Exercise 2: Why "Modern Computers Are Fast Enough" Breaks Down — Possible Solution
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THE CLAIM BEING CHALLENGED
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The colleague argues that raw hardware speed makes complexity analysis
unnecessary. This chapter's own growth-rate table shows exactly why
this reasoning fails once input size grows, regardless of how fast the
underlying hardware is.

WHY FASTER HARDWARE DOESN'T RESCUE O(n^2)
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Per this chapter's own table, an O(n^2) algorithm's work grows as the
SQUARE of the input size: going from n=5 to n=20 (a 4x increase in
input) inflated the operation count from 25 to 400 - a 16x increase in
work. A faster computer multiplies how many operations it can do per
second by some constant factor, but it cannot change the SHAPE of that
growth curve - doubling the input will always roughly quadruple the
work for an O(n^2) algorithm, no matter how fast each individual
operation runs. Eventually, input size grows faster than any constant
hardware speedup can compensate for.

WHY THIS IS EVEN MORE SEVERE FOR O(2^n)
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This chapter's own table showed O(2^n) growing from 32 operations at
n=5 to 1,048,576 operations at n=20 - a jump of over 32,000x from only
a 4x increase in input size. No realistic amount of hardware speedup
can keep pace with growth this explosive; even a supercomputer
millions of times faster than a laptop would only postpone the
problem by a small, fixed number of additional input elements before
running into the same wall.

THE GENERAL PRINCIPLE
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Faster hardware effectively multiplies the largest input size you can
handle in a reasonable amount of time by some constant amount for a
POLYNOMIAL algorithm - but for an EXPONENTIAL algorithm, it barely
moves that limit at all, since each additional unit of input size
doubles (or worse) the required work. This is exactly why choosing a
better algorithm is fundamentally different from - and often far more
impactful than - simply buying faster hardware.

WHY THIS WORKS AS AN ANSWER
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The explanation is grounded directly in the specific numbers from this
chapter's own growth-rate table, distinguishing the genuinely
different consequences for O(n^2) (a bad but survivable growth rate)
versus O(2^n) (a growth rate no realistic hardware speedup can
outrun), rather than dismissing the colleague's claim in only general
terms.