The reason why the game does not progress even though all numbers are grey, it's because it greys out when the blocks are in the correct position, however, it doesn't mean your puzzle is correct. This has happened to me before because I placed an additional block to a row which I wasn't suppose to, but I assumed it was right because the number was grey. Reset the puzzle or carefully look for you mistake.
I've noticed people still having trouble with the hard badge even with my guide! People are saying to not buy the gold nail and the random factor. I replayed the game (good times :P) and I still managed to complete it within 11 days doing exactly what I wrote in my guide. I do believe there IS a random factor, but I don't think it'll affect you too much since I won again with over 2k gold left. I still believe the gold nail is a must. Letting monsters hit you will reduce the amount of gold you could have potentially had. One new final tip that I can offer which I think is REALLY important is to not go crazy with artillery as soon as you see the red golems. Those golems are really slow. Wait for the semi-fast birds to enter the map before using your artillery! Using your artillery as soon as you see a red golem enter the map is a complete waste of 20 gold since you could have potentially used a bird's bomb later. That's all I could help after playing the game a second time. Good luck!
Hard badge guide:
1. Only buy the artillery for those red golems and the gold nail for the floating balloons.
2. After buying those two upgrades, just repair as soon as you can.
3. I was able to repair my ship within 11 days with no enemies hitting my ship.
Tips:
1. Cage the birds when they are above large groups of enemies! The bombs they drop do as much damage as your artillery, so instead of spending $20, cage the birds at the right moment! (bird's bomb can't hit the floating balloons, which is why you should upgrade your nail to make your life easier)
2. Get the fast moving units first such as the red flames that fly.
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