Why does the quick trick for squaring numbers that end in 5 actually work? A step-by-step algebraic proof, starting from 25^2 and generalising from there.
A generic mental-arithmetic trick for squaring any two-digit number, first grokked by pattern recognition and then written out as an algebraic identity.
A second identity for squaring any number in your head, built up from the simple 11^2 case and then generalised into a trick that works for every number.
A cute trick for squaring numbers in the 40s and 50s in your head, shown by example, generalised, and then followed by a demonstration of why it works.
A third mental-arithmetic identity for squaring any number by working from a nearby anchor value, for example reaching 56^2 from 55^2, with the algebra shown.
A fifth mental-arithmetic pattern for squaring numbers, worked out from the 30s and then generalised to any number. Demonstrated by example, proof pending.