[superprismatic]

The polynomial X²³+X+1 can be

expressed as the product of 2 factors.

If the powers of X, excluding the

constant term, in one of these factors

is converted into letters (1=A, 2=B,

etc.), it is observed that these letters

can be anagrammed into a three-word

phrase meaning, with a certain stretch

of the imagination, "a peculiarity of a

seat in a church gallery." What is the

phrase?

SUPERPRISMATIC CLARIFICATION: Perhaps

the word "powers" should be replaced

by "exponents" to better nail down the

intent of the puzzle.

SUPERPRISMATIC OBSERVATION: Walter

Penney did not have access to computer

algebra software!

[bushindo]

The polynomial X²³+X+1 can be

expressed as the product of 2 factors.

If the powers of X, excluding the

constant term, in one of these factors

is converted into letters (1=A, 2=B,

etc.), it is observed that these letters

can be anagrammed into a three-word

phrase meaning, with a certain stretch

of the imagination, "a peculiarity of a

seat in a church gallery." What is the

phrase?

SUPERPRISMATIC CLARIFICATION: Perhaps

the word "powers" should be replaced

by "exponents" to better nail down the

intent of the puzzle.

SUPERPRISMATIC OBSERVATION: Walter

Penney did not have access to computer

algebra software!

No computer algebra software was used in the solving of this puzzle. Though for most practical purposes, I did the same algorithm that an algebra software would have used.

x²³ + x + 1 = (x ² + x + 1) * ( x²¹ - x²⁰ + x¹⁸ - x¹⁷ + x¹⁵ - x¹⁴ + x¹² - x¹¹ + x⁹ - x⁸ + x⁶ - x⁵ + x³ - x² + 1 )

The exponents that make up the anagram must be (21 20 18 17 15 14 12 11 9 8 6 5 3 2), which correspond to the following letters.

"u" "t" "r" "q" "o" "n" "l" "k" "i" "h" "f" "e" "c" "b"

Here's where I need a tag team. I'm not much of an anagram person, so feel free to jump here.

Edited August 17, 2009 by bushindo

[Guest]

loft/bench/quirk

[bushindo]

loft/bench/quirk

Good teamwork. I have never been good at anagrams, so I'd like to know how you did it so quick. Is there a particular algorithm or approach that you use to construct/deconstruct anagrams?

[bushindo]

Belch Fork Quint

Belch Font Quirk

Blench Fork Quit

Blench Oft Quirk

Bench Folk Quirt

Bench Fork Quilt

Bench Loft Quirk

Blotch Fen Quirk

Herb Flock Quint

Bilk Quench Fort

Brink Clef Quoth

Cob Elf Nth Quirk

bonanova, I'd like to ask you the same question I asked tpaxatb. Is there a non-computer particular algorithm or approach that you use to construct/deconstruct anagrams?

[Guest]

bonanova, I'd like to ask you the same question I asked tpaxatb. Is there a non-computer particular algorithm or approach that you use to construct/deconstruct anagrams?

(if you could call it a method <g>)

I play a *lot* of scrabble. (OK, I may have been slightly exaggerating on the speed...it took about 15 mins or so )

For this one, the Qu is pretty obvious...

In addition, you could see that you run out of vowels really quickly. I was operating under the assumption that no letter was repeating. so we're talking (discounting the U paired with the Q) 1 long word, 2 words (1 with 2 vowels and 1 with 1) or 3 words, each with one vowel (i discarded the "scrabble words" that are silly in the context of something like a walter penney puzzle <g>)... In any of the cases you realize that you will have more than one few consonant connectors. You're talking stuff like NT, LF, FT, CH, TH, LT, RK etc.

I operated under the assumption that it was at least 2 words. When dealing with Qu, I tend (just habit, I guess) I look for words that START with QU.

You can see rather quickly that

QUI

QUO

QUE

are your starters. Plus I had the puzzle in mind, and it happened that I was able to get quirk quickly (I ran through Quilt, Quick, Quench,

(eliminated because of two vowels + u) and hit Quirk, which fit the puzzle. (although I was thrown and didn't see it right away because of the lack of "Y" I was looking for an adjective)

that gave me:

E O B C F H L N T

Still operating under the assumption that there's two words left, I was still looking for consonant connectors.... CH, NT, FT, TH. I'm also looking for a word that has something to do with church seat. At first I thought pew, but obviously no P or W (<g>)...Some quick rearranging gave me BENCH (actually I saw BENCH because I had previously seen QUENCH)

So now I have O L F T - Only thing that makes sense is LOFT.

Now, I wouldn't have gotten it without the letters (qualify: it would have taken me a LOT longer...you did the "hard" part

[bushindo]

(if you could call it a method <g>)

I play a *lot* of scrabble. (OK, I may have been slightly exaggerating on the speed...it took about 15 mins or so )

For this one, the Qu is pretty obvious...

In addition, you could see that you run out of vowels really quickly. I was operating under the assumption that no letter was repeating. so we're talking (discounting the U paired with the Q) 1 long word, 2 words (1 with 2 vowels and 1 with 1) or 3 words, each with one vowel (i discarded the "scrabble words" that are silly in the context of something like a walter penney puzzle <g>)... In any of the cases you realize that you will have more than one few consonant connectors. You're talking stuff like NT, LF, FT, CH, TH, LT, RK etc.

I operated under the assumption that it was at least 2 words. When dealing with Qu, I tend (just habit, I guess) I look for words that START with QU.

You can see rather quickly that

QUI

QUO

QUE

are your starters. Plus I had the puzzle in mind, and it happened that I was able to get quirk quickly (I ran through Quilt, Quick, Quench,

(eliminated because of two vowels + u) and hit Quirk, which fit the puzzle. (although I was thrown and didn't see it right away because of the lack of "Y" I was looking for an adjective)

that gave me:

E O B C F H L N T

Still operating under the assumption that there's two words left, I was still looking for consonant connectors.... CH, NT, FT, TH. I'm also looking for a word that has something to do with church seat. At first I thought pew, but obviously no P or W (<g>)...Some quick rearranging gave me BENCH (actually I saw BENCH because I had previously seen QUENCH)

So now I have O L F T - Only thing that makes sense is LOFT.

Now, I wouldn't have gotten it without the letters (qualify: it would have taken me a LOT longer...you did the "hard" part

That's a wonderful approach. Thanks for the description, I learned a lot from it.

Edited August 17, 2009 by bushindo

[Guest]

That's a wonderful approach. Thanks for the description, I learned a lot from it.

No problem (hehe I was just thinking that I feel honored now that someone actually learned from me...all these puzzles recently have been making me delve into math/comp sci recesses I haven't touched in years,since I don't deal with them daily.?) And this set of puzzles has been driving me batty

I think (for me) the biggest help for this one (for me at least)...were the QU, the limited vowels, and the clues about the word...getting "quirk" helped ALOT

[bonanova]

bonanova, I'd like to ask you the same question I asked tpaxatb. Is there a non-computer particular algorithm or approach that you use to construct/deconstruct anagrams?

Just what comes to mind when I look.

My parents were addicted to a game called anagrams.

It seems some people are gifted in that area - a glance at letters scattered on the table is all they need.

I didn't inherit the gift, but I do play a decent game of Boggle... and spent several years as an editor, so I'm kind of a word guy.

I looked up [wordsmith.org] the solutions, posted them, then decided it didn't add anything to the discussion and deleted it.

You replied before I hit the button ...