1. Let us say that cosmonaut Ijon Tichy has encountered a
space-time-anomaly which traps him into a boundless, infinite space
that luckily contains an oxygen-nitrogen atmosphere. The anomaly also
instantly duplicates a countably infinite number of Tichys into the
space, and alters their metabolisms in such a way that they don't
require most nutrients and they don't age. Additionally, the ambient 
lighting in the space imbues the Tichys with instant teleportation 
abilities (within the anomaly only) as well as telepathy, so that any
logistical problems in arranging them can be ignored.

Now, while the Tichys don't require most nutrients, they do have to
consume food for one thing: energy. Luckily, the human body has lots
of available biochemical energy, and there are countably infinite bodies
in the space. Each Tichy also has a cheap Swiss army knife replica,
which astonishigly never dulls within this anomaly.

The Tichys therefore need a fresh Tichy corpse now and then to eat
(flesh does decay in this space due to internal bacteria). To be fair,
they arrange so that whenever they get hungry, each Tichy stabs
another Tichy simultaneously (mediated via the telepathy
group-mind). Those that were not stabbed are left alive to eat the
flesh of their prey.

a) Show that the Tichys can, if they're smart, maintain this cycle so
that an infinite number of Tichys live forever in this space.

b) Show that the Tichys also can, if they're not so smart, arrange
the killings to occur in such a way that not one of them lives for very
long. (Simply not killing other Tichys for food doesn't count.)

2. Just as he is stabbed, Tichy #4588731 comes up with the idea that
they don't have to resort to full-body cannibalism, but perhaps only
take some blood off of another Tichy every once in a while. Sadly, he
didn't have time to communicate the specifics of the idea through
telepathy before he died.

a) Show that all the Tichys could, in fact, survive indefinitely by
drinking other Tichys' blood. Assume conservation of energy. It may 
simplify a bit to note that every Tichy needs not get blood on every 
round, but should they miss one, they will be hungry and require 
nourishment on the following round.

b) Show that this method also can be badly managed to the detriment of
all the Tichys. Assume conservation of energy.