Tasking Open AI Strawberry To Solve a Complex Math Problem

OpenAI Strawberry just got released, and I was wondering how it performs on math problems. As the latest advancement in AI language models, Strawberry is touted to have significantly enhanced capabilities in mathematical reasoning and problem-solving. From tackling complex integrals to solving differential equations, this model aims to bridge the gap between human intuition and machine computation in the realm of mathematics.

In this blog post, I’ll put OpenAI Strawberry through its paces by challenging it with a intricate mathematical problem that test its understanding, accuracy, and reasoning skills. I’ll analyze the results to see if this would have been the same approach I would take to solve the problem.

Let’s take the following equation

and see how strawberry navigates the complexities involved in finding the value of t that satisfies this equation. By examining its step-by-step solutions, we can uncover the strengths and limitations of Strawberry’s mathematical prowess.

Here are the screenshots of the results.

Initial Thoughts

Wow, this is a significant improvement in the capabilities compared to 4o. The model arrived at the right answer and with decent steps. However, this is not how I would have solved the problem manually. When it comes to solving math problems, rigor is important. Here are the following things that this solution could have improved on:

Provide a Rigorous Solution: Derive explicit formulas and use appropriate mathematical tools to find an exact or highly accurate numerical solution, rather than relying on approximations.

Include Verification Steps: Verify the obtained solution by substituting it back into the original equation to ensure its correctness.

Explore Mathematical Properties: Utilize special functions like the gamma function or numerical integration techniques to handle the integral more effectively.

Clarify Assumptions and Domains: Discuss the domain of t and ensure that all steps are valid within that domain, addressing any potential issues with negative bases or undefined expressions.

Rigorous Analysis of the Approach

Here is what I would have done to solve the problem:

Missing Elements:

Exact Evaluation Instead of Approximation:

• Lack of Exact Solution: The Strawberry solution uses linear interpolation between I(2) and I(3) to estimate t. This method provides an approximate value but doesn’t yield an exact solution (which is ok btw. Approximations are generally acceptable, but quite dangerous. It somehow worked in this problem.)
• No Verification: The AI model didn’t verify whether t=7/3 actually satisfies the original equation I(t) = 2/3.
• Lack of Explicit Expression: Strawberry didn’t derive an explicit formula for I(t) that could help in solving for t exactly.

Improvements to the Approach

One significant area where Strawberry could enhance its solution is by employing a series expansion. Instead of relying solely on recurrence relations, Strawberry could expand e^x into its Taylor series:

Expand e^x as a power series:

Substitute back into the integral:

Evaluate the integral:

So:

Now, set I(t)=2/3:

Boom, now I have a explicit expression derived that is very useful (unlike linear interpolation). Note that this is a transcendental equation in t that generally cannot be solved algebraically, but we can find a numerical solution using methods like Newton-Raphson or other root-finding algorithms.

Use Numerical Methods for an Accurate Solution:

The strength of the the explicit expression derived is that we can compute I(t) numerically for various values of t and use a root-finding algorithm to find the precise value of t. We can choose how precise the solution needs to be to our domain requirement. None of this was possible with the Strawberry’s approach.

Using Newton-Raphson Method:

From above, define the function as follows:

Compute the derivative f′(t) numerically or analytically if possible, and apply the Newton-Raphson iteration:

The derivative of f(t) with respect to t is:

Start with an initial guess t_0 = 2.3333 (from the previous approximation) and iterate until convergence.

Verify the Solution:

• Substitute Back into the Original Integral: Once we have a numerical value for t, substitute it back into the integral to verify that it satisfies the equation I(t)=2/3​ within an acceptable margin of error.
• Check for Multiple Solutions: Ensure that there are no other values of t that satisfy the equation within the domain of interest.

Conclusion with Improved Solution

By deriving an explicit expression for I(t) and applying numerical methods, we can find a more accurate value for t that satisfies the equation.

After applying a numerical method (e.g., Newton-Raphson), we find that:

t≈2.3081

This value can be verified by substituting back into the original integral:

Improvements to Strawberry’s Approach

By incorporating the series expansion method, Strawberry can offer a more precise and mathematically rigorous solution. This approach allows for:

• Explicit Expression: An explicit representation of I(t) in terms of known functions, facilitating easier manipulation and analysis.
• Numerical Solvability: The ability to compute numerical values for t using well-established numerical methods.
• Enhanced Accuracy: Improved precision over approximations made through linear interpolation or recurrence relations alone.

Through this exercise, we’ve explored how Strawberry approaches complex mathematical problems and identified areas for improvement. I must say, Strawberry’s response is remarkably close to that of a human expert. Given the rapid pace at which AI models are advancing, I half-jokingly exaggerate that by the time I post this blog and you finish reading it, the results will have probably improved even more! Just saying 🤷‍♂️

Amazing work though.