Exercise 3: Why Contradiction-Testing Beats Blind Brute Force — Possible Solution
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Blind brute force means trying every single one of the ~10^114
possible daily keys (rotor choice/order, ring settings, starting
positions, and plugboard pairs all together) one at a time, fully
decrypting the ciphertext with each, and checking whether the result
looks like real language. That number is so large it remains
completely infeasible even today.

The Bombe's contradiction-based approach is faster for two combined
reasons:

1. IT SEPARATES THE PROBLEM INTO SMALLER PIECES. Rather than testing
   the full combination of rotor order + ring setting + starting
   position + plugboard all at once, the Bombe (per Chapter 3's
   keyspace breakdown) focused mainly on rotor order and starting
   position first -- a MUCH smaller space (60 rotor orders x 17,576
   starting positions, roughly a million combinations) rather than the
   full 10^114. Plugboard and ring settings could often be resolved
   afterward, once a consistent rotor order/position had already been
   found.

2. IT REJECTS BAD CANDIDATES INSTANTLY, WITHOUT FULLY TESTING THEM.
   For each candidate rotor order/starting position, the Bombe didn't
   need to fully decrypt anything to check if it was wrong -- it only
   needed to find ONE contradiction (most directly, a chain in the
   menu implying some letter would have to map to itself, which
   Chapter 3's reflector rule makes impossible). The instant a
   contradiction appeared, that entire candidate -- along with every
   plugboard/ring-setting variation built on top of it -- was thrown
   out at once, without ever needing to individually test any of
   those variations.

Put together: instead of blindly checking 10^114 fully-specified keys
one at a time, the Bombe checked roughly a million rotor-order/
starting-position candidates, discarding the vast majority almost
instantly via a single contradiction, and only pursued full plugboard
and ring-setting determination for the small number of candidates that
survived. That's the difference between an intractable exhaustive
search and a tractable, targeted one.

WHY THIS WORKS AS AN ANSWER
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This connects the "How the Bombe Actually Worked" section directly
back to Chapter 3's own keyspace breakdown (60 rotor-order
combinations from that chapter's Exercise 3, multiplied by starting
positions) -- showing concretely how "rule out entire families of
settings the moment a single contradiction appears" turns an
astronomically large number into a genuinely searchable one, which is
the chapter's central technical point about why the 10^114 figure,
while true, was never actually the real barrier it appeared to be.