Solving the logic puzzle gives the following 3x3 grid of people and their numbers:
| ELSIE (3) |
FREYA (4) |
HOLLY (4) |
| APRIL (5) |
COREY (2) |
GOPAL (1) |
| BILLY (2) |
DAISY (4) |
IZUMI (4) |
Since all names are five letters long, you can index into the name by the number, to get the answer SYLLOGISM.
A full solution of the logic puzzle follows.
Note that each person has either three, five, or eight neigbors.
We call the three types of positions corner, face, and center respectively.
- Corey must be in the center, because a neighbor sum of 27 is too high for a corner or a face. So Corey is everyone's neighbor.
- Corey cannot have distinct numbers for each neighbour. If anyone on a face has distinct numbers for all their neighbors, then their neighbor sum is 15 = 1 + 2 + 3 + 4 + 5.
- By Billy and April's statements, this means that at most one face person has distinct neighbor numbers, so all corner people have distinct neighbor numbers.
- By Daisy's statement, we need exactly five people with distinct neighbor numbers. So Freya must be a face person with distinct neighbor numbers, which have to be 1, 2, 3, 4, and 5.
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Further, Freya's statement means that Freya has to be on the north face.
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April can now only be on the west face. (The northwest corner is ruled out because corner people have distinct neighbor numbers.) This implies that Corey's number (in the center) is 2, and the SW corner is also 2.
- Elsie's number must be 3, because the sum of all numbers is now determined to be 29.
- Holly has to be in a corner, otherwise her neighbor sum would be too large. Since her neighbor numbers are distinct, they have to be 4 and 1 in some order, not including Corey.
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By Izumi's statement, Holly is either in the NE corner or the NW corner.
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By Daisy and Gopal's statements, we see that they are both face people.
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If Gopal were on the south face, then at least one more of Freya's neighbors would need to have a 2, which is impossible. So Gopal is on the east face, and Daisy is on the south face.
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Freya's number cannot be 2 (otherwise the NW corner does not get distinct neighbor numbers). If all of Gopal's other neighbors had 2s, then Freya would not have distinct neighbor numbers. So all four of Gopal's neighbors excluding Corey have the same number not equal to 2.
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We can determine that Gopal's neighbors (excluding Corey) all have the number 4, as follows. They cannot have the number 1 because the total sum would be too small. They cannot have the numbers 3 or 5, otherwise there is no place for Holly to go.
- In particular, Freya and Daisy are both 4.
- This determines that the NW corner must be a 3 (by April's neighbor sum), and so that is the only possible place for Elsie.
- Holly can now only be in the NE corner, and Izumi in the SE corner. This determines that Gopal's number is 1.
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By elimination, Billy is in the SW corner.
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Finally, April's number must be 5 because of Freya's neighbor sum (or by deducing from the total sum).