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A traditional inode uses a fixed number of direct pointers to file data blocks, followed by singly, doubly and sometimes triply indirect block pointers. A singly indirect block pointer reaches one address block whose entries point to file data blocks. A doubly indirect block pointer reaches a block of pointers to singly indirect address blocks; a triply indirect one adds another level. The file data blocks need not be physically contiguous.

This multilevel inode indexing keeps the inode compact. A small file uses direct pointers without allocating address blocks, while a large file allocates deeper address blocks only as needed. If each address block holds $q$ addresses, the successive levels cover $q,q^2,q^3$ file data blocks. The logical file offset selects a path arithmetically, so random access does not require walking through all preceding file data blocks.

With a resident inode, the deepest allowed path uses two address blocks and one file data block, so every byte through the end of the doubly indirect region meets the three-read bound. There are ten directly addressed file data blocks, 512 singly addressed ones and $512^2$ doubly addressed ones. Therefore the largest contiguous file satisfying this bound isThe next byte requires the triply indirect region, with three address-block reads and a fourth read for the file data block.

The resident-inode convention is essential. If the question's three reads must also include fetching a previously uncached inode, only two further reads remain; the maximum is thenNeither figure includes arbitrary pathname-resolution reads. These are worst-case counts in the traditional fully allocated multilevel inode indexing model; existing buffer cache entries can reduce them.