Magic Squaresproof atlas
alpha · 0.6.0
← Back to the atlas

General proof chapter

Yellow lemma: equality of two sums of squares

A yellow four-cell support comes from composition of the Gaussian norm and gives an equality between two pairwise cell sums.

01u=ac+bd,v=ad−bc,w=ac−bd,z=ad+bcu=ac+bd,\quad v=ad-bc,\quad w=ac-bd,\quad z=ad+bc
02u2+v2=w2+z2=(a2+b2)(c2+d2)u^2+v^2=w^2+z^2=(a^2+b^2)(c^2+d^2)
03(U,V,W,Z)=(u2,v2,w2,z2)⟹U+V=W+Z(U,V,W,Z)=(u^2,v^2,w^2,z^2)\Longrightarrow U+V=W+Z
Each family proof specifies the exact permutation of U, V, W, and Z; the color marks precisely the participating cells.