Step 1: Capha sees Ewina, Feach, Garou, Gnote and Ameno, and reports three red hats. Dabro sees Feach, Garou, Gnote, Ameno and Arplu, and reports only two red hats. Note that they both see Feach, Garou, Gnote and Ameno. So Ewina must be wearing a red hat and Arplu’s hat must be green.
Step 2: Likewise, Dabro and Dondu both see Garou, Gnote, Ameno and Arplu plus one other. Dabro’s other is Feach, Dondu’s is Bbyan. Dondu reports one more red than Dabro does, thus Bbyan’s hat is red and Feach’s is Green.
Step 3: Similarly, Feach and Garou both report three red hats. The elves they see have Bbyan, Bowan, Capha and Dabro in common, and one sees Arplu while the other sees Dondu. This means Arplu and Dondu must be wearing the same color. Since we already know Arplu’s hat is green, Dondu’s must also be green.
Step 4: Using logic similar to step 3, by considering Bbyan and Bowan’s reports, we can deduce that Dondu and Gnote are wearing the same color, thus Gnote’s hat is also green.
Step 5: Returning to Dabro’s report, Dabro reports three green hats among Feach, Garou, Gnote, Ameno and Arplu. We’ve already deduced that Feach, Gnote and Arplu are wearing green, so Garou and Ameno must have red hats.
Step 6: We’ve now deduced the color of four of the elves in Bowan’s report, so we can deduce the fifth. Eddle’s hat is red.
Steps 7-9: Using logic similar to step 6, Eddle’s report tells us Bowan’s hat is red; Ewina’s report means Capha’s is Green; and Feach’s tells us Dabro’s is red.
Abro-ca-Dabro. Puzzle solved!
Addtional Info: The puzzle has an analogy in music theory. The names of the elves correspond to notes in the scale — the first letter of their names are the note names, and the elves with flat topped hats are the flat notes. For instance, Dondu is a D and Dabro is a D flat. Gnote is, in fact, the G note.
The arrangement of the red hats corresponds to the notes of the key of B, with Bowen as the root.
Combining the reports from neighboring elves, as in steps 1-4, provides inferences about elves at an interval corresponding to a musical 5th. It is possible to follow these inferences similar to the musical circle of fifths. Steps 3 and 4 do just that, but after step 4 we've deduced enough to allow shorcuts for the remaining steps.
Even Ameno and Arplu would see three hats if they opened their eyes.
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