$3 \cdot 29$ Label the faces of a cube with $1,2, \cdots, 6$ such that the sum of the numbers on opposite faces is 7. Now consider a chessboard with $50 \times 50$ squares, each square being the same size as a face of the cube. The cube is to be rolled from the southwest corner of the chessboard to the northeast corner, each roll being along an edge of the cube, moving the cube into the eastern or northern adjacent square (no moves to the south or west are allowed). During the rolling process, each square the cube lands on is marked with the number on the face of the cube that is in contact with the square. Find the maximum and minimum possible sums of all the numbers marked on the squares.
[Solution] Suppose a number $a$ is printed after the cube rolls north, then the face numbered $a$ becomes the south-facing side of the cube. In the subsequent rolling process, if it does not roll north, the face $a$ will always face south; if it rolls north, the number printed will be $7-a$. It can be seen that in the sequence of numbers printed by the cube, $a$ and 7-a alternate (other numbers can be in between). Thus, in all 99 numbers, $a$ and 7-a either appear the same number of times or differ by one time.

Since the average value of the numbers on opposite faces of the cube is 3.5, the average value of the sum of the 99 numbers printed on the chessboard is $3.5 \times 99 = 346.5$. The pairs \{1,6\}, \{2,5\}, \{3,4\} can differ in their number of appearances by at most one, so the sum of the 99 numbers is at most 351 and at least 342.

Place the cube on the square at the southwest corner of the chessboard, with the 1 face down, the 2 face facing east, and the 3 face facing north. Then roll the cube east continuously until it reaches the southeast corner. At this point, the 2 face is down, and the 3 face still faces north. Then roll the cube north continuously until it reaches the northeast corner. It is easy to verify that the sum of the 99 numbers printed is 342. When the 3 pairs of opposite faces of the cube are swapped and the above process is repeated, the sum of the 99 numbers is 351.
In summary, the minimum value of the sum of the 99 numbers printed is 342, and the maximum value is 351.