| A group of researchers are designing an experiment to
test the IQ of a monkey. They will hang a banana at the roof of a building, and
at the mean time, provide the monkey with some blocks. If the monkey is clever
enough, it shall be able to reach the banana by placing one block on the top
another to build a tower and climb up to get its favorite food.
The researchers have n types of blocks, and an
unlimited supply of blocks of each type. Each type-i block was a
rectangular solid with linear dimensions (xi, yi, zi).
A block could be reoriented so that any two of its three dimensions determined
the dimensions of the base and the other dimension was the height.
They want to make sure that the tallest tower possible
by stacking blocks can reach the roof. The problem is that, in building a tower,
one block could only be placed on top of another block as long as the two base
dimensions of the upper block were both strictly smaller than the corresponding
base dimensions of the lower block because there has to be some space for the
monkey to step on. This meant, for example, that blocks oriented to have
equal-sized bases couldn't be stacked.
Your job is to write a program that determines the
height of the tallest tower the monkey can build with a given set of blocks.