Extra Large

Which is bigger, a jillion or a zillion? No one’s quite sure, though we all use these terms pretty readily. In 2016 Wayne State University linguistic anthropologist Stephen Chrisomalis cataloged the first appearance of 18 “indefinite hyperbolic numerals” — here they are in chronological order:

forty-leven
squillion
umpteen
steen
umpty
umpty-ump
umpty-steen
zillion
skillion
jillion
gillion
bazillion
umptillion
kazillion
gazillion
kajillion
gajillion
bajillion

The Oxford English Dictionary’s first cited usage of gajillion occurred in 1983, and they don’t yet have an entry for bajillion. So maybe that’s largest?

(Stephen Chrisomalis, “Umpteen Reflections on Indefinite Hyperbolic Numerals,” American Speech 91:1 [2016], 3-33, via Math Horizons.)

| Language · Science & Math

Progress

In 2012 I mentioned that Helen Fouché Gaines’ 1956 textbook Cryptanalysis: A Study of Ciphers and Their Solution ends with a cipher that’s never been solved. Reader Michel Esteban writes:

I think I found what kind of cipher Helen Fouché Gaines’ last challenge is.
In my opinion, it is a seriated Playfair of period 5 with two peculiarities:
– Zs are nulls in the ciphertext,
– Z is the omitted letter in the cipher square (instead of J).
If I am right, period 5 is the most likely reasonable period: we can observe no coincidences between upper and lower letters.
On the other hand, six reciprocal digrams appear: FD-DF, EC-CE, JN-NJ, JB-BJ, QL-LQ and GW-WG. These are almost certainly cipher counterparts of common reciprocal digrams (ES-SE, EN-NE, IT-TI, etc.).
I did not solve this cipher, because it is too short to use statistics. The only way to solve it is to use some metaheuristics (like Hill Climbing), but I have no computer!
I have no doubt you know someone that will be able to unveil the plaintext after having read these considerations.

Can someone help? I’ll add any updates here.

| Puzzles · Science & Math

All the Way Down

https://commons.wikimedia.org/wiki/File:Geometric_series_14_square2.svg

The infinite series 1/4 + 1/16 + 1/64 + 1/256 + … was one of the first to be summed in the history of mathematics; Archimedes had found by 200 BC that it totals 1/3. There are two neat visual demonstrations that make this fact immediately apparent. In the unit square above, the largest black square has area 1/4, the next-largest black square has area 1/16, and so on. Regions of black, white, and gray make up equal areas in the total figure, so the black squares, taken together, must have area 1/3.

The same argument can be made using triangles (below). If the area of the largest triangle is 1, then the largest black triangle has area 1/4, the next-largest 1/16, and so on. Areas of black, white, and gray make up equal parts of the total figure, so the black regions must total 1/3.

https://commons.wikimedia.org/wiki/File:Geometric_series_triangle.svg
Image: Wikimedia Commons
| Science & Math

Misc

https://commons.wikimedia.org/wiki/File:Lady_Godiva_by_John_Collier.jpg

  • Lady Godiva’s horse was named Aethenoth.
  • UGHA in BROUGHAM is silent.
  • 7 × 58 × 73 × 28 = 7587328
  • APHELIOTROPISMS is an anagram of OMPHALOTRIPSIES.
  • “The French for London is Paris.” — Ionesco

“No general proposition is worth a damn.” — Oliver Wendell Holmes Jr. (a general proposition)

| Language · Quotations · Science & Math · Trivia

Midy’s Theorem

The decimal expansion of 1/7 is

0.142857142857 …

Interestingly, if you split the repeating decimal period in half and add the two complements, you get a string of 9s:

142 + 857 = 999

It turns out this is true for every fraction with a prime denominator and a repeating decimal period of even length:

1/11 = 0.090909 …
0 + 9 = 9

1/13 = 076923 …
076 + 923 = 999

1/17 = 0.0588235294117647 …
05882352 + 94117647 = 99999999

1/19 = 0.052631578947368421 …
052631578 + 947368421 = 999999999

It was discovered by French mathematician E. Midy in 1836.

| Science & Math