Human outsmarts Google DeepMind AI, solving centuries-old ‘kissing problem’
A human has surpassed one of Google’s super-powerful artificial intelligence systems. The feat, however, is not in the realm of romance. Instead, this victory lies in the intellectual realm of advanced mathematics. Although largely conceptual in nature, the ramifications could soon help spur advances in telecommunications and satellite networks.
What’s the problem with kissing?
The “kissing problem” is not the term used for a junior high dance conundrum; this is actually a reference to a famous mathematical riddle. The setup is simple: how many circles or spheres can be arranged so that each individual simultaneously touches or “kisses” a single rounded shape in the center?
The answer is relatively simple when it comes to three dimensions or less. The answer for one dimension is 3, two dimensions gives you 6 and a three dimensional situation can support 12 kissing spheres. In 2003, mathematician Oleg Musin proved that the number of kisses for four dimensions is 24. While this concept is difficult to understand, it only gets stranger as the dimensions increase.
Stuck in the 16th dimension
Kissing problem experts have been stuck for about two decades. Despite their efforts, no one had established a new lower limit of objects for dimensions smaller than dimension 16.
However, in May 2025, Google’s DeepMind lab announced that its AlphaEvolve artificial intelligence system had successfully increased the lower limit of kissing objects in the 11th dimension to 593. As in many other fields, the news seemed to indicate that the future of investigating kissing problems belonged to AI.
But thanks to the work of doctoral student Mikhail Ganzhinov of Aalto University in Finland, humans still do well when it comes to kissing. Ganzhinov’s recent thesis work showed three new lower bounds: at least 510 in the 10th dimension, 592 in the 11th dimension, and at least 1,932 in the 14th dimension. Crucially, Ganzhinov outperformed AlphaEvolve in two out of three cases.
“Far from being all-powerful”
Understanding how Ganzhinov felt that these solutions were beyond most people’s mathematical prowess, but he still attempted to distill his approach for his university’s October 23 announcement.
“I reduced the size of the problem by only looking for arrangements with a high degree of symmetry,” he said for what it’s worth. Either way, the implications are much easier to digest.
“Artificial intelligence can do amazing things, but it is far from omnipotent,” added Patric Östergård, Ganzhinov’s thesis advisor.
His former student probably isn’t finished either. According to Ganzhinov, the current lower limit of the 11th dimension is “still quite weak” and can probably extend “well beyond 600”.
“The game could still turn in Mikhail’s favor in dimension 11 as well,” Östergård said.
Ganzhinov is not the only one questioning the capabilities of AI. MIT mathematicians are preparing a paper that pushes the limits of number in dimensions 17 to 21. Their work marks the first progress in these dimensions in more than half a century.
“This enigma has challenged mathematicians since the famous conversation between Newton and [17th century mathematician David] Gregory,” Ganzhinov explained. “But solving them also has a practical purpose: understanding the connections with spherical codes has real-world implications in the field of communications.”
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