Same votes. Different winner.

Twenty-five voters. Ten of them back the city party. Nobody changes their mind, ever. Move only the lines between districts and that party wins three seats of five — or none.

What you're seeing

Twenty-five voters live on this grid. Ten of them — the filled circles — back the city party, and they live together in the two left-hand columns, the way city voters usually do. The other fifteen back the country party. That is the entire electorate, and it never changes. No voter here ever switches sides, stays home, or moves house.

The grid has to be cut into five districts. The rules are the ordinary ones: each district holds exactly five voters, and each district has to be one connected piece. Whoever has three or more voters inside a district wins that district's seat.

Start with five columns, which is what the geography suggests. The city block fills two whole columns, so the city party wins two seats out of five. Ten votes in twenty-five, two seats in five. That looks fair.

Now press five rows. Nothing about the voters has moved. But every row now cuts across the city block and picks up two city voters and three country ones, so the country party wins all five districts by the same narrow margin. Forty per cent of the electorate has just been reduced to nothing. That is cracking — spread a group thin enough and it is a minority everywhere.

Then press packed & cracked. Here the map hands one district entirely to the country party, which wastes five of their votes on a seat they were going to win anyway, and then takes three of the remaining four by the smallest margin that works. The city party now holds a majority of the chamber with forty per cent of the vote. That is packing — concentrate your opponents where their extra votes are thrown away. Every district is still exactly five voters, still one connected piece. Nothing has been broken.

Press squares yourself and you will feel the real lesson quickly: the outcome is easy to move and the votes are impossible to move. Whoever draws the lines is choosing the answer before anyone votes.

The rule, exactly. A map is legal when the grid is cut into five contiguous: one connected piece, reachable square to square without lifting your finger districts of exactly five voters, and a district goes to whoever holds three of its five. Fixing the electorate at ten city voters to fifteen country ones, the question is what the legal maps can do: seats(city) ∈ {0, 1, 2, 3} over all 4,006 legal maps This is settled by exhaustive enumeration, not by example (improve/verify/97-gerrymander.js): every one of the 4,006 ways to cut a 5×5 grid into five connected pentominoes is generated, checked against the page's own legality test, and scored. The count 4,006 is a known combinatorial quantity, so reproducing it independently checks the enumerator itself rather than merely reporting a number. Exactly one legal map leaves the city party with nothing — the five rows — and 276 of them, close to one map in fourteen, hand it a majority of the seats on 40% of the vote. Notice also what the rules do constrain: no legal map gives the city party four or five seats. The rigging has a ceiling; it just sits far above what its votes deserve. Four negative controls. A map with a six-voter district must be rejected; a map whose district falls into two separate pieces must be rejected; a legal but ugly S-shaped district must still be accepted, so the test rejects disconnection rather than mere untidiness; and the comfortable intuition that 40% of the votes buys 2 of 5 seats is refuted by counting — only 3,033 of the 4,006 legal maps, 76%, are proportional at all.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 97.