To everyone who had fun, ask questions ChatGPT You will see that he is quite good at what he does, but he is not infallible either. Sometimes he invents something he doesn’t know, but he invents it very well. Very good. That’s why you need to be careful, because everything is very coordinated. The same thing happens with algorithms for solving mathematical problems. Something known as excellent language modelswhich are effective but may have hallucinations which lead to erroneous results. However, a group of scientists from Google DeepMind has developed a new tool that allows you to reject these falsified results. Thanks to this, they achieved for the first time that a language model makes a real scientific discovery. solve a math problem which until now no one has been able to answer.

This is a problem known as boundary set. It was proposed decades ago, and while it is true that with a small number of variables it could be solved in 70sUntil now, it has not been possible to solve this mathematical problem of greater complexity.

A DeepMind tool called FunSearch, gave an answer that no mathematician has ever received. And, if that wasn’t enough, he then followed up with another math problem, demonstrating that his first success was no mere coincidence.

## Keys to Solving a Math Problem

The limit set problem is to find largest set of points on a multidimensional gridwhere there are no three points on a straight line.

Like any good combinatorics problem, its difficulty increases as the complexity of the sets of numbers involved increases. FunSearch found a solution for many 512 points. But how did he achieve this?

At the heart of the solution, as we have already seen, are large language models, which are also used to develop chatbots such as ChatGPT. These are language models consisting of neural network with many parameterstrained on large volumes of unlabeled text using self-supervised or semi-supervised learning.​

The problem with these neural networks is that they are not looking for an answer, but rather patterns. That is, if they find a pattern that makes sense, they assume it is true without checking to see if it actually is true. This is the cause of typical ChatGPT failures. Solving a math problem this way is not easy. That’s why, FunSearch groups two programs to solve the problem. The first is Cody,a coding model based on large language models. As for the second, this is algorithm that checks and evaluates Kodi’s actions. Said very roughthere is a watchman who makes sure that the chatbot doesn’t screw up.

So these are the steps. Scientists write a line of code that tells them how to solve a math problem, but leaves the commands that actually provide the solution free. Cody will take care of suggest these lines; but before they are approved, the review algorithm sends them for review. Many of them will be false decisions that need to be abandoned. But out of all of them, some may provide the answer. Thus, it became possible to solve a mathematical problem that had remained unanswered for many decades.

## New demo

Now that there is an answer to this problem, the developers FunSearch They wanted to conduct a second experiment to ensure that their results were not the result of a happy coincidence. So they used this tool to solve another math problem: the problem container packaging.

Essentially, this is a problem that involves finding the most efficient way to pack large quantities of items into containers. Again this is a problem combinatorics, where the complication begins when the number set becomes very large. But for FunSearch this was not a difficult task.

These results are interesting, but they may also worry mathematicians who are afraid of losing their jobs. Fortunately, at least for now, they have nothing to fear. This tool is only useful for combinatorics problems with very specific characteristics. In addition, it is necessary human supervision. Like many professions, they may change in the future, but they don’t have to disappear. At the end of the day, machines are very useful and empowering, but they still need to be looked after.

Source: Hiper Textual