# The Crossing Yard

A brick factory of the future, floating above a neon city at night\. Plasma kilns send cargo carts along maglev rails to every storage depot; where two rails cross, the carts take turns through the junction and jolt across in a burst of sparks\. Drag the kilns and depots anywhere on the deck: the rails stretch, the carts keep running and a hologram counts the crossings against the proven minimum\.

By Numa

Topic: Maths

[Open the interactive Jig](https://ask.jig.com/j/1L6XsBL1RQoi)

## Guide

### 1. The Crossing Yard

A brick factory of the future\. Every kiln sends carts to every storage depot, and where two rails cross, the carts must take turns and jolt through, throwing sparks\. The fewer crossings, the smoother the yard\. How few can there be? The question is from 1944, and the answer was proven only this year\. Drag the city or the deck to look around\.

Scroll or pinch to move closer\.

### 2. Turan's carts

In 1944 the mathematician Pal Turan was forced to work in a brick factory near Budapest\. Rails ran from every kiln to every storage yard, and wherever two rails crossed, the carts jumped the track and the bricks fell out\. Watching the carts, he asked how the rails could be laid with the fewest crossings\. Here a cart waits for the junction to clear, then jolts across\.

With m kilns and n yards there are m times n rails, and crossings can only happen between rails that go to different kilns and different yards\.

### 3. Lay the rails yourself

Drag the kilns and the depots anywhere on the deck\. The rails stretch with them, the carts keep running, and the hologram counts the crossings\. The faint number beneath is the lowest any layout can reach\. Try to get there, or shuffle the yard and try again\.

Two rails that share a kiln or a depot meet at that building, and that does not count as a crossing\.

### 4. Zarankiewicz's cross

In 1954 Kazimierz Zarankiewicz proposed a layout\. Put the kilns on one line, half on each side of the centre, and the depots on a line across it, half on each side\. Join every kiln to every depot by a straight rail\. Lay the cross and watch the count fall to the minimum\.

For 4 kilns and 4 depots the cross has 4 crossings: one in each quarter of the yard\.

### 5. Bigger factories

The cross works at every size, and its crossings follow one rule\. Count the kilns on each side of the centre and multiply those two numbers; do the same for the depots; then multiply the two results\. Three by three gives 1, four by four gives 4, six by six gives 36, and ten by ten already needs 400\. Slide the factory up and watch the count race ahead of the buildings\.

The number grows roughly with the fourth power of the factory size: double the factory and the crossings multiply by about sixteen\.

### 6. Seventy years of doubt

Zarankiewicz also claimed that no layout could do better, but around 1965 a gap was found in his proof\. Daniel Kleitman proved the rule in 1970 for factories with up to six kilns, and there it stayed\. In September 2026 a proof for every size was produced by an AI model at OpenAI and checked line by line by Lean, a computer program that accepts only flawless logic\.

The same release holds hundreds of other claimed results\. Mathematicians are still reviewing them; this one is among the few with a complete machine\-checked proof\.

### 7. Everyone to everyone

Now let every building connect to every other, as the wires in a chip or the links in a network do\. The best layout here puts the buildings along one avenue: some rails arch over it as bridges, the rest curve along the deck\. With seven buildings the minimum is 9 crossings, a rule guessed by Harary and Hill in 1963 and proven in the same 2026 paper series\.

The bridges and the ground rails are two pages of a book, and the avenue is its spine\. Rails on different pages never cross\.

### 8. Crossings everywhere

Every circuit board faces the same problem\. Two copper tracks cannot cross on one layer, so each crossing needs a via: a tiny plated hole that carries the signal down to another layer and back\. Here the yard has become a circuit board: the kilns are chips, the depots are connectors, the rails are copper, and every crossing is a via\. The proven minimum is now the fewest vias any engineer can get away with\. Drag the chips and connectors to get there, or switch back to the yard to compare\.

A modern chip stacks more than ten layers of wiring to keep its billions of connections from crossing\.