Dancing links algorithm solutions Right now I am working on my implementation of Knuth's DLX algorithm.

I did understand the exact cover problem. But I didn't fully understand how Dancing links works. Here is Knuth's paper.
The question is about the solutions presented in the paper. I don't understand how these solutions make sense taking into account the matrix that he uses.

Matrix for the exact cover problem

     A B C D E F G
  1  0 0 1 0 1 1 0
  2  1 0 0 1 0 0 1
  3  0 1 1 0 0 1 0
  4  1 0 0 1 0 0 0
  5  0 1 0 0 0 0 1
  6  0 0 0 1 1 0 1


And solutions are:

A D
B G
C E F


But how come these solutions are valid? Isn't it supposed to be answered like A B E? Where in each row there is only one 1.
Knuth is trying to cover the columns using rows. The solution you describe consists of rows 4, 5, 1, in this order (counting from 1). Each row is represented by its 1-columns. For example, row 4 is 1001000, and its 1-columns are AD.