Can any one try to help prove why in the game of conway's solitaire army that,
C(4) > or equal to 19??
At each step, show that any other solution would add more pieces. That will prove it using numbers. So, once you have a piece on level 3 and one on level 2, you have 2 choices. Either, you can get the one on level 2 back with a minimal number of piece (4) leaving one on level 3 or you can get the one back on level 3 first (but you need to move the one on level 2 out of the way). Each time you make a decision, show how any other decision will result in more pieces on the board than the decision you make. Each soldier on level 2 requires at least 4 pieces to start to get there. Each soldier on level 3 requires at least 8 pieces to get it there.
Here are some possible positions for the first few moves. {e8}, {e7,e6}, {e7,d6,c6}, {e6,e5,d6,c6}, {e6,d6,c6,f5,g5}, {d6,c6,e5,f5,g5,e4}, {c6,d5,e5,f5,g5,d4,e4}, {c5,d5,e5,f5,g5,c4,d4,e4}, ...
At this point, there are 5 pieces on level 1 with 3 pieces in the way.