TLDR

Bryzcki looked at how much weight strong lifters could lift for 1 rep and for several reps. Then he made a formula to predict the 1 rep max from the other rep numbers.

Formula

https://wikimedia.org/api/rest_v1/media/math/render/svg/e28d2dc4b3dcb9060f19bfb966cf78fcc6442635

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Background

The Brzycki 1RM formula was derived empirically by Dr. Matt Brzycki through analyzing large amounts of actual 1RM and sub-maximal repetition data from trained lifters. The basic process he followed was:

1. Collect data on hundreds of successful 1RM attempts across the main lifts (squat, bench, deadlift) from experienced lifters.
2. Also collect data on how many reps those same lifters could do with certain sub-maximal weights (e.g. 80-95% of their 1RM).
3. Use regression analysis to find the equation that best predicted the 1RM value from the sub-maximal reps and weight.
4. Refine the equation coefficients through multiple iterations on the dataset to minimize the error between predicted and actual 1RMs.

This empirical analysis found that the relationship between sub-maximal reps/weight and 1RM could be well modeled by the specific form: 1RM = Weight / (a - b*Reps)

Where a and b are coefficients derived from the regression.

The final values of a=1.0278 and b=0.0278 in Brzycki's formula provided the closest fit to the actual data and minimized over/under-prediction errors.

So while not derived from first principles, the formula coefficients were determined through iterative analysis of real-world strength data across a large sample of trained lifters. This empirical approach is why it is fairly accurate for estimating 1RM levels.