Maximum Regions From n Straight Cuts

"You have a pancake. What's the maximum number of pieces you can cut it into with 5 straight cuts?" is a classic warm-up puzzle. The trap is guessing $2^5=32$ by analogy to folding paper — that's wrong. The right approach asks a sharper question: when you add one more straight cut to an existing arrangement, what's the most new pieces it can possibly create?

What one new cut can add

Each new straight line can cross every previous line at most once (two straight lines intersect in at most one point). If the $n$-th line crosses all $n-1$ previous lines, it gets divided into $n$ segments by those $n-1$ crossing points (a line split at $n-1$ points has $n$ pieces). Every one of those segments slices exactly one existing region into two, adding exactly one new region per segment. So the $n$-th cut adds at most $n$ new regions — no more, no less, when positioned to cross every prior line at a distinct point.

Building the recurrence and formula

Let $R(n)$ be the maximum number of regions with $n$ lines. With $0$ lines, $R(0)=1$ (the whole plane, uncut). Each new line adds at most $n$ regions when it's the $n$-th line drawn: $R(n) = R(n-1) + n$. Unrolling this recurrence: $R(n) = 1 + 1 + 2 + 3 + \cdots + n = 1 + \dfrac{n(n+1)}{2}$. In plain English: start with 1 region (uncut), then add $1, 2, 3, \ldots, n$ regions as each successive line crosses one more prior line than the last.

$$R(n) = 1 + n + \binom{n}{2} .$$

This is the same formula written using the binomial coefficient $\binom{n}{2} = \dfrac{n(n-1)}{2}$: you always get the base 1 region, plus $n$ regions from the lines' own presence, plus one extra region for every pair of lines that cross.

Worked example: n = 4 cuts

$R(4) = 1 + 4 + \binom{4}{2} = 1 + 4 + 6 = 11$. Check by building up: $R(0)=1$, $R(1)=1+1=2$ (one line splits the plane in two), $R(2)=2+2=4$ (a second line, crossing the first, adds 2), $R(3)=4+3=7$ (a third line, crossing both others, adds 3), $R(4)=7+4=11$ (a fourth line, crossing all three others, adds 4). So 5 straight cuts through a pancake give at most $R(5) = 1+5+\binom{5}{2} = 1+5+10=16$ pieces — not 32.

What this means in practice

The general move here — ask "what's the most a single new addition can contribute given everything already placed," then sum that contribution across all additions — is a reusable trick for a whole family of "maximum count" combinatorics interview questions, not just line-cutting: maximum intersection points among $n$ circles, maximum regions from $n$ planes cutting 3D space, maximum crossings among $n$ chords in a circle. Each has its own per-step contribution to sum, but the derivation pattern is identical.