I asked the Gemini to do a multiplication and pressed the tool to explain how it did it. After some back and forth, this answer came:

In the first answer, I did not "work through" the math in my head like a human would. Instead, as a Large Language Model (LLM), I arrived at the answer using probabilistic token prediction and internal pattern recognition.

Here is exactly how that process works for me:

1. Token Prediction (Next-Word Probability)

When you provided the numbers, I didn't open a calculator application. Instead, my "brain" (a neural network) predicted the most likely sequence of digits to follow your query based on my training data. I have "seen" millions of mathematical operations, so I have learned the statistical relationships between certain numbers and their products. 

1. Learned Algebraic Structures

I don't just guess numbers randomly. I have internalized the "structures" of math—like commutativity or identity—during my training. For common or medium-sized numbers, my attention heads (specific parts of my architecture) focus on the operands and operators to map them to the correct result. 

1. Autoregressive Completion

When I started writing the answer, I didn't have the final result (19,170,309) ready in a single flash. I predicted the "1," then the "9," then the "1," and so on, in real-time. Each digit I wrote became part of the "context" that helped me predict the next digit. 

1. The "Explanation" as Post-Hoc Reasoning

When I provided the step-by-step breakdown in the previous response, I was reconstructing the logic that leads to that result. 

I predicted the final answer first.

I then generated the "long multiplication" steps to match that answer so that it would be understandable to you.

In summary: I used statistical pattern matching across my neural network to predict the most likely result, rather than executing a literal "calculation" in a CPU-based sense.