5
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Arrange numbers 2 to 20 to the hexagons, with rules :

  1. The difference between 2 connected hexagons (side by side) is more than 4
  2. Numbers inside the yellow hexagons are prime numbers.

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3
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Should be it

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Approach

Fixing the Prime numbers did the trick. Rest fell in place with a few trials

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5
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The answer is

First row: 2
Second row: 14 12
Third row: 4 7 6
Fourth row: 19 18
Fifth row: 9 13 11
Sixth row: 3 5
Seventh row: 17 20 16
Eighth row: 8 10
Ninth row: 15

Full solution:

  1. You start by noting that the three primes that border 9 must be 3, 17, 19. The location of 3 is then set since it must take the spot that borders 20.
  2. Next, note that 5 must be in one of the two yellow hexagons that are furthest to the right, or else it would border 2 or 3. The 7 must then be directly below the 2, or else it would border either 3 or 5.
  3. After that, note that the 11 must also be in one of the furthest two yellow hexagons, or else it would border 7 or 9. This fixes 13 as the one right below 7.
  4. From there, you can deduce that 5 is to the right of 13, and 11 is the prime furthest right (since the other way would have 11 border 13).
  5. And finishing off the primes, due to 13's location, the prime between 9 and 7 must be 19, and the last yellow hexagon is where 17 is located.
  6. Note that because 5,7,11,13 are all adjacent to a square and we already know the location of both 19 and 20, this square contains 18.
  7. From here, note that the only possible number above 18 left is 12, and the only possible number above 19 is 14.
  8. Next, note that the only number left that can be above 9 is 4.
  9. Next, note that the only number left that can border 11,12,18 is 6.
  10. 16 must then be below 11, since otherwise it would border 20.
  11. 15 must be below 20 or else it borders 16 or 17.
  12. 8 must then be below 3 (or else it would border 5), and 10 takes the remaining spot.

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  • $\begingroup$ Probably the ninth row that you've mentioned is a typo :) $\endgroup$ – Kalaivanan Sep 2 '17 at 7:09
  • $\begingroup$ Explanation incoming; currently writing out all of the logic $\endgroup$ – Dennis Meng Sep 2 '17 at 7:15

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