The following rule for asserting the existence of a predecessor for any nonzero number is merely admissible:
Every nonzero number in the spectrum is an eigenvalue.
In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9.
The sum of the reciprocals of all the nonzero triangular numbers is:
Any nonzero number raised by the exponent 0 is 1; one interpretation of these powers is as empty products.
So a nonzero number is a residue mod 8, 16, etc., if and only if it is of the form 4(8n + 1).
The standard ordering of the natural numbers is a well-ordering and has the additional property that every nonzero natural number has a unique predecessor.
The digital root of a nonzero number is 9 if and only if the number is itself a multiple of 9.
Every nonzero real number has a multiplicative inverse (i.e. an inverse with respect to multiplication) given by (or ).
The goal is to fill all the cells with nonzero single-digit numbers (1 through 9) such that:
