Silhouette Riddle

It would be easier to start off with a plain cube of side length $11$ , which will have the same silhouette of a $11\times 11$ square in every view, and from the question, every visible silhouette has an area of a $4\times 5$ rectangle missing. This rectangle will act as a base of the missing cuboid.

Thus, we can visualize its chunk being chopped off as a $4\times 5\times 11$ cuboid. This will result in the desired silhouette as shown in grey, below:

Next, we proceed similarly by cutting another cuboid of dimension of $4\times 5\times (11-5) = 4\times 5\times 6$ ; this new cut cuboid has shorter length than previous one because a part of it was already cut in the first process. This again will result in the silhouette as shown in grey, below:

Finally, the last cuboid of $4\times 5\times (11-4) = 4\times 5\times 7$ will be taken out, and the three boxes will be arranged as followed:

Even though there are many possible outcomes for the box combinations, the total volume will always equal:

$11^3- 4\times 5\times 11 - 4\times 5\times 6 - 4\times 5\times 7 = 11^3 - 4\times 5\times 24 = 1331 - 480 = \boxed{851}$

Plotting a 3-D drawing basing on the top, front and side views, we find that the three cuboids are $6\times 6 \times 11$ , $5 \times 7 \times 7$ , and $5\times 6 \times 7$ . And their total volume is $6\times 6 \times 11 + 5 \times 7 \times 7 + 5\times 6 \times 7 = \boxed{851}$ .