I started off using a 'trial and error' approach but this wasn't working so I started to think about what the total in each pile needed to be. I added up all the numbers 1-20 to reach a total of 210. I then divided this by 6 to find the total for each pile (210/6 = 35).
Once I had realised this, I found different ways of making 35 (for example 17 + 18). I then placed the numbers in groups, marking off which ones I'd used along the way. It didn't work first time because I was left with numbers that were too big for the final total. I spotted that I could use number bonds to help (e.g. 1 + 2 + 3 + 4 = 10), this made it easier.
Eventually, I found a solution:
Pile 1: 20, 15
Pile 2: 17, 18
Pile 3: 16, 19
Pile 4: 11, 14, 1, 2, 3, 4
Pile 5: 12, 13, 10
Pile 6: 5, 6, 7, 8, 9
I wonder if there are any other solutions?