This is a twist on those old ditloid puzzles where you are given “equations” that use only the initial letters of the significant words, like “200 = D for P G in M” and you have to work out that this means “Dollars for Passing Go in Monopoly.” (This type of puzzle was originated by Will Shortz in the May-June 1981 issue of Games Magazine, where he called it Equations Analysis Test.)

The variation here is that there is a bijective cryptarithm used to map the initial letters into the digits 0-9 and vice versa.

Solvers first need to figure out the cryptarithm mapping used here, which is:

0 1 2 3 4 5 6 7 8 9

P A F W T D S B H C

Solvers must apply this two-way mapping to the given equations to work out the full words in each one. The subscripts (which naturally become digits when you apply the mapping) specify an index into the associated word, indicating which letter from that word to use.

N: 52 = Playing₆ Cards in a Deck
I: 32 = Degrees Fahrenheit₉ at which Water Freezes
N: 4 = Horsemen₈ of the Apocalypse
E: 37 = Plays Written by William Shakespeare₅
O: 99 = Bottles₂ of Beer on the Wall
N: 76 = Trombones₇ that led the Big Parade
E: 1000 = Words that a Picture₇ is Worth
Z: 13 = Bakers Dozen₃
E: 7 = Wonders of the Ancient₅ World
R: 24 = Blackbirds₈ Baked in an Pie
O: 6 = Degrees of Separation₉
F: 7 = Dwarfs₅ in Snow White
O: 168 = Hours₂ in a Week
U: 28 = Days in February₅
R: 64 = Squares on a Chess Board₄
N: 9 = Planets₄ in the Solar System (including Pluto!)
I: 7 = Brides₃ for 7 Brothers
N: 13 = Stripes on the American₈ Flag
E: 100 = Cents (or Pennies₂) in a Dollar
E: 7 = Days in a Week (or 8, in the Beatles₂ Song)
I: 12 = Days of Christmas₄
G: 180 = Degrees in a Triangle₆
H: 23 = Pairs of Chromosomes₂ in the Human Body
T: 101 = Dalmatians₆
O: 11 = Players on a Football₂ Team
N: 4 = Weddings and a Funeral₃
E: 1 = Point for a Free Throw in Basketball₅

The sequence of letters selected in this way, in order, spells out NINE ONE ZERO FOUR NINE EIGHT ONE. Applying the cryptarithm mapping from digits to letters to 9104981 yields CAPTCHA, which is the answer to this puzzle.

Note that a CAPTCHA is a kind of “screen test”, in that it’s a test presented on a (computer) screen, and it is intended to stop “bad actors” (in the sense of “people who do bad things”): bot programmers who try to hack in to web sites.