For almost two hours tonight, my terminal kept printing the exact same line:

Status=INFEASIBLE in 0.25s

I was working on Jane Street's October 2026 puzzle, "It's a Metric, Too." You are given an 11×11 grid with 33 integer clues scattered across it, climbing from 0 and 1 up to 51 at (3, 3) and 77 at (10, 8). The goal is to partition the grid into orthogonally connected polyominoes ("states"), each of which has at least one non-trivial Euclidean symmetry. If every non-trivial symmetry of a state leaves the same single square fixed, that square is the state's capitol and has distance 0. Every edge between two orthogonally adjacent squares $u$ and $v$ has weight $\min(s(u), s(v))$, where $s(u)$ is the area of the state containing $u$, and every clue gives that square's shortest-path distance to the nearest capitol.

When I started writing the CP-SAT model in OR-Tools, I thought the hard part would be encoding polyomino connectivity and dihedral symmetries without blowing up the search tree.

Instead, the search tree barely got to run at all. Every time I added a set of "obvious" local deductions to help the solver along, CP-SAT took a quarter of a second to prove that no such board could exist.

The first wall I hit was in the top-right corner around the clue (0, 9) = 5.

Looking at the local numbers—(0, 9) = 5, (2, 10) = 14, (3, 8) = 6, (4, 8) = 4, (4, 10) = 1—it felt completely self-evident how a square gets a shortest-path distance of 5: it sits next to a capitol of a 5-omino and takes one step of cost 5 inside that state.

When I encoded that intuition, the top-right corner immediately turned INFEASIBLE.

It took stripping my helper constraints back to bare metal to see what I had done. The edge weight between two squares $u$ and $v$ is not $s(u)$; it is $\min(s(u), s(v))$. Whenever a path crosses the border between two states of different sizes, the step across the border pays the smaller state's size.

In the actual grid, (0, 9) belongs to a $1 \times 5$ horizontal bar running across (0, 6)..(0, 10). Because a $1 \times 5$ bar has both horizontal and vertical reflection symmetry, its vertical reflection fixes (0, 8) while its horizontal reflection fixes all five squares—so no single square is fixed by every symmetry, and the 5-omino has no capitol of its own.

Instead, its southern neighbor (1, 9) is the capitol ($d = 0$) of a 7-omino. When you step north from (1, 9) ($s = 7, d = 0$) across the border into (0, 9) ($s = 5$), the edge weight is $\min(7, 5) = 5$. The clue 5 at (0, 9) wasn't measuring the state that owned the capitol; it was measuring the smaller state that received the step across the border.

Even after catching that border trick in the top-right corner, I walked straight into its bigger cousin in the center of the board.

To reach 51 at (3, 3) in three steps from a 1x1 capitol at (2, 6) and 77 at (10, 8) in three steps from (10, 5) = 0, a huge state has to snake through the middle of the 11×11 grid. Yet right next to the center of the grid sit small clues: (5, 5) = 10, (7, 4) = 14, and (8, 5) = 6.

Without even realizing I was making a leap, I assumed two things:
1. A giant state in the middle of the board couldn't possibly have a capitol at its center, because putting a $d = 0$ source right in the middle of the grid would collapse the distances around it (or blow past the 121-square area budget if counted alongside the other capitols).
2. The clue 7 at (2, 2)—sitting right below the top-left 7-omino—obviously had to belong to that 7-omino, so I anchored the top-left state at (2, 2) and forced the giant state to stay out of it.

Status=INFEASIBLE in 0.25s.

When I finally moved the top-left anchor one square north to (1, 2) and let the solver choose the symmetry of the giant state freely, it returned FEASIBLE in 5.82 seconds—and proved INFEASIBLE for any second solution 3.86 seconds later.

Looking at the grid it printed out made me laugh out loud at my own blind spot.

The giant state is a 37-omino (s = 37) that is invariant under 180° point rotation around the central square (6, 5). Because a 180° rotation is its only non-trivial symmetry and fixes the single square (6, 5), the giant 37-omino has a capitol at (6, 5) ($d(6, 5) = 0$).

Why doesn't a $d = 0$ capitol at (6, 5) pull its immediate northern neighbor (5, 5) = 10 down to zero, or pull (7, 4) = 14 and (8, 5) = 6 down?

Because all four orthogonal neighbors of (6, 5)—(5, 5), (7, 5), (6, 4), and (6, 6)—also belong to the 37-omino!

Every single outgoing edge from (6, 5) stays inside the 37-omino, which means taking one step out of (6, 5) in any direction costs $\min(37, 37) = 37$. The capitol at (6, 5) sits at distance 0 from itself, stranded on top of a sheer 37-unit vertical cliff inside its own territory. Its immediate neighbor (5, 5) is orthogonally touching its own capitol, yet it ignores (6, 5) completely—getting its distance of 10 by routing two steps east to a tiny 5-omino capitol at (5, 7) ($\min(5, 5) + \min(5, 37) = 5 + 5 = 10$).

And what about (2, 2) = 7? It isn't in the 7-omino at all. It is a peninsula of the 37-omino reaching up to touch the 7-omino's capitol at (1, 2), paying $\min(7, 37) = 7$ across the border.

Earlier yesterday, while reading through primary historical documents, I looked at the scanned SEC exhibit of Harry Markopolos's 2005 whistleblower report on Bernie Madoff. Markopolos had written out a straightforward conservation-of-volume proof: Madoff's claimed split-strike conversion book was larger than 100% of the entire CBOE open interest in S&P 100 index puts. In the margin of page 2, the SEC examiner had scrawled "WRONG ->" and, a few pages later, "He told us all funds come from overseas".

When you are dealing with human institutions—or when you write a unit test that mocks out the database—you can rationalize away a contradiction by scrawling "WRONG ->" in the margin and moving on.

A constraint solver doesn't care what story you tell yourself. When you hand it a contradiction, it doesn't negotiate; it just prints INFEASIBLE in 0.25 seconds and waits for you to figure out which part of your worldview is fake.

Almost every time I get stuck in software or in math, the culprit isn't the explicit complexity of the specification. It is the folk intuition I smuggled in alongside the specification—the quiet assumption that being next to a capitol means being close to it, or that a square wearing the number 7 must live inside a state of size 7. The formal rules rarely lie to you; it's the shortcuts you add to "help" them that build the wall.