ing the dice in which they spun frantically but did not tumble; this
method was so profitable that the gambler was banned from casinos, and
it is now a rule in craps games that both dice must hit the wall, which
contains numerous bumps that presumably randomize the outcome of
the throw.
But does it? If we roll a fair die, will the number 1 (and all the other num-
bers) come up one-sixth of the time? After all, it seems reasonable that
once the die is thrown, only one possible outcome is in accordance with
the laws of physics and the initial conditions of the problem—how the
gambler held the die, whether his hands were dry or damp, and so on. And
so, if the universe knows what’s going to happen, why shouldn’t we?
Let’s grant this argument, temporarily, that given sufficient informa-
tion and sufficient computational capability, we can determine the out-
come of a thrown die. Does that leave anything that can be said to be
perfectly random—in the universe, or in mathematics?
One possibility that occurs to us is the randomness that appears in
quantum mechanics, but randomness in quantum mechanics, though
it has been confirmed to an impressive number of decimal places, is still
an infinite number of decimal places short of perfectly random. Maybe
mathematics can deliver something ultimately and perfectly random,
something that we cannot, under any circumstances, predict.
The Search for the Ideal Random Penny
Let’s try to construct a sequence of f lips for a penny that conforms to our
intuitive idea of how a random penny should behave. We would certainly
expect that an ideal random penny should occasionally come up heads
three times in a row—and also occasionally (but much more rarely) come
up heads 3 million times in a row. This leads us to the realization that
there must be an infinite sequence of f lips in order to determine whether
or not the penny is truly random. Notwithstanding that there are certain
technical problems in what we mean by “half” when dealing with an infi-
nite set (those familiar with probability can think of it as having a proba-
bility of 0.5), we can try to construct such a sequence. If we use H to
denote heads and T to denote tails, the sequence H,T,H,T,H,T,H,T...
obviously satisfies the restriction that half the f lips are heads and half are
tails. Equally obviously, it is not a random sequence; we know that if we
keep f lipping, sooner or later heads or tails will occur twice in a row, and
they never do in this sequence. Not only that, but this sequence is perfectly
predictable, which is about as far from perfectly random as one can get.
OK, let’s modify this sequence, so that each of the possible two-f lipThe Disor ga nized Universe 171