# Three monotonically decreasing word sequences

A simple puzzle: given below are three word sequences. Each of them decreases monotonically (and two of them also decrease linearly). What is the next word in each of these word sequences?

INSIST, ACCIDENTAL, WHY, WAR, PIE, LAG, BEE, ???

FUNKY, TOUGH, ENIGMA, ATTIC, MIX, EYE, TEN, ???

TECHNOLOGISTS, THERMOSTATIC, RECORDISTS, PROJECTS, ATTRACT, GOATS, KITE, ???

The next word in the first sequence was already found by Techidiot, I'll include the answer here to make this post complete.

If we give every letter a numerical value according to $\text{A}=1,\dots,\text{Z}=26$, we can give each word a numerical value by taking the sum of the values of its letters. We get the following sequence. \begin{align*}\text{INSIST}&\to 90 \\ \text{ACCIDENTAL}&\to 72 \\ \text{WHY} &\to 56\\ \text{WAR} &\to 42\\ \text{PIE} &\to 30 \\ \text{LAG} &\to 20 \\ \text{BEE} &\to 12 \end{align*} So the numerical sequence is \begin{align*} 90,\ 72,\ 56,\ 42,\ 30,\ 20,\ 12\dots \end{align*} The differences between these numbers are \begin{align*} 18,\ 16,\ 14,\ 12,\ 10,\ 8\dots \end{align*} This suggests that the last word must have a numerical value of $12-6 = 6$. There are 32 possible strings of letters with this numerical value, namely F, AE, BD, CC, DB, EA, AAD, ABC, ACB, ADA, BAC, BBB, BCA, CAB, CBA, DAA, AAAC, AABB, AACA, ABAB, ABBA, ACAA, BAAB, BABA, BBAA, CAAA, AAAAB, AAABA, AABAA, ABAAA, BAAAA, AAAAAA. An excellent band is a possibility, but there is only one string that can really be considered a word. So for completeness: \begin{align*} \text{CAB} \to 6 \end{align*}

The next word in the second sequence.

The $n$th word in the second sequence uses $9-n$ dots and $9-n$ dashes when the word is written in Morse.
\begin{align*} \text{FUNKY} &\to\ ..-.\ ..-\ -.\ -.-\ -.-- \\ \text{TOUGH} &\to\ -\ ---\ ..-\ --.\ .... \\ \text{ENIGMA} &\to\ .\ -.\ ..\ --.\ --\ .- \\ \text{ATTIC} &\to\ .-\ -\ -\ ..\ -.-. \\ \text{MIX} &\to\ --\ ..\ -..- \\ \text{EYE} &\to\ .\ -.--\ . \\ \text{TEN} &\to\ -\ .\ -. \end{align*} So the last word should use one dot and one dash. A dot translates to the letter E and a dash to the letter T, so if we view them as separate letters we can make either TE or ET. If we string the dash and the dot together we find that $-.$ translates to the letter N and $.-$ translates to the letter A, which is also a word. So for completeness:\begin{align*}A \to\ .- \end{align*}

The next word in the third sequence.

If we write all the words (in lowercase) in braille we get the following. A character in braille has six positions where there can either be a dot or not. If we count for each word the number of dots appearing in each position we get the following sequence. \begin{align*} \left(\begin{array}{cc} 8 & 8 \\ 8 & 8 \\ 8 & 0 \\ \end{array}\right),\ \left(\begin{array}{cc} 7 & 7 \\ 7 & 7 \\ 7 & 0 \\ \end{array}\right),\ \left(\begin{array}{cc} 6 & 6 \\ 6 & 6 \\ 6 & 0 \\ \end{array}\right),\ \left(\begin{array}{cc} 5 & 5 \\ 5 & 5 \\ 5 & 0 \\ \end{array}\right),\ \left(\begin{array}{cc} 4 & 4 \\ 4 & 4 \\ 4 & 0 \\ \end{array}\right),\ \left(\begin{array}{cc} 3 & 3 \\ 3 & 3 \\ 3 & 0 \\ \end{array}\right),\ \left(\begin{array}{cc} 2 & 2 \\ 2 & 2 \\ 2 & 0 \\ \end{array}\right) \end{align*} So we are looking for a word that only has one dot in these five positions. There is one letter that satisfies this, namely Q. There are 8 combinations of two letters satisfying this, namely AT, ES, IO, JK, KJ, OI, SE, TA. There are no combinations with more letters that satisfy the condition. Now I think AT is the only possibility that can be called a word.

For completeness: • Ah, GHOST is not correct I see, should have been GOATS, I'll make amends. As for the two options: if you count all positions separately you'll see there's only one possibility. You already picked the right one, though ;) I think it's OK to include Techidiot's answer to the other part, as long as you give him credit for it. – Levieux Mar 8 '17 at 13:12
• @Levieux Ah, I see :). Thank you for the fun puzzle, I will edit my answer soon to include this new information. – Pjotr5 Mar 8 '17 at 13:22

Partial

INSIST, ACCIDENTAL, WHY, WAR, PIE, LAG, BEE, ???

Difference between each word value is 18,16,14,12,10,8, so the next difference should be 6.
INSIST $\rightarrow$ 9 + 14 + 19 + 9 + 19 + 20 = 90
ACCIDENTAL $\rightarrow$ 1 + 3 + 3 + 9 + 4 + 5 + 14 + 20 + 1 + 12 = 72
WHY $\rightarrow$ 23 + 8 + 25 = 56
WAR $\rightarrow$ 23 + 1 + 18 = 42
PIE $\rightarrow$ 16 + 9 + 5 = 30
LAG $\rightarrow$ 12 + 1 + 7 = 20
BEE $\rightarrow$ 2 + 5 + 5 = 12
CAB $\rightarrow$ 3+1+2 = 6
Hence, CAB is the next word in the sequence.