Doing all the powers-of-two additions sets you up to think "they want
me to pick a power of two".  So when it asks you to pick a number
between 12 and 5, although you <strong>do</strong> first think of 8, you change it.  The
way integers are represented in your head (okay, big hypothesis) makes
it easiest to quickly change by going up or down 1.

<P>

The reason you change it to 7 instead of 9 is that 12 - 5 = 7, and this
is subtle enough that you don't notice it, and therefore don't resist
it.

<P>

There's a competing theory that people always pick the number 7, and
the other stuff is just mumbo-jumbo to disguise that fact.  Clearly,
we need a control experiment, where you ask people to pick a number
between 5 and 12 (or maybe two controls, also asking 1 to 10).  Then
we could compare the statistics.

<P>

Another theory just suggested is that you add 12 and 5, get seventeen,
and drop the "teen".

<P>

Check out the  <A
HREF="cool-trick-stats">current statistics</A>.