8
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Arrange numbers 1 to 25 to each cell, so :

enter image description here

  • Difference between 2 adjacent numbers (horizontally and vertically) is bigger than 1 and smaller than 6
  • The bigger-smaller sign is correct between 2 adjacent numbers.
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4
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  4 >  2 <  5 >  1 <  3
  ^    ^    ^    ^    ^
  9 >  7 < 10 >  6 <  8
  ^    ^    ^    ^    ^
 14 > 12 < 15 > 11 < 13
  ^    ^    ^    ^    ^
 19 > 17 < 20 > 16 < 18
  ^    ^    ^    ^    ^ 
 24 > 22 < 25 > 21 < 23

Reasoning:

the lower rows numbers are bigger that the upper rows for each column.
This means we can split the numbers in groups of 5 like this:
(1,2,3,4,5) goes on the first row, (6,7,8,9,10) go on the second row and so on. and on each column row[i] = row[i-1] + 5.
We just need to find a way to match the columns to the signs.
Most probably we can permute the columns 2 & 4 and 1 & 5 (both permutations at once).

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  • $\begingroup$ columns 2 and 4 cannot be permuted without 1 and 5 permuted as well, because then number 2 would be adjacent to number 3, which is not allowed $\endgroup$ – elias Aug 23 '16 at 8:18
  • $\begingroup$ @elias. That's what I meant. I thought that and would clear it. Apparently I need to change this. $\endgroup$ – Marius Aug 23 '16 at 8:19
  • $\begingroup$ oh, sorry. maybe it was just my misinterpretation $\endgroup$ – elias Aug 23 '16 at 8:19
3
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Besides the solutions listed in @JamalSenjaya's answer, I think

  4  2  5  1  3
  9  7 10  6  8
 14 12 15 11 13
 19 17 20 16 18
 24 22 25 21 23

,

  4  2  5  3  1
  9  7 10  8  6
 14 12 15 13 11
 19 17 20 18 16
 24 22 25 23 21

,

  5  2  4  1  3
 10  7  9  6  8
 15 12 14 11 13
 20 17 19 16 18
 25 22 24 21 23

and

their mirrorings around a vertical axis.

are

six correct solutions.

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  • 1
    $\begingroup$ There is 2 more correct answer. $\endgroup$ – Jamal Senjaya Aug 23 '16 at 9:50
2
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These 2 are the last another correct answers

  3  1  4  2  6
  8  5  9  7 11
 13 10 14 12 16
 18 15 19 17 21
 23 20 24 22 25

and it's mirror arround vertical axis

  6  2  4  1  3
 11  7  9  5  8
 16 12 14 10 13
 21 17 19 15 18
 25 22 24 20 23

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