Halves and Fair Shares

A fair share is about size, not about the number of pieces. Cutting a sandwich into two pieces only makes halves when the pieces are the same size. Four matching pieces are fourths; each friend of four gets exactly one.

Halves of the same whole can look different and still be fair. A square cut corner to corner makes two triangles; cut down the middle it makes two rectangles. Different shapes — but each piece is half the square, because the two pieces of each cut match in size.

One half is written 1/2: one piece out of 2 equal pieces. The bottom number names how many equal pieces the whole was cut into; the top counts how many of them you take. Equal is the working word — unequal pieces have no fraction name at all.

Fairness also divides groups. Eight grapes shared between 2 friends is 4 each: same count for everyone, nothing left over. More sharers means smaller shares — 8 grapes among 4 friends is only 2 each. The whole never changes; only the way it is split does.

Cutting a whole into fair shares is a partition: pieces that do not overlap and together make the entire whole — nothing left over, nothing counted twice. Same size means equal measure (equal area, equal count), not equal shape: congruent pieces are sufficient but not necessary.

This is why 1/2 is a number rather than a picture. Any half of any whole stands in the same relation — one part in two of equal measure — and that shared relation is the value one-half, which earns its own point on the number line.

Fold a square of paper so the corners meet. Press the fold. Open it. Two pieces, and they match exactly. Each piece is a half.

Now look at a cut that does not match: one big piece, one small piece. Two pieces — but no halves. Halves must match.

A partition into equal-measure parts is the combinatorial core; the relation belongs-to-the-same-share is an equivalence relation whose classes are the parts. For divisible goods, fairness is a theorem, not a hope: with two claimants, divide-and-choose guarantees each at least half by their own measure — and the two measures need not even agree.

That last clause is the deep move: fairness is defined per claimant, using each one's own valuation — formally, a non-atomic measure on the cake. Proportionality, everyone receiving at least 1/n by their own lights, is always achievable, by the Dubins–Spanier moving-knife argument.

Envy-freeness — no claimant prefers another's share — is strictly harder than proportionality once n ≥ 3. Selfridge and Conway settled n = 3 with a bounded discrete protocol; Stromquist's moving-knife theorem gives an envy-free division into connected pieces for three; and Aziz–Mackenzie (2016) produced a bounded envy-free protocol for every n, closing a forty-year open problem.

The kindergarten fold survives at full strength: divide-and-choose is exactly the n = 2 protocol, and it is optimal in query complexity under the Robertson–Webb model. Fair division is one of the few fields where the five-year-old's algorithm still sits on the research frontier.