Question 163
At the beginning the board contains $31$ vectors
$$
(1,0,0,\dots,0),\ (0,1,0,\dots,0),\ \dots,\ (0,0,0,\dots,1)
$$
each having $31$ components. At each step we choose two vectors $(a_1, a_2, \dots, a_{31})$ and $(b_1, b_2, \dots, b_{31})$ written on the board and write their sum $(a_1 + b_1, a_2 + b_2, \dots, a_{31} + b_{31})$ to the board. Find the minimal number of steps which should be made to get all vectors
$$
(0,1,1,\dots,1),\ (1,0,1,\dots,1),\ \dots,\ (1,1,1,\dots,0).
$$
on the board.
$$
(1,0,0,\dots,0),\ (0,1,0,\dots,0),\ \dots,\ (0,0,0,\dots,1)
$$
each having $31$ components. At each step we choose two vectors $(a_1, a_2, \dots, a_{31})$ and $(b_1, b_2, \dots, b_{31})$ written on the board and write their sum $(a_1 + b_1, a_2 + b_2, \dots, a_{31} + b_{31})$ to the board. Find the minimal number of steps which should be made to get all vectors
$$
(0,1,1,\dots,1),\ (1,0,1,\dots,1),\ \dots,\ (1,1,1,\dots,0).
$$
on the board.