# What is a Trinity Number™?

This is inspired by the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles, and the Number™ series.

If a number conforms to a special rule, I call it a Trinity Number™. Use the examples below to find the rule.

$$% set Title text. (spaces around the text ARE important; do not remove.) % increase Pad value only if your entries are longer than the title bar. % \def\Pad{\P{1}} \def\Title{\textbf{ Trinity }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Number}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ }\displaystyle \smash{\lower{29px}\bbox[lightblue]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ 666 }&\text{ 777 }\\ \hline \text{ 1988 }&\text{ 2020 }\\ \hline \text{ 2048 }&\text{ 64 }\\ \hline \text{ 404 }&\text{ 200 }\\ \hline \text{ 4096 }&\text{ 1024 }\\ \hline \text{ 1048576 }&\text{ 2097152 }\\ \hline \text{ 89 }&\text{ 97 }\\ \hline \text{ 20200520 }&\text{ 20201225 }\\ \hline \end{array}$$

The puzzle satisfies the series' inbuilt assumption that each number can be tested for whether it is a Trinity Number™ without relying on the other numbers. These are not the only examples of Trinity Numbers™; many more exist.

What is the special rule these numbers conform to?

CSV version:

Trinity Number™, Not Trinity Number™
666, 777
1988, 2020
2048, 64
404, 200
4096, 1024
1048576, 2097152
89, 97
20200520, 20201225


Hint:

If you were to write down all of the numbers on paper, one of the Trinity numbers™ will possibly be not found in the set of Trinity Numbers™ (i.e. depending on how it is written, it could become a non-Trinity Number™).

• I like the trademark – Ankit May 20 at 18:04

It appears that each Trinity Number™ encloses at least three separate planar regions inside its digits. Depending on font/writing conventions, a '4' can contribute zero or one to this total, and a '0' one or two. If assuming '4'->+1 and '0'->+1, this explains the given lists. '404' would be disqualified if '4's are not enclosed, and slashed '0's would make each NTN containing them ->TN.

• Could you illustrate this with a diagram? I'm not sure I follow... Thanks :) – Stiv May 21 at 15:44
• @Stiv, he means count the 'holes' of the digits :) – Prim3numbah May 21 at 17:41
• @Prim3numbah That's infinitely clearer, thanks. 'Planar regions' is not a phrase I've ever come across... – Stiv May 21 at 18:37