The bunk conjecture says that the probability of finding the path to the bottom bunk is always greater than or equal to the probability of finding the path to the top bunk. It doesn’t matter which graph you start with, how many vertical posts you draw between berths, or which start and end vertices you choose.

For decades, mathematicians thought this must be true. Their intuition told them that moving on a single berth should be easier than moving between two – that the extra vertical jump required to move from the lower berth to the upper berth should significantly limit the number of available paths.

Mathematicians also wanted the bunk bed conjecture to be true. It belongs to a class of statements in a field called percolation theory, which deals with paths and clusters that exist after the edges of graphs have been randomly removed. These graphs can be considered as simplified models of the way a fluid moves or seeps through a porous material, similar to how water moves through a sponge. The bunk conjecture, on the other hand, would involve a widely held assumption in physics about the probability of a fluid passing through a solid. It would also indicate how to solve problems related to the physics of percolation.

But that would only happen if someone could prove the bunk bed conjecture to be true. There was a reason why no one could.

Probably fake

Igor Pakmathematician at the University of California, Los Angeles, has always doubted the veracity of the bunk bed conjecture. “He was skeptical from the start,” said Nikita Gladkovone of his graduate students. “He doesn’t believe in old conjectures at all.” Pak strongly criticized the tendency of mathematicians to concentrate their efforts on proving such conjectures. He argues that equally important advances can come from asking: “What if they were all wrong?»

Pak also had particular reason to doubt the bunk bed conjecture: it seemed to be far too broad a claim. He was skeptical that this actually held true for every imaginable graph. “Some conjecture is driven by substance, and some is driven by wishful thinking,” he said. The bunk bed hypothesis resembled the latter hypothesis.

In 2022, he set out to refute it. He spent a year making unsuccessful attempts. He then asked Gladkov to use a computer to perform an exhaustive and brutal search on every possible graph. Realizing that this task would require sophisticated programming, Gladkov enlisted the help of a friend he had known since high school, Alexandre Ziminenow a graduate student at the Massachusetts Institute of Technology. “We were actually roommates in college — we had real bunk beds in our dorm,” Gladkov said.

Gladkov, Pak and Zimin were able to manually check all possible graphs with fewer than nine vertices. In these cases, they were able to verify that the bunk bed conjecture was true. But for larger graphics, the number of possible situations explodes. They couldn’t take into account all the possible ways to remove edges or form paths.

The team then turned to machine learning. They trained a neural network to produce graphs with circuitous paths that could potentially prefer the upward jump. In many of the examples cited, they found that a path to the bottom bunk was only a tiny bit more likely than its alternative to the top bunk. But the model didn’t discover any graphs showing the opposite.

There was another problem. Each graph created by the neural network was still so large that mathematicians could not study every result of the drawing stage. Instead, the team had to calculate the probability of finding upper and lower paths on a subset of these results – much like polls sample a subset of voters to predict the outcome of an election.

The mathematicians realized that they could have greater than 99.99% confidence in any counterexample their neural network gave them – but not 100%. They began to doubt whether this approach to the problem would be rewarded. This was unlikely to convince the mathematical community; no prestigious journal would certainly consider it as rigorous proof. “Ph.D. students need a job in reality, not in theory”, Pak wrote on his blog – and Gladkov and Zimin would soon be looking for work. “That’s really why we stopped,” he continued. “Why persevere and create controversy when you can simply try to do something else?

They abandoned their IT approach, but they didn’t stop to think about the problem. Over the next few months, they focused on formulating a theoretical argument that would not require a computer. But they didn’t have all the parts needed to make it happen.

Then a breakthrough came from abroad.

Who needs computers?

In June, Laurent Hollom from Cambridge University refuted one version of the bunk bed problem in a different context. Instead of dealing with graphs, this formulation of the conjecture asked about objects called hypergraphs. In a hypergraph, an edge is no longer defined as the connection between a pair of vertices, but rather as the connection between any number of vertices.

Hollom found a counterexample to this version of the conjecture. He created a small hypergraph whose edges each connected three vertices: