At the end of the proof of lemma 10, lemma 8 is cited. In order to use it and finish the contradiction, we need to show $n$ is not a multiple of $3.$ However, I don't see any contradiction in having $n \equiv \pm 2 \mod 8$ and $n \equiv 0 \mod 3.$ I also asked on MSE but not even a halfpenny of thoughts were given.

Proof: https://sci-hub.se/https://doi.org/10.1080/00029890.1990.11995558

At the end of the proof of lemma 10, lemma 8 is cited. In order to use it and finish the contradiction, we need to show $n$ is not a multiple of $3.$ However, I don't see any contradiction in having $n \equiv \pm 2 \mod 8$ and $n \equiv 0 \mod 3.$ I also asked on MSE but not even a halfpenny of thoughts were given.

Proof: Anglin - The square pyramid puzzle.