Map coloring and the vector cross product (Q1099183)

This paper proves that the Four Colour Theorem is equivalent to a combinatorial problem about the three-dimensional vector cross product algebra. Letting i, j, k denote the usual basis for this algebra \((i\times i=j\times j=k\times k=0\), \(i\times j=k\), \(j\times i=-k\), etc.) the problem is: Given two associations of a product \(X_ 1X_ 2...X_ n\) called L and R respectively, is there a nonzero solution to \(L=R\), (i.e. \(L\neq 0)\) choosing \(X_ 1,...,X_ n\) from \(\{\) i,j,k\(\}\) ? [For example \(L=(X_ 1X_ 2)(X_ 3X_ 4)\), \(R=X_ 1(X_ 2(X_ 3X_ 4))\) then \(L=R=-i\) for \(X_ 1=k\), \(X_ 2=i\), \(X_ 3=j\), \(X_ 4=i.]\) The solvability of this problem for all n and all choices of L and R is equivalent to the Four Colour Theorem.