# What is a Felt Word™?

This is in the spirit of the What is a Word/Phrase™ series inaugurated by JLee with his original Phrase™ and Word™ puzzles.

If a phrase adheres to a certain rule, then I call it a Felt Word™.

Use the examples below to find the rule.

Felt Words™ Non Felt Words™
zit boil
vex bother
baby grandparent
saw axe
wild zoo
ordinary uncanny
violet blue
urchin waif
clue hint

In case you want it in CSV:

Felt Words™, Non Felt Words™
zit, boil
vex, bother
baby, grandparent
saw, axe
wild, zoo
ordinary, uncanny
violet, blue
urchin, waif
clue, hint


The puzzle relies on the series' inbuilt assumption, that each word can be tested for whether it is a Felt Word™ or not on its own.

Hint 1

Let's just say Felt Words™ offer their readers a complete experience.

A Felt word is a word ...

... whose Braille representation uses all six dots, that is: Overlaying all letters yields the full 2×3 rectangle. (This is equivalent to or'ing all the bits with a result of 63.)

(@Deusovi already made the connection that "felt" isn't a fuzzy material, but refers to the sense of feeling needed to read Braille. I had tried several things like "A felt word is still a valid word when mirrored or turned on its head", but to no avail. Only the new hint made the penny drop.)

Evidence section:

Felt Words™
zit — ⠵⠊⠞ → ⠿
vex — ⠧⠑⠭ → ⠿
baby — ⠃⠁⠃⠽ → ⠿
saw — ⠎⠁⠺ → ⠿
wild — ⠺⠊⠇⠙ → ⠿
ordinary — ⠕⠗⠙⠊⠝⠁⠗⠽ → ⠿
violet — ⠧⠊⠕⠇⠑⠞ → ⠿
urchin — ⠥⠗⠉⠓⠊⠝ → ⠿
clue — ⠉⠇⠥⠑ → ⠿

Non Felt Words™
boil — ⠃⠕⠊⠇ → ⠟
bother — ⠃⠕⠞⠓⠑⠗ → ⠟
grandparent — ⠛⠗⠁⠝⠙⠏⠁⠗⠑⠝⠞ → ⠟
axe — ⠁⠭⠑ → ⠽
zoo — ⠵⠕⠕ → ⠵
uncanny — ⠥⠝⠉⠁⠝⠝⠽ → ⠽
blue — ⠃⠇⠥⠑ → ⠷
waif — ⠺⠁⠊⠋ → ⠻
hint — ⠓⠊⠝⠞ → ⠟

A bit of statistics:

I didn't see the solution earlier, because I (falsely) assumed that Felt words would be special and therefore rare. A Felt word must use one of the letters of the third decade (u, v, y, x, z) or a w. Otherwise, it won't use the bottom right dot.

The longer a word is, the higher the probability that it uses all dots. The words in the list are rather short, so the puzzle uses examples from a range where Felt words are rare. (The outlier is "grandparent", which is not Felt, because it doesn't use any letters from u to z. I had seen that all Felt words use one of the letters u to z, but failed to, um, connect the dots.)

Running the Feltness test on a dictionary of English words shows that the ratio of Felt words increases with word length as expected:

length    Felt   non F   Felt%

2     116    1118     9.4
3    1134    5135    18.1
4    3834    9708    28.3
5   10912   16379    40.0
6   20268   25750    44.0
7   28494   32400    46.8
8   35116   34867    50.2
9   37806   32354    53.9
10   36142   26943    57.3
11   30738   20785    59.7
12   24603   14982    62.2
13   18461    9875    65.2
14   12938    6402    66.9
15    8500    3795    69.1
16    5295    2071    71.9
17    3107    1123    73.5
18    1657     568    74.5
19     882     277    76.1
20     434     122    78.1
21     199      49    80.2
22      87      22    79.8
23      35      15    70.0
24      19       8    70.4

total  244808  280803    53.4

Overall, the Felt rule seems to divide the words into two halves of nearly equal size. (That metric isn't very good, because the short words are more frequently used than the long words, of course..) With a list of letter frequencies such as the one on Wikipedia, we can also calculate the frequency of dots:

      75.1%    49.4%
49.7%    59.0%
50.8%     8.4%


Unfortunately, ...

... there are no "perfectly Felt" words, where each dot is used exactly once. "Juju" comes close: It uses each dot exactly twice.

• Ah, nicely done! – Deusovi Jan 23 '17 at 7:22
• The reason I didn't add longer words was that It'd soon become obvious they didn't contain u-z. Similarly I tried to generate Non Felt Words™ with u-z to avoid this detection. Great job though, and a very interesting analysis. – Karan Atree Jan 23 '17 at 12:42
• Good job on your part to chosse the appropriate words. "Zit" and "vex" seemed uncommon enough and had me stumped. I needed the hint to solve this. – M Oehm Jan 23 '17 at 12:51

Felt Words™ have...

an even number of dots when encoded in Braille.

• Naturally. Why didn't I think of that?? ;) – Bailey M Jan 21 '17 at 9:04
• Doesn't WAIF have an even number of dots in Braille? – BaSzAt Jan 21 '17 at 13:56
• Or about half of the Non Felt Word™ products? – BaSzAt Jan 21 '17 at 14:06
• @BaSzAt: Hm, you're right. I was too hasty in posting this. Maybe this is part of the answer, though? – Deusovi Jan 21 '17 at 18:09
• Likely a property that requires an even number of dots. – greenturtle3141 Jan 21 '17 at 18:32