Whoooo, finally got the hard badge! Here's a couple tricks I used: 1. find maps where there is two or more locked hexagons adjacent to a piece. The more the better. 2. Keep following that path and keep adding, time is of the essence, you can change later. 3. Eliminate possible pieces by keeping track of how many colors are on a side.
Holy... finally completed the challenge badge, tip for the 4th challenge, have the sword follow shield and control the shield only. Always stand slightly to the in between the sword and dragon. Look for strafe movements, this is where you deal the most damage. I followed this guide:http://www.kongregate.com/forums/3-general-gaming/topics/6623-bot-arena-3-hard-badge-guide
There is an item later in the game called "booster". It enables anyone to always get the first move in any fight. If you max a mage with an aoe, you can make all non-boss battles trival.