Just solved it, what a fun puzzle.
You can cheese this one pretty easily. What really matters is what the possible letters are for the 4 special squares (+, ++, —, ----). Once you limit these, you can make a very simple python script which will compare the possible 4 letter combinations with a list of common 4 letter words. When I did this, there were only a handful of possible answers.
I got right answer using similar method as gwarbeh.
I tried to solve it in normal way, but I’m always getting some inconsistences.
Even knowing the answer (hence the results of +/- ) I still cannot solve it fully.
Can someone who has the whole grid ping it to me once the contest is done, please?
I’ve not spent a lot of time on this so far, though it’s a fun puzzle. I’m trying the conventional approach to see how many possibilities I can eliminate and abstract logic is so far proving quite helpful. I’m nowhere near solving it yet … I will almost certainly resort to brute forcing the letters as others have done, if I haven’t made progress by this time tomorrow!
EDIT: Got the solution by forcing the letters. Now to solve the puzzle!
Another fun puzzle. Wrote a script to find and work through the valid shapes. Once the bugs were ironed out, finds the solution in a few minutes.
I did it the conservative method. At first, I got only meteor squares left which didn’t connect, then started shuffling things around. I couldn’t get to the ending, but from what I had I could guess the answer. I’m still gonna try finding the complete layout by using the answer as a hint.
So assuming I want to actually solve this puzzle the “intended” way, what’s the right approach? I’ve been using the elimination method, but it’s been a very tedious process.
Edit: Still haven’t solved the puzzle but I randomly submitted a word that has what I believed might be the first and last letters and it worked out 
Are the 4 letters in some order on the puzzle?
I would absolutely love to find out more about this. Please do consider sharing, after the contest is all over!
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Yes. Exactly as stated in instruction. My initial mistake was to go from top to bottom, but once put in right order, i found right answer.
so it is: +, ++, —, ----.
That’s what I thought at first too, top to bottom. Thanks.
I finally solved the grid by hand today… it was tricky but fun!
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I think I won’t be able to solve this one…
I understood the explanations (not completely I guess), but when I look at the grid I don’t get how, based on the rules, I’m supposed to fill it…
Do I have to change the shape of the grid? Or I must let it as it is, a 10x10 square? I have no idea 
You still have more than a day to solve this… you can do it!
I found it confusing at first too but the concept is pretty simple once you understand it.
You divide up the 10x10 grid into tetromino-like shapes of differing sizes, constructed of individual grid squares, connected by their edges (i.e. shapes cannot have squares attached diagonally by their corners alone).
Each shape contains two numbered squares and the size of the shape is any value in between the lowest and highest numbers in the shape.
Example with three invalid red shapes and one valid (random - not a hint) green shape:
6-4 shape is correct size but invalid shape.
1-3 shape is valid shape but incorrect size.
3-3 shape contains an invalid pair of values.
5-2 shape is valid (shapes of size 3 could also be created with these values).
NOTE: No shape can overlap any other and every square on the grid is used.
Once you’ve successfully paired up all of the numbers into valid shapes, the instructions for finding the solution are pretty self-explanatory.
As for how to tackle the puzzle? It’s not easy but you know that every shape must contain values which have a difference of at least 2, as otherwise there’s no ‘in-between’ value determining the size of the shape. So you could start by noting which numbers cannot pair and thus which numbers must (or most likely must) pair.
The only concrete hint I can offer is that the area with all the larger numbers (around the middle of the grid) was the key for me to unlock this, as it’s a very congested area and, if you can get that section to work, the rest of the grid should follow.
I hope that helps!
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Those shapes are also called ‘polyominoes’.
Polyomino
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Finally got all the shapes. When the puzzle ‘clicks’ it’s a great sensation and the frustration melts away. I had the correct answer just by process of looking at the possibilities for letters just around the +/- squares. What threw me off was an invalid answer upon entering the correct word. I think the page required refreshing??? When I tried much later, that same word was now correct. Also worked out the shapes on a spreadsheet and forgot to copy 1 number onto sheet. Had me thinking this puzzle was impossible at first.
I must say this was a tough one.
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Thank you SO MUCH @igorgetmeabrain for your help!!
It made me realize that I didn’t understand the puzzle at all; I thought we were supposed to fill the blank squares with numbers 
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I’ve tried looking at this several times and something about it just… isn’t clicking. I can’t seem to figure out how to approach it or where to start, or if there are any segments that can be definitively identified. There’s one single number that I’ve identified must pair with a specific other number because it’s boxed in by non-options, but even then there are too many constellations of blanks that could form that block. On the verge of giving up and just accepting that I’ll be missing out on these entries (not that I’m likely to win anyway).
I didn’t count how many iterations I tried with each getting within one or two left over that didn’t fit. I though of saying oh well, I will miss this one. Then got re-inspired resulting in success. Walk away, refresh, then re-try if you have the time.
Were you just going at it by trial and error? Because I can’t think of any other way that would work here.