red one of these horizontal tubes open at
both ends, I was greatly struck by what happened. The series consisted
of ten cocoons. It was divided into two equal batches. The five on the
left went out on the left, the five on the right went out on the right,
reversing, when necessary, their original direction in the cell. It was
very remarkable from the point of view of symmetry; moreover, it was
a very unlikely arrangement among the total number of possible
arrangements, as mathematics will show us.

Let us take n to represent the number of Osmiae. Each of them, once
gravity ceases to interfere and leaves the insect indifferent to either
end of the tube, is capable of two positions, according as she chooses
the exit on the right or on the left. With each of the two positions
of this first Osmia can be combined each of the two positions of the
second, giving us, in all, 2 x 2 = (2 squared) arrangements. Each of
these (2 squared) arrangements can be combined, in its turn, with each
of the two positions of the third Osmia. We thus obtain 2 x 2 x 2 = (2
cubed) arrangements with three Osmiae; and so on, each additional
insect multiplying the previous result by the factor 2. With n Osmiae,
therefore, the total number of arrangements is (2 to the power n.)

But note that these arrangements are symmetrical, two by two: a given
arrangement towards the right corresponds with a similar arrangement
towards the left; and this symmetry implies equality, for, in the
problem in hand, it is a matter of indifference whether a fixed
arrangement correspond with the right or left of the tube. The previous
number, therefore, must be divided by 2. Thus, n Osmiae, according as
each of them turns her head to the right or left in my horizontal tube,
are able to adopt (2 to the power n - 1) arrangements. If n = 10, as in
my first experiment, the number of arrangements becomes (2 to the power
9) = 512.

Consequently, out of 512 ways which my ten insects can adopt for their
outgoing position, there resulted one of those in which the symmetry
was most striking. And observe that this was not an effect obtained by
repeated attempts, by haphazard experiments. Each Osmia in the left half
had bored to the left, without touching the partition on the right; each
Osmia in the right half had bored to the right, without touching
the partition on the left. The shape of the orifices and the surface
condition of the partition showed this, if proof were necessary