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

Suggestion

As to your method of work, I have a single bit of advice, which I give with the earnest conviction of its paramount influence in any success which may have attended my efforts in life — Take no thought for the morrow. Live neither in the past nor in the future, but let each day’s work absorb your entire energies, and satisfy your widest ambition. That was a singular but very wise answer which Cromwell gave to Bellevire — ‘No one rises so high as he who knows not whither he is going,’ and there is much truth in it. The student who is worrying about his future, anxious over the examinations, doubting his fitness for the profession, is certain not to do so well as the man who cares for nothing but the matter in hand, and who knows not whither he is going!

— William Osler, advice to students, McGill College, 1899

| Science & Math · Society