In a distant empire, quarry workers are given a choice of multiple square boxes with various integer lengths and a height of 1 to pack cubes of granite, each of volume 1, into.

They are taking these blocks to the capital city of Sacul, where they will construct a pyramidal monument for the emperor Edouard II.

The pyramid is constructed with a square base\(^{*}\) such that the center of each cube above aligns vertically with each point that four cubes meet beneath. **Every block of granite must be used.**

Given the above conditions, what is the maximum length of square box the quarry workers can choose?

*\(^{*}\)Note: The pyramid should look something like this:*

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