5
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Arrange numbers $2$ to $20$ in the hexagons using the following rules:

  1. The difference between two adjacent hexagons is greater than $4$.
  2. Numbers inside the green hexagons are even numbers.
  3. Numbers inside the yellow hexagons are prime numbers.

enter image description here

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4
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I think there are no unique solutions, but two.

Solution
3 and 5 can't be filled by logic.

Sorry for the hand drawn numbers. I was too lazy to open my Photoshop.


Below is how I named each cells.

Cell names

Once again, sorry for the hand drawn letters.


And the steps below are how I solved this. Basically I kept tracking what numbers can go into a cell, and eliminating those numbers.

1. Two white cells should be one of 9, 15.
2. C1 != 5 ~ 19, and 2 is gone. Therefore C1 is 4.
3. E1 != 2 ~ 8 + 11 ~ 13 (possible: 17, 19)
4. If D1 is 15, both 17 and 19 can't go into E1. Therefore D1 is 9, and B1 is 15.
5. E2 is 14 by adjacent cells.
6. A1 != 11 ~ 20 (possible: 3, 5, 7)
7. B2 != {2 ~ 9, 16 ~ 20} (possible: 10, 12)
8. D2 != 8 ~ 20 (possible: 3, 5, 7)
9. C3 != 3 ~ 12 (possible: 16, 18)
10. E3 != {3 ~ 7, 14 ~ 20} (possible: 11, 13)
11. F2 != {3 ~ 7, 9 ~ 18}. Therefore F2 is 19, and E1 is 17.
12. G2 != {2 ~ 6, 10 ~ 20}. Therefore G2 is 7.
13. H1 != 2 ~ 11. Therefore H1 is 13, and E3 is 11.
14. I1 != 3 ~ 17. Therefore I1 is 18, and C3 is 16, and B2 is 10.
15. G1 is 8, and G3 is 6, and H2 is 12.

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  • $\begingroup$ Superb and clear read. Just suggesting a notational improvement, instead of labeling the hexes {A_i, ..., H_i}, why not label the yellows {P_i} for primes, greens {E_i} for evens, and the white {O_i} for odd (composite) numbers? (of course the adjacent interconnections become slightly trickier) $\endgroup$ – smci Sep 2 '17 at 11:49
4
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Eureka!! Eureka!!

enter image description here

Challenge faced

Cracking the cluster of prime numbers near 20 was bit tricky - Once that was fixed to 2 combinations - placement of 2 helped the choice

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  • $\begingroup$ Oh, I was few seconds late while double-checking my solution. $\endgroup$ – Otami Arimura Sep 2 '17 at 8:30
  • $\begingroup$ @Otami, Yep.. but you gave a better explanation.. upvote for that :) $\endgroup$ – Kalaivanan Sep 2 '17 at 8:33

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