Grok 4.20 beta reveals first look at advanced math problem solving capability

xAI has recently opened internal beta testing for Grok 4.20, an upcoming Grok version, for some users, and one of them has revealed new math problem solving capabilties.

Grok 4.20 is developed on the building blocks of earlier models, like Grok 4 and 4.1. This internal beta represents a step forward in AI capabilities, especially in handling complex tasks that require deep reasoning and quick response.

Recently, a University of California, Irvine math professor, Paata Ivanisvili, received an internal beta access for Grok 4.20 and highlighted how this AI tackled a challenging problem in just minutes.

The problem centers on something called a Bellman function, which is a mathematical tool used to find the best possible outcomes under certain rules or constraints.

The math the tester showcased is very high level, and since I am not a mathematician, I asked Grok to break it down in the simplest term and below you can check the explanation of what the equation that was solved.

Imagine you’re trying to measure the maximum “wiggliness” or variation in a system that follows random paths, like a particle bouncing around unpredictably. Professor Ivanisvili and his student N. Alpay had been working on this in their research paper. They focused on a function U(p, q), where “p” is a starting point between 0 and 1, and “q” is an extra value that affects the calculation.

Previously, the team had shown a lower limit for U(p, 0)—meaning when q is zero—using a concept from Gaussian isoperimetry. This gave an estimate that behaved like p times the square root of the logarithm of 1 over p, as p gets very small. Think of it as a way to guess how much “bounce” or change you can expect in the system, but it wasn’t the sharpest possible bound.

What Grok 4.20 did was remarkable: in about five minutes, it derived an exact formula for U(p, q). The AI proposed that U(p, q) equals the expected value (or average) of the square root of (q squared plus τ), where τ is the “exit time” for a Brownian motion starting at p.

Brownian motion is like the random jiggle of a tiny particle in water—it moves unpredictably until it hits the boundaries of 0 or 1. The exit time τ is simply how long it takes on average.

This new formula improves the understanding of U(p, 0), showing it behaves like p times the logarithm of 1 over p when p is tiny. That’s a better estimate because it removes the square root from the log factor, making the bound tighter and more accurate. In everyday language, it’s like upgrading from a rough sketch to a precise measurement of how wild the system’s variations can get.

The advancement ties into harmonic analysis, a branch of math that studies functions and their breakdowns. Specifically, it provides a sharp lower bound for the L1 norm of the dyadic square function on indicator sets. Indicator sets are basic “on/off” functions for parts of an interval, like marking yes or no for whether a point is in a certain area. The dyadic square function measures squared differences at power-of-two scales, and the L1 norm adds up the absolute values to gauge overall size.

This result builds on earlier bounds, like the Burkholder-Davis-Gundy inequality, which gave a simpler estimate of p times (1 minus p). The professor’s paper improved it to include a square root log term, but Grok’s version uses a full log, proving it’s the best possible.

Broader implications include better insights into quadratic variations—analogs of derivatives in random processes—for Boolean functions, which are yes/no decision rules. Unlike the fractal Takagi function (linked to the unsolved Riemann hypothesis, involving prime numbers), this new profile is smooth and relates to isoperimetric shapes, like efficient boundaries in geometry.

That’s what Grok said about the math improvement in Grok 4.20’s internal beta.

To conclude, it demonstrates AI’s growing role in probabilistic inequalities. By solving such problems efficiently, Grok 4.20 could provide big help in research, exploring ideas that were once too time-consuming.

However, this isn’t the final release, and xAI has yet to make the release date official for this new AI model. Also, if you want to join the internal beta, reach out to Tony, one of xAI’s co-founders, on X.