Level 1½ · builds on chapter 5 (switches & AND/OR). No math beyond counting to three.
{kids ? <>Chapter 5 ended with gates in a row (AND) and gates side-by-side (OR). Engineers got tired of drawing the whole plumbing every time, so they gave each pattern a name and a symbol. Same circuits — new vocabulary. : <>Chapter 5 built AND (switches in series) and OR (switches in parallel) from plumbing. Name those patterns, treat each as a sealed box, and you can compose them without re-deriving the pipes every time. That move — name it, box it, stack it — is the whole secret of digital design.}
{kids ? <>One newcomer: XOR, the "different detector." It lights only when the two inputs disagree — one on, one off. (Two switches working a hallway light from both ends of the hall: flip either one and the light changes!) : <>One newcomer: XOR — exclusive-OR, the disagreement detector. High when inputs differ, low when they match. In switch form it's the two-way hallway light: two 3-way switches, either one toggles the outcome. Keep XOR in your pocket; it's about to do something remarkable.}
{kids ? <>Add two one-digit binary numbers by hand: 0+0=0, 0+1=1, 1+0=1… and 1+1 = 10 (that's "two" in binary — a 0, carry the 1). Look at the answer's LAST digit: 0,1,1,0. Now look at XOR's column: 0,1,1,0. They're the same! : <>Write the addition table for two bits: 0+0=00, 0+1=01, 1+0=01, 1+1=10. Now stare at the two output columns. The low bit goes 0,1,1,0 — that's XOR. The high bit goes 0,0,0,1 — that's AND. Addition was hiding inside the gates all along.}
{kids ? <>And the carry digit goes 0,0,0,1 — exactly AND's column! So one XOR plus one AND is an adding machine for two digits. Nobody "programmed" it. The arithmetic is built out of the wiring itself. : <>So: one XOR (sum) + one AND (carry) = a half adder. No program, no instructions — the arithmetic is the wiring. This is the moment "circuits" become "computers," and it's also the answer to chapter 5's cliffhanger about why switches matter so much.}
{kids ? <>One half adder handles one digit. For bigger numbers, line adders up like columns in pencil-and-paper addition — each column passes its carry to the next, like a little bucket brigade. : <>To add multi-bit numbers, chain adders column-by-column, each passing its carry leftward — exactly your grade-school algorithm, in copper. (Columns past the first need a full adder — a half adder upgraded to accept the incoming carry as a third input.)}
{kids ? <>This two-digit machine is built from just FOUR gate-patterns from chapter 5. Your computer does the same thing — with 64 columns instead of 2, billions of times every second. : <>This 2-bit ripple adder is a handful of gates. A CPU's arithmetic unit is the same idea at 64 bits with cleverer carry plumbing — but nothing conceptually new. From here, multiplication is repeated addition, memory is gates holding their own output (feedback again!), and the rest is scale.}