  We provide a natural answer to Lewis Carroll's pillow problem of what is the
probability that a triangle is obtuse, Prob(Obtuse). This arises by
straightforward combination of a) Kendall's Theorem - that the space of all
triangles is a sphere - and b) the natural map sending triangles in space to
points in this shape sphere. The answer is 3/4. Our method moreover readily
generalizes to a wider class of problems, since a) and b) both have many
applications and admit large generalizations: Shape Theory. An elementary and
thus widely accessible prototype for Shape Theory is thereby desirable, and
extending Kendall's already-notable prototype a) by demonstrating that b)
readily solves Lewis Carroll's well-known pillow problem indeed provides a
memorable and considerably stronger prototype. This is a prototype of, namely,
mapping flat geometry problems directly realized in a space to shape space,
where differential-geometric tools are readily available to solve the problem
and then finally re-interpret it in the original `shape-in-space' terms. We
illustrate this program's versatility by posing and solving a number of
variants of the pillow problem. We first find Prob(Isosceles is Obtuse). We
subsequently define tall and flat triangles, as bounded by regular triangles
whose base-to-median ratio is that of the equilateral triangle. These
definitions have Jacobian and Hopfian motivation as well as entering Kendall's
own considerations of `splinters': almost-collinear triangles. We find that
Prob(Tall) = 1/2 = Prob(Flat) is immediately apparent from regularity's
symmetric realization in shape space. However, Prob(Obtuse and Flat),
Prob(Obtuse is Flat) and all other nontrivial expressions concerning having any
two of the properties mentioned above, or having one of these conditioned on
another, constitute nontrivial variants of the pillow problem, and we solve
them all.
