon.
If Naruto didn't bring Senbon, that leaves only Shuriken available for him. That means we can eliminate Shuriken from any other lines.
Naruto, Shuriken, (Rain/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Shikamaru/Hinata/Lee), (Senbon/Explosives), Mist
(Shikamaru/Hinata), (Senbon/Explosives), Sand
7. The one with the Senbon went to the Mist Village.
If the Mist village is the one with Senbon, that also means that the Sand village is the one with Explosives. Referring to clue #2 shows us that Shikamaru is the one with the Explosives.
Naruto, Shuriken, (Rain/Leaf)
(Sakura/Lee), Kunai, (Rain/Sound/Leaf)
(Sakura/Hinata/Lee), Nunchaku, (Rain/Sound/Leaf)
(Hinata/Lee), Senbon, Mist
Shikamaru, Explosives, Sand
8. Lee did not go to the Village Hidden in Mist nor bring any Kunai.
If Lee didn't bring Kunai, then Sakura must have, and if Lee didn't go to the Mist village, then Hinata must have. That leaves only one line that Lee could be on. We also know from clue #5 that Sakura went to the village in Sound.
Naruto, Shuriken, (Rain/Leaf)
Sakura, Kunai, Sound
Lee, Nunchaku, (Rain/Leaf)
Hinata, Senbon, Mist
Shikamaru, Explosives, Sand
9. The one who went to the Village Hidden in Rain brought the Nunchaku.
We know where the Nunchaku is, so that must be the person who went to Rain, leaving Leaf for the one remaining person.
Naruto, Shuriken, Leaf
Sakura, Kunai, Sound
Lee, Nunchaku, Rain
Hinata, Senbon, Mist
Shikamaru, Explosives, Sand