22
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A librarian browsing the basement of a library came across an empty box. Its contents were missing, and to her annoyance even its label had been vandalized.

The label did tell her that the box had once contained a series of five volumes, but other important lines had been obliterated.

Above one of the obliterated lines, some words had been written, and below it were some numbers:

food ice cream sugar bio gas
#################################
  14    11    ?    22    33

What is the missing number?

Hint 0:

There are already a couple of hints above.

Hint 1: Here is the (almost) complete label:

Title: ######################### Volumes: 5 food ice cream sugar bio gas ################################# 14 11 ? 22 33 Sections: 3 perfect inner love ###################### 14 11 7

Hint 2: And this is the last line:

Dewey Class: 516

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  • $\begingroup$ Are the categories Food, Ice, Cream, Sugar, Bio Gas or Food, Ice Cream, Sugar, Bio Gas or something else? The formatting makes it hard to see the five category names $\endgroup$ – SGR Apr 4 '16 at 8:16
  • 2
    $\begingroup$ The formatting is deliberate - that's what the label looked like too. So hypothetically, it could be read as 4, 5 or 6 terms. I'll only confirm that there are exactly 5 terms. ;-) $\endgroup$ – KeyboardWielder Apr 4 '16 at 13:22
  • 1
    $\begingroup$ 516 is the Dewey Decimal System number for geometry. So my guess would be the section numbers have something to do with it. "14, 11" and "22, 33" look like patterns to me. $\endgroup$ – ASCIIThenANSI Apr 7 '16 at 18:09
  • $\begingroup$ @ASCIIThenANSI: Your first sentence is correct. $\endgroup$ – KeyboardWielder Apr 7 '16 at 18:59
14
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Them's five volumes and three sections alright.

           Title:
    .---   #########################
    '--->  How Much #### Things Hold   (Revised edition, discontinued)
    .---   S####s I#######d ## # ###   (From partial records in a comment)
    '--->  Solids Inscribed in a Box   (Pirate edition, discovered by ffao)
    '--->  Shapes Inscribed in a Box   (Ur-edition, restored by puzzle's poser)

           Volumes: 5
           food ice cream sugar bio gas
    .---   #################################
    '--->  pyramid cone cube sphere cylinder
    .---     14    11    ?    22    33
    '---------------->  42

           Sections: 3
           perfect inner love
    .---   ######################
    '--->  square circle triangle
              14    11    7
 

Blurb for the Volumes and Sections:

The numbers of the five Volumes are indeed the volumes of the 1×1×1 solids mentioned in their titles, and the numbers of the three Sections are likewise the cross-sectional areas of 1×1×1 prisms. (Note that the volumes and areas are converted ×42A3 and ×14A2 into cubic and square answer units, then rounded to integers.) $$ \require{begingroup} \begingroup \def \T#1{{ \small\textsf{#1}} = \kern -8mu } \def \A { \kern -18mu {\small\sf ~\times 14 \sf\small A^2} ~~ = } \def \V { \kern -18mu {\small\sf ~\times 42 \sf\small A^3} ~~ = } \begin{array}{rclcrcl} \T{(food) pyramid} & \frac13 l w h &\V 14 && \T{(perfect) square} & l w &\A 14 \\[5mu] \T{(ice cream) cone} & \dfrac\pi{12} d^2 h &\V 11 && \T{(inner) circle} & \dfrac\pi4 d^2 &\A 11 \\[5mu] \T{(sugar) cube} & l w h &\V 42 && \T{(love) triangle} & \frac12 l w &\A 7 \\[5mu] \T{ (bio) sphere} & \dfrac\pi6 d^3 &\V 22 \\[5mu] \T{ (gas) cylinder} & \dfrac\pi4 d^2 h &\V 33 \\[5mu] \end{array} \endgroup $$


Colophon. This puzzle's apparent construction.

Missed clues:   (Would the hints have been necessary in the long run?)

The term volumes for what was missing from the box.
The association between $\frac{\Large\pi}2 {\approx} \frac{11}7$ and the volumes' numbers' being multiples of 7 and 11.
The suggestiveness of sugarcube right next to the compelling association ice creamcone.

Clues that registered:

The words box and inscriptions in the puzzle's very title. (caught by ffao)

Hint 2:Dewey Class: 516   →   geometry, category 516 in the Dewey Decimal subject classification system   →   ice cream+cone   →   pyramid cube sphere cylinder square circle triangle.

The 11 22 33 multiples of 11 are paired with circular shapes cone sphere cylinder circle   →   those numbers relate to $\pi$ more strongly than to other characteristics such as numbers of letters.

The ratio cone:cylinder = 11:33 matches the 1:3 ratio of unit-sized conical and cylindrical volumes   →   pyramid:cube = 14:? matches the 1:3 ratio of unit-sized pyramidal and cubical volumes   →   cube=42.

Reinforcing clues:

The repeated use of the word box in the puzzle statement.
The words Volumes and Sections in Hint 1.

Bonuses:

Obfuscation of 6 words by relating them to only 5 volumes.
If this had been solved quicker we wouldn't've been treated to the perfect inner love hint.
Perhaps the box has dimensions 1×1×1(×42A3) and the series of volumes had fit into it singly, not all at once.

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  • 2
    $\begingroup$ Can I upvote this puzzle / answer more than once? No? :( $\endgroup$ – ffao Apr 8 '16 at 6:46
  • $\begingroup$ @humn: That's correct! For the complete explanation, think of more wordplays in the same context for the title of the puzzle and for "Sections". The Title on the label could be anything - it's not necessary to figure out: But if you're interested, the one I had in mind is the puzzle title reworded as "S####s i#######d ## # ###" (noun past-participle preposition article noun). $\endgroup$ – KeyboardWielder Apr 8 '16 at 6:51
  • 1
    $\begingroup$ LOL at your title translation xD $\endgroup$ – ffao Apr 8 '16 at 8:21
  • 1
    $\begingroup$ *Snorts* The editions tho $\endgroup$ – feelinferrety Apr 8 '16 at 16:16
  • 1
    $\begingroup$ @humn: I think you caught everything. I was a little surprised that after Hint 1, nobody latched on to the numbers all being multiples of either 7 or 11 (a strong whiff of pi). $\endgroup$ – KeyboardWielder Apr 10 '16 at 15:15

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