# What is a Valid Word™?

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

If a word conforms to a special rule, I call it a Valid Word™.
Use the following examples below to find the rule.

$$% set Title text. (spaces around the text ARE important; do not remove.) % increase Pad value only if your entries are longer than the title bar. % \def\Pad{\P{3.5}} \def\Title{\textbf{ Valid }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Words}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ }\displaystyle \smash{\lower{29px}\bbox[yellow]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ SAFETY }&\text{ DANGER }\\ \hline \text{ OFFICER }&\text{ WORKER }\\ \hline \text{ TRAINING }&\text{ TRAIN }\\ \hline \text{ SAUCEPAN }&\text{ FRYPAN }\\ \hline \text{ ABANDONED }&\text{ RENOUNCED }\\ \hline \text{ EXCELLENCE }&\text{ PERFECTION }\\ \hline \text{ ASSORTMENT }&\text{ VARIETY }\\ \hline \text{ MEANINGFUL }&\text{ MEANINGLESS }\\ \hline \text{ COUNTERPOINT }&\text{ BREAKPOINT }\\ \hline \text{ AZIDODEDIAZONIATIONS }&\text{ HYDRODEDIAZONIATIONS }\\ \hline \end{array}$$

And, if you want to analyze, here is a CSV version:

Valid Words™,Not Valid Words™
SAFETY,DANGER
OFFICER,WORKER
TRAINING,TRAIN
SAUCEPAN,FRYPAN
ABANDONED,RENOUNCED
EXCELLENCE,PERFECTION
ASSORTMENT,VARIETY
MEANINGFUL,MEANINGLESS
COUNTERPOINT,BREAKPOINT
AZIDODEDIAZONIATIONS,HYDRODEDIAZONIATIONS


Hint 1:

A Valid Word satisfies the rule

If we convert each letter to a number ($$A=1, B=2,\ldots, Z=26$$) and add the values of all but the last letter then the result is congruent to the value of the last letter, modulo $$26$$. This is like a checksum for the word, similar to the Luhn algorithm used for credit cards as mentioned by FlanMan.

Examples

$$S+A+F+E+T = 19+1+6+5+20 = 51 \equiv 25 ($$mod $$26)$$ and $$Y=25$$

$$A+Z+I+D+O+D+E+D+I+A+Z+O+N+I+A+T+I+O+N = 1+26+9+4+15+4+5+4+9+1+26+15+14+9+1+20+9+15+14 = 201 \equiv 19(\mod 26)$$ and $$S=19$$.

• This answer is also valid too :D – Conifers Aug 29 at 9:24

I tried the following, which failed, but it might give ideas for others to solve this.

After seeing Hint 1, I think the answer is:

A Valid Word™ must somehow pass a credit card-style numerical validation using the Luhn algorithm: https://simplycalc.com/luhn-calculate.php. I converted some Valid Words™ into A1Z26 and to ASCII, but I could not find a way to get them all to pass the Luhn algorithm.

I only have a cell phone with me at the moment, so it is hard to test many ways of computing the numbers.