# What is a Foldable Word™?

In the spirit of the What is a Word™/Phrase™ series started by JLee

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

$$% set Title text. (spaces around the text ARE important; do not remove.) % increase Pad value only if your entries are longer than the title bar. % \def\Pad{\P{0.0}} \def\Title{\textbf{ Foldable }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Words}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ } \smash{\lower{29px}\bbox[teal]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ acted }&\text{ reacted }\\ \hline \text{ beside }&\text{ aside }\\ \hline \text{ chrome }&\text{ firefox }\\ \hline \text{ doubted }&\text{ hesitated }\\ \hline \text{ emoji }&\text{ emoticon }\\ \hline \text{ freedoms }&\text{ rights }\\ \hline \text{ gloria }&\text{ monica }\\ \hline \text{ hiking }&\text{ skiing }\\ \hline \text{ ideas }&\text{ thoughts }\\ \hline \text{ karate }&\text{ judo }\\ \hline \text{ loss }&\text{ gain }\\ \hline \text{ motto }&\text{ logo }\\ \hline \text{ niggle }&\text{ trifle }\\ \hline \text{ obtuse }&\text{ acute }\\ \hline \text{ pesos }&\text{ euro }\\ \hline \text{ quash }&\text{ extinguish }\\ \hline \text{ rudeness }&\text{ braveness }\\ \hline \text{ smell }&\text{ taste }\\ \hline \text{ trumpeter }&\text{ tromboner }\\ \hline \text{ university }&\text{ college }\\ \hline \text{ vacant }&\text{ present }\\ \hline \text{ wipe }&\text{ clean }\\ \hline \text{ yacking }&\text{ talking }\\ \hline \hline \end{array}$$

The CSV version:

Foldable Words™,Not Foldable Words™
acted,reacted
beside,aside
chrome,firefox
doubted,hesitated
emoji,emoticon
freedoms,rights
gloria,monica
hiking,skiing
ideas,thoughts
karate,judo
loss,gain
motto,logo
niggle,trifle
obtuse,acute
pesos,euro
quash,extinguish
rudeness,braveness
smell,taste
trumpeter,tromboner
university,college
vacant,present
wipe,clean
yacking,talking


These are not the only examples of Foldable Words™, many more exist.

What is the special rule these words conform to?

Hint 1

foldable, foldable... it is about folding words. How many ways are there to fold a word?

Hint 2

The process of telling whether a word is foldable or not involves three words

Hint 3

The folding process is a little bit different depending on the parity of number of letters.

Hint 4

a word which length is an odd number can be written as aaaacbbbb (where c is the central letter). When you "fold" the bs over the as, you get ????c, when you fold the as over the bs you get c????. But what are the question marks?

Consider the following binary operation on letters of the alphabet

Convert each letter to its corresponding position in the alphabet (A=1, B=2, ..., Z=26).
Then we can add two letters by adding their corresponding indices and taking the result modulo 26 to get a new letter. For example, D (4) + S (19) = W (23).

Now consider folding the given words so that when two letters combine the result is yielded by the binary operation above. We can either do this left over right (LR) or right over left (RL) and I contend that the Foldable Words always yield a new word when folded in this way.
For example, ACTED -> T(C+E)(A+D) = THE when folded left over right (LR)

Similarly,
BESIDE -> BIG (LR)
CHROME -> HUG (RL)
DOUBTED -> BOTH (LR)
EMOJI ->OWN (LR)
FREEDOMS -> YETI (RL)
GLORIA -> HUG (RL)
HIKING -> TWO (LR)

While, for example,
REACTED -> VJUC or CUJV

• That's a classic, I thought I had created a good puzzle (not the best ever, just a good one) but nobody can solve it until I add tons of hints. Then hexomino arrives and solves it in minutes. do you think it was solvable even without the hints? Jul 19 '20 at 10:29
• @melfnt Yes, I think it was. Sorry I did look at it before and thought it was about identifying a symmetry rather than constructing something new and I think it's easy to get stuck on that idea. Nice puzzle! Jul 19 '20 at 11:09

My idea: if you write the words out

on a seven segment display, all the letters in the Foldable Words are symmetric in some way. This sadly doesn't hold true for Y.

• Or presumably the 'j' in 'emoji'... In fact only X and Z are missing from the LHS words. Since they don't appear in every RHS word I'm pretty sure this can't be it! :)
– Stiv
Jul 5 '20 at 21:34
• @Stiv J is symmetric if you make it a square.
– matt
Jul 5 '20 at 22:00
• Sorry, this is not the intended answer Jul 6 '20 at 2:01