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: