By Erica Newman and Justin Melvin

The flavor text recalls a famous conversation between the mathematicians Hardy and the dying Ramanujan. In the original story, Hardy remarks to Ramanujan that he saw a dull number on the way to visit that was on a taxicab, the number 1729. Ramanujan convinced Hardy that this was indeed an interesting number, and 1729 became the known as the smallest of the “taxicab numbers.”

In the flavor text, the “Oh. Er, I See” is meant to indicate to solvers that they should use the OEIS—the Online Encyclopedia of Integer Sequences.

The presentation of the tax form for “carriages and similar vehicles” is another clue that the players are looking for taxicab (“tax cab”) numbers.

If the solver does not understand the Ramanujan connection, they can instead search for the word “cab” in the OEIS and they will find taxicab numbers.

In each minipuzzle that the solvers resolve, they will come up with an “A” series. Looking the A series up in the OEIS will give them a sequence of numbers. Each indicated sequence contains the number 1729 somewhere in it, in the 1-26 position. The position of 1729 in the sequence then corresponds to the letter that they should extract for the puzzle.

Each of the minipuzzles relates to facts about Ramunajan or relatively famous results of his from different branches of mathematics, which are all documented on his Wikipedia page.

Section I (Filing status) generates three separate series. Parts 1, 2, and 3 are solvable by checking Ramanujan’s Wikipedia page. His age at death was 32 (Part 4). This section is meant to establish that the whole set of puzzles is about Ramanujan and his work, including, in this part, infinite series. The answers within the minipuzzles correspond to numbers highlighted in green, and Parts 5, 6, and 7 indicate how to calculate the numbers that are the A series answers.

Section II (Blank) indicates that the solver should leave this answer blank (this is the break between two words).

Section III (Income) says “This section to be continued.” This is a clue that solvers should rewrite the fraction as it is presented as a continued fraction. This step can be done online by searching for “continued fractions” and using a solver. The sequence of denominator terms (excluding repeating 1’s, as indicated in the minipuzzle) will resolve to an A series. If the sequence is not in the 6-digit format of an A series, the solver is meant to pad the answer with leading zeroes. So, for example, 2498 would become A002498 in the OEIS.

Section IV (Adjustments) is a magic square (another of Ramanujan‘s interests) where the solver fills out the numbers in the colored squares to make the sum of each column and the sum of each row the same. Then the solver organizes the resulting numbers in rainbow order (red to purple) and the result is the A series.

Section V (Exemptions) presents Death Cab for Cutie song titles, mildly altered. This is related to the overall puzzle theme because it brings up cabs again (cluing taxicab numbers). Each song contains a number or the word “nothing” (indicating zero) or “one” (1).

Section VI (Payments) says “This section is a bit derisive.” This minipuzzle flavor text is meant to indicate Ramanujan‘s mock modular forms. The text mimics an equation that is on Ramanujan‘s Wikipedia page that is related to the quantity Pi. The solver can easily calculate the answer by checking that equation.

Section VII (Refund) is a minipuzzle where the solver takes the sequence of numbers given, searches for the OEIS series based on that sequence, and therefore finds the A series directly.

“Total Amount Owed” has enough spaces for the solver to input all of the extracted letters including the blank from Section II to spell out `TAX DOCUMENTS`.

Solved forms follow.

 Department of the Treasury—Internal Revenue Service Estate Forms Form “A” IRS use only—Do not write or staple in this space. See separate instructions. For the year of Jan 1–Dec 31st 1920 Carriages and Similar Vehicles P.1 Code A Series I. Filing Status [This section may diverge off topic] Instructions: Check status boxes appropriately. Note the Code. 1 Natural numbers, summation of* *in your own idiosyncratic way -1/12 X Negative 099500 Positive 005590 Undefined 095900 2 Potential afflictions (past and present) (a) Aquagenic urticaria 87 (b) X Dysentery (0) 0 (c) X Hepatic amoebiasis (3) 3 (d) X Hydrocele testis (35) 35 (e) Progeria 49 (f) X Vitamin deficiency (02) 02 3 Recurring forms (a) 1/2 + 1/4 + 1/8 + 1/16 +… (04) (b) 1 - 1 + 1 - 1 + 1… in your own style! (6) (c) Even number (25) (d) All known real valued parts to non-trival zeroes of the Riemann zeta function (6) Instructions: Answer above with boxes provided below. Note the Code. (a–d) -1 7 0 25 ½ 6 1 04 Other/unknown 18 4 Age at expected death 32 5 Sum answer in part [4] with Code [1] A 099532 T 6 Concatenate Codes part [2] A 033502 A 7 Concatenate Codes part [3] (a)–(d) A 046256 X II. Blank 8 [This section intentionally left blank] BLANK [] III. Income [This section to be continued] [continued fractions] Instructions: Calculate your fraction of use. Ignore repeating “1”s. 9 [7708061 - √5] / 7014298 A 110921 D 10 [166495 - √5] / 51362 A 003476 O 11 [153333195 + √5] / 18836890 A 087788 C
Department of the Treasury—Internal Revenue Service
Estate Forms
Form “A”
IRS use only—Do not write or staple in this space.
See separate instructions.
For the year of Jan 1–Dec 31st
1920
Carriages and Similar Vehicles
P.2
Code
A Series
[magic square] Instructions: Pull some numbers out of your hat
12
 18 7 8 3 2 9 12 13 10 15 0 11 6 5 16 9
A 072360 U
V. Exemptions [This section is adorable]
[Death Cab for Cutie lyrics]
13
“Home’s a’fire” insurance Nothing
[use numbers in songs]
14
“Tiny vessels” saving for surgery fund 2
15
St. Peter’s Cathedral restoration charity Nothing
16
“Codes and keys” musical cryptography after school program 1
17
Scientist studies research and development credit 4
18
Kelly Huckaby memorial fund 5
19
Total exemptions A 020145 M
VI. Payments [This section is a bit derisive]
[famous equation on Ramanujan wiki page is a solution for pi. Rerranged, it becomes this number]
20
Yo momma is so numeric that she multiplied pi by fifty-four point eight six nine three five oh six eight six nine and multiplied this by the sum from k equals zero to infinity of four k factorial times the quantity one thousand one hundred and three plus twenty-six thousand three hundred ninety k all over k factorial to the fourth power times three hundred ninety-six to the four kth power.
Total payments A 190132 E
VII. Refund [This section is refunded to you]
[type series into OEIS for unique series] Instructions: Found objects; find more
21
121, 286, 561 A 122780 N
22
19096, 23001, 26335 A 231569 T
23
4606, 250, 765 A 061147 S
Total Amount Owed
T A X D O C U M E N T S
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