Some of your friends are interning at the small high-tech company Web-Exodus. A running jo...

Some of your friends are interning at the small high-tech company Web-Exodus. A running joke among the employees there is that the back room has less space devoted to high-end servers than it does to empty boxes of computer equipment, piled up in case something needs to be shipped back to the supplier for maintainence.

A few days ago, a large shipment of computer monitors arrived, each in its own large box; and since there are many different kinds of monitors in the shipment, the boxes do not all have the same dimensions. A bunch of people spent some time in the morning trying to figure out how to store all these things, realizing of course that less space would be taken up if some of the boxes could be nested inside others.

Suppose each box i is a rectangular parallelepiped with side lengths equal to (i1 i2 i3); and suppose each side length is strictly between half a meter and one meter. Geometrically, you know what it means for one box to nest inside another: It's possible if you can rotate the smaller so that it fits inside the larger in each dimension. Formally, we can say that box i with dimensions i2, i3) nests inside box j with dimensions j2, j3) if there is a permutation a, b, c of the dimensions {1,2,3} so that ia1, and ib2, and ic3. Of course, nesting is recursive: If i nests in j, and j nests in k, then by putting i inside j inside k, only box k is visible. We say that a nesting arrangement for a set of n boxes is a sequence of operations in which a box i is put inside another box j in which it nests; and if there were already boxes nested inside i, then these end up inside j as well. (Also notice the following: Since the side lengths of i are more than half a meter each, and since the side lengths of j are less than a meter each, box i will take up more than half of each dimension of j, and so after i is put inside j, nothing else can be put inside j.) We say that a box k is visible in a nesting arrangement if the sequence of operations does not result in its ever being put inside another box.

Here is the problem faced by the people at WebExodus: Since only the visible boxes are taking up any space, how should a nesting arrangement be chosen so as to minimize the number of visible boxes?

Give a polynomial-time algorithm to solve this problem.

Example. Suppose there are three boxes with dimensions (.6, 6, 6), (.75, 75, 75), and (9, 7, 7). The first box can be put into either of the second or third boxes; but in any nesting arrangement, both the second and third boxes will be visible. So the minimum possible number of visible boxes is two, and one solution that achieves this is to nest the first box inside the second.