Exercise 1: Why the Points Become Separable in (h1, h2) Space — Possible Solution
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WHAT nn1-2 PROVED ABOUT THE ORIGINAL SPACE
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Per nn1-2, in the original (x1, x2) space, the two "output 1" points
((0,1) and (1,0)) sit on one diagonal, and the two "output 0" points
((0,0) and (1,1)) sit on the other diagonal — an arrangement no single
straight line can correctly separate.

WHAT THE HIDDEN LAYER ACTUALLY PRODUCES, PER THIS CHAPTER'S OWN TABLE
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Per this chapter's own worked table, applying the hidden layer's OR and
NAND computations to each of the four original inputs produces new
(h1, h2) coordinates: (0,0)→(0,1), (0,1)→(1,1), (1,0)→(1,1), (1,1)→(1,0).
Per this chapter, "in the new (h1, h2) space the hidden layer produces,
the four points become (0,1), (1,1), (1,1), (1,0)."

WHY THIS NEW ARRANGEMENT IS GENUINELY DIFFERENT FROM THE ORIGINAL
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Critically, two of the four original inputs — (0,1) and (1,0), the two
that should both output "1" — map to the EXACT SAME new coordinate,
(1,1). This is a real structural change: in the original space, these
two points were far apart (opposite corners), but the hidden layer's
own OR/NAND computation happens to bring them together onto a single
point in the new space. Meanwhile, the two "output 0" points map to
(0,1) and (1,0) respectively — two different points, but both with a
combined h1+h2 value of 1, distinctly lower than the "output 1" point's
own combined value of 2.

WHY A SINGLE LINE NOW WORKS IN THIS NEW SPACE
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With the "output 1" class reduced to a single point (1,1) and the
"output 0" class occupying two points that both share h1+h2=1, a single
straight line — precisely the output neuron's own step(h1+h2-1.5)
boundary — can now cleanly separate them: everything with h1+h2 > 1.5
(only the point (1,1), where h1+h2=2) is classified 1, everything with
h1+h2 ≤ 1.5 (both remaining points, where h1+h2=1) is classified 0.
This is a real, valid single-line separation, something Exercise
material from nn1-2 proved was impossible in the original (x1, x2)
space specifically because of how those four original points were
arranged.

WHY THIS ISN'T A CONTRADICTION OF nn1-2's OWN PROOF
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nn1-2's proof applies specifically to the ORIGINAL input space — it
never claimed no transformed space could ever be linearly separable, only
that the original (x1,x2) coordinates themselves couldn't be split by
one line. The hidden layer doesn't violate that proof; it sidesteps it
entirely by computing an entirely different set of coordinates first,
in which the geometry is genuinely different.

WHY THIS WORKS AS AN ANSWER
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It traces exactly how the hidden layer's own OR/NAND computations
reposition all four points into a new (h1,h2) space, explains why two
of the original points collapse onto the same new coordinate, and
explains why the resulting arrangement is one a single line can
separate — without contradicting nn1-2's own proof about the original,
different space.