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In Hexologic, there are bonus levels that you can unlock after completing each 15 levels. (although it doesn't matter if it's on Easy or Hard mode) I have been able to complete all of the bonus levels on both Easy and Hard mode except for 2 of them, which I am attempting to beat on Hard. This is the puzzle layout:

enter image description here

I-

Anyways, there are some things that I am going to be able to do:

  1. The bottom left 4 must be a 2-2 pair. If the 4 was a 1-3 pair, the 9 would not be able to be satisfied, and if it was a 3-1 pair, the 3 could not be satisfied. Some other deductions leads me to this board state:

enter image description here

  1. Let's now focus on the green cell. If that were to be a 3, then the 4 must be a 1-3 pair, which then according to Hexologic rules, is impossible to satisfy. So the green cell must be a 1 or a 2. However, if the green cell was a 2, then the 5 makes a 2-3 pair, and then in the row containing 17, we would have $3+2+4+7=16$ which then we would need $x+y=1$ which is not satisfiable with our conditions. So the green cell must be a 1, which means that the 9 must be a 3-1-5 pair. Then we can satisfy the 4s, which gives this grid state:

enter image description here

however I'm unsure if I can logically deduce the solution from here. So my question is:

How should I progress in solving the Hard mode version of the 2nd bonus puzzle unlocked after completing Level 45 in Hexologic?

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The simplest way I can see to break into the central section is:

  • Look at the down-left 20. It needs 5 more from two empty cells, so they must be 2,3 in some order.
  • That means the other cell of the leftward 4 must be 1 or 2.
  • Just above that, the leftward 10 needs 3 from two empty cells, which must be 1,2 in some order.
  • That gives you two cells of the down-left 7 that sum to at most 4.
  • So the other cell in the 7 must be a 3, and both of those cells must be a 2.
    enter image description here

  • From there, you only have single unknowns for the 4 and 10 rows.
  • And filling those in leaves single unknowns in the 6, 17 and 20 that complete the puzzle.
    enter image description here

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