# What is a Trio 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 Trio Word™.
Use the following examples below to find the rule.

Trio Words™ Not Trio Words™
BANANA APPLE
EAST WEST
NORTH SOUTH
UNFORTUNATELY FORTUNATELY
DOGS DOG
CAT CATS
DOOMSDAY ARMAGEDDON
THOUSAND HUNDRED
PAINTING DRAWING

For Trio Word™ marked star above, it's also the Perfect Trio Word™, please also specify what condition makes the perfection. :P

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

Trio Words™,Not Trio Words™
BANANA,APPLE
EAST,WEST
NORTH,SOUTH
UNFORTUNATELY,FORTUNATELY
DOGS,DOG
CAT,CATS
DOOMSDAY,ARMAGEDDON
THOUSAND,HUNDRED
PAINTING,DRAWING


Hint 1:

https://www.calculator.net/big-number-calculator.html -> For verification

Hint 2:

ARK > BED

• shouldn't play be perfect as well? Jul 31 '19 at 10:37
• Yes, PLAY is Trio but not perfect. Jul 31 '19 at 10:41
• Added one more example PAINTING as a Perfect Trio Word~ Jul 31 '19 at 12:31
• hey there, any hints? Aug 17 '19 at 5:24
• @OmegaKrypton Just really close to the Perfect Trio Word... in comments need a little adjustment to get it :) Aug 18 '19 at 14:02

A Trio word:

has its value divisible by 3 when decrypted with A1Z26

A Not-Trio word:

has its value indivisible by 3 when decrypted with A1Z26

A Perfect Trio Word:

has its value equal to 45 or 90 when decrypted with A1Z26. only from the examples, it may not be able to determine the most suitable criteria for such category. @Bananenkopp proposed a very propable case that Perfect Trio Words have their value being a Triangular Number. @Hugh also proposed a possible solution that their values are divisible by 9. Another possible case is that their values are divisible by 5/15/45.

• UNSTABLE is 21+14+19+20+1+2+12+5 = 94 = not value divisible by 3. I was about to post my answer, but found that. Jul 31 '19 at 10:07
• @Jan Ivan I have changed an example pair :P Jul 31 '19 at 10:29
• maybe the reason for perfection is its value is a rot13(Gevnathyne ahzore) Jul 31 '19 at 11:25
• Not close to the rot13(Gevnathyne ahzore) :P Jul 31 '19 at 12:54

Just a small expansion of OmegaKrypton answer:

A Perfect Trio Word:

is where the number obtained by A1Z26 is divisible by 9 (it's my first answer), but as noticed in comments then play also should be perfect word while it's not, so we can add another divisor. For the perfect words we see their decomposition as
$$1) 3*3*5$$
$$2) 3*3*5$$
$$3) 2*3*3*5$$
We can conclude that common prime factors are 3*3*5, so perfect word is word which A1Z26 cipher sum divisible by 3*3*5=45

• not necessarily... Jul 31 '19 at 10:53
• play = 54 but not perfect Jul 31 '19 at 11:00
• Nice attempt, need be more perfect :P Jul 31 '19 at 13:22
• Just one step closer to the perfection~ Aug 5 '19 at 3:36
• This seems arbitrary. Okay, how about that a perfect trio word is divisible by 3*3*(((33/3) mod 3)+3)? Aug 5 '19 at 9:56

A Trio Word is:

a word for which the sum of all its letters turned to A1Z26 is divisible by 3

EAST5+1+19+2045

A Perfect Trio Word is:

a Trio Word that also has the resulting number of concatenating the letters turned to A1Z26 divisible by 3, at least three times

EAST5119201706405688018960

Wouldn't it be more perfect if they were only divisible by 3 exactly three times (just as DOGS and PAINTING)?