# Your goal is to maximize the absolute difference of the numbers marked ?.

(? marks indicate region areas. They may or may not belong in the same region which another number or ?. They may or may not be shaded.)

Answers must prove their claim (and provide a construction).

Normal Chocolate Banana rules apply.

Rules:

1. Paint some of the cells black.

2. Black cells linked orthogonally must always form a rectangle (or a square).

3. Non-blacked cells linked orthogonally must not form a rectangle (or a square).

4. A number indicates the area (the number of cells) of the orthogonally region containing this cell.

Click on the image to use the online solver and view the rules.

Hint: this builds upon my last puzzle.

• Wow really nice puzzle shame I am a little busy atm!
– PDT
Commented Apr 23 at 12:14

The difference of 73.

Long story short (look below if you want to see the deductions):

Using deductions without taking maximization into consideration you can only get this far: So whatever is the final solution it has to resemble this.

To maximize:

We see that we can easily make the bottom right square 1 without taking anything away from the top left and from here we need to make top left yellow and everything else too to maximize the difference.

From here on out you can have two ways to complete the grid.

However

if we want the higest difference plus a unique solution we can make the bottom right 2 making 72 the difference!

The solution path:

Starting off with the ones being obviously shaded it forces the 2 to be shaded for if it was yellow then it would lead to a >2 region. 3 has to be yellow so as to not imprison the 2. Then the square to the right of the 2 has to be shaded otherwise the 3 region would be too large:

Next

The 4 cannot be yellow as it leads to the 5 being shaded but this leads to a no hoper for a 5 squared rectangle as the only two ways to make one makes the 4 region too big and so 4 needs to be shaded… This forces:

Then

For the 6, a similar issue arises as for the 4 if it was yellow- it is a no hoper for the 7 rectangle as the only two ways of making one makes the 6 region too big: Then this forces:

Then

8 being yellow leads for no possibility for a 7 and 8 segment to co-exist: And so this forces:

Finishing off

The 7 segment then needs to me made like so so that it does not violate the rule that the shaded regions have to form a rectangle. The bottom right question-marked region has to be grey otherwise there would not be enough for a 9 sized region and the other shaded area is forced on R1C7 to prevent the size of the region being too big:

• I could've sworn I made it unique... smh
– Sny
Commented Apr 23 at 13:51
• Hmm, but if I said maximum it could also have been 9...
– Sny
Commented Apr 23 at 13:53
• Assign values to the question marks such that the solution is unique? You gave me a new idea :). I think bottom right 2 will also make it unique?
– Sny
Commented Apr 23 at 14:02
• Haha yes it would d’oh!
– PDT
Commented Apr 23 at 14:03
• It’s not too late to change to question, I will just just re-edit this slightly. But it’s up to you…
– PDT
Commented Apr 23 at 14:22