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Carnival of Mathematics #125

This month I have the honour of hosting the Carnival of Mathematics (Episode 125).

“What is this Carnival of which you speak?” The Carnival is monthly blog round-up hosted by a different guest blogger each month. This rotating carnival gives bloggers a chance to cross-pollinate their readerships, and a chance to share stories, puzzles, anecdotes, reviews and comments.

To any new readers, welcome. To returning friends, welcome back.

My Blog

I've been blogging now for about five years. I started off slowly, but now have a fairly regular cadence of publishing about four to five articles a month. If you like what you read, you can subscribe to a newsletter alert service (to receive an email when I publish a new article), or subscribe to my RSS feed.

For a comprehensive list, see here for a complete list of awesome articles!

Carnival 125

This is episode 125 of the Carnival, and it is traditional to start of with some little pieces of trivia about the number 125:

125 is the fifth perfect cube. 125 = 5 × 5 × 5 = 5³.
125 is the third Friedman number (in base 10). This means it can be represented by all its own digits using just the four basic arithmetic operators (+ , − , × , ÷), additive inverses, parentheses, and exponents:

125 = 5⁽¹⁺²⁾

Interestingly; 126, 127 and 128 are also Friedman numbers. This is the only time four consecutive numbers are Friedman numbers for four digits (or five, or six!)

126 = 6×21 127 = -1+2⁷ 128 = 2^(8-1)

127 is also a 'nice' Friedman number because the digits in the solution occur in the same order as they do in the target.

Here's an example of a more complicated Friedman number, that has 125 in it (it's also nice).

210125 = (2¹⁰+1)²/5

Finally, here's a pandigital Friedman number. It uses all the digits 1-9 (by definition, on both sides!)

157326849 = 12543^(7+9-6-8)

You can find more complete listings of Friedman numbers here.
It's the twelfth value in the Tribonacci Sequence (A001590) defined by T₁=0, T₂=1, T₃=0.

0, 1, 0, 1, 2, 3, 6, 11, 20, 37, 68, 125, 230, …

A Tribonacci Sequence is a generalization of a Fibonacci sequence, taken with 3 elements. T_n = T_n-1 + T_n-2 + T_n-3
125 can be expressed as a sum of two different sets of perfect squares.
125 = (10² + 5²) 125 = (11² + 2²)
125 and 126 form a Ruth-Aaron pair. The sum of the prime factors of these consecutive numbers are the same.

Prime Factors (125) = 5 × 5 × 5

Prime Factors (126) = 2 × 3 × 3 × 7

5 + 5 + 5 = 2 + 3 + 3 +7
The Roman numerals for 125 is CXXV

CXXV

There is no word in the English language that contains the letters 'CXXV' in that order. In fact, there is no word that contains one 'C', two 'X's and one 'V' in any order!

Less mathematical references to 125

Here are a couple of other references to 125 from popular culture:

In Star Trek, Species 125 was the Borg designation for a humanoid race which, by the 2360s, had been assimilated into the Collective.

One notable member of Species 125 was the Borg Queen, who was first encountered by Captain Jean-Luc Picard in 2366, before the Battle of Wolf 359. In 2375, the Queen stated the designation of her species, when she kidnapped Seven of Nine. (Star Trek: First Contact; VOY: "Dark Frontier"). See here for more details.

The British Aerospace 125 is a twin-engine, low-wing, mid-size corporate jet. Originally developed by de Havilland and initially designated as the DH125 Jet Dragon, it entered production as the Hawker Siddeley HS-125. It features a perfectly cylindrical fuselage and a slightly swept wing.

The HS-125 has the dubious distinctions of being the only business jet to have ever been hijacked, and also being the only business jet to have survived being hit by an air-to-air missile! (Two separate incidents).

Image - Wikipedia

In 1967, a chartered 125 carrying the former Congolese Prime Minister Moise Tshombe was diverted to Algeria by armed hijackers on board.

Then, in 1988, a British Aerospace 125-800 transporting Botswanan President Quett Masire was stuck by a missile which had "inadvertently" been launched by an Angolan MiG-23. Despite being badly damaged by the direct hit* (which resulted in the loss of an engine, the cabin decompressing, and the fuel tanks being ruptured!) the aircraft was successfully landed by BAe demonstrator co-pilot Arthur Ricketts. Interestingly, it was later rebuilt, and returned to service!

*The Angolan MiG-23 Flogger pilot fired two R-60 (AA-8 Aphid) missiles at the plane. One missile hit the No. 2 engine, causing it to fall off the aircraft. The second missile then hit the falling engine. The captain of the business jet was incapacitated when the cabin steward was blown forward, onto him. The co-pilot made a successful emergency landing on a bush strip at Cutio Bie.

(For an overview over how missile guidance systems work, and the history of their development, you might be interested in this article.)

Image - Wikipedia

When I was growing up, in England, 125 referred to the InterCity 125; the high speed train (HST).

The InterCity 125 was the brand name of British Rail's High Speed Train (HST) fleet, which was built from 1975 to 1982 and was introduced in 1976. The train operates at speeds of up to 125 mph (200 km/h) in regular service. It has an absolute maximum speed of 148 mph (238 km/h), making it the fastest diesel-powered train in the World when it was introduced, and a record it still holds today!

Image - Wikipedia

STS-125 was the fifth (and final) Space Shuttle servicing mission to the Hubble Telescope.

The flight lasted a total of 13 days, and was performed by the shuttle Atlantis. This was the 30th flight by Atlantis, and the Launch occurred on 11 May 2009. NASA managers and engineers declared the mission a complete success. The shuttle had a crew of seven astronauts, three of whom this was their first flight into space.

Submissions

The following links are article suggestions submitted by readers of the Carnival for inclusion in this posting. You can make your own submissions and suggestions by completing this form. I'll update this list as articles are suggested.

#	Description:
1	The annoying boxes puzzle http://blog.plover.com/math/logic/annoying-boxes.html An annoying logic puzzle about two boxes, one of which contains a treasure.
2	High-dimensional integration http://www.johndcook.com/blog/2015/07/19/high-dimensional-integration/ Overview of how, in practice, high dimensional integrals are computed.
3	My Mathematical Journey http://www.flyingcoloursmaths.co.uk/my-mathematical-journey/ My Mathematical Journey.
4	How Much Pi Do You Need? http://blogs.scientificamerican.com/roots-of-unity/how-much-pi-do-you-need/ Scientific American.
5	Infinite hotels in swirling beams of light http://skullsinthestars.com/2015/07/24/infinite-hotels-in-swirling-beams-of-light/ Infinite hotels in swirling beams of light.
6	The Singular Mind of Terry Tao http://mobile.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html?_r=1&referrer= NYT Magazine (subscription may be required).
7	Hadley Wickham, the Man Who Revolutionized R http://priceonomics.com/hadley-wickham-the-man-who-revolutionized-r/ Priceonomics.
8	Useful generalizations, Part II http://voices.norwich.edu/daniel-mcquillan/2015/07/11/useful-generalizations-part-ii/ How generalizing a problem can actually lead to a simpler solution.
9	Spelling Matters http://mathmisery.com/wp/2015/07/27/comic-34-spelling-matters/ A letter here, a letter there.
10	Triangulations are rigid. You can do better using pseudo-triangles http://mappingignorance.org/2015/07/27/triangulations-are-rigid-you-can-do-better-using-pseudo-triangles/ Pointed pseudo-triangulations: 2D rigid structures with fewest bars.
11	Hair Band Sierpinski Tetrahedra at the MoSAIC Festival http://blog.andreahawksley.com/the-mosaic-festival-and-sierpinski-tetrahedra/ Hair Band Sierpinski Tetrahedra at the MoSAIC Festival.
12	Train tracks and graph theory https://possiblywrong.wordpress.com/2015/07/30/train-tracks-and-graph-theory/ Train tracks and graph theory.
13	How to get to the fourth dimension http://www.scientificamerican.com/article/how-to-get-to-the-fourth-dimension/ Scientific American.
14	Advice on how to calculate area … https://reflectivemaths.wordpress.com/2015/08/02/b-and-q-help-you-with-maths/ … but not how to do percentages.
15	Problems with reductio proofs: "jumping to conclusions" http://m-phi.blogspot.nl/2015/07/problems-with-reductio-proofs-jumping.html Part III of a series of posts on reductio ad absurdum arguments from a dialogical perspective.
16	Parallel Weighings Solution http://blog.tanyakhovanova.com/2013/08/parallel-weighings-solution/ Detecting coins of different weights with more than one set of scales.
17	The Connoisseur of Number Sequences https://www.quantamagazine.org/20150806-neil-sloane-oeis-interview/ For more than 50 years, the mathematician Neil Sloane has curated the authoritative collection of interesting and important integer sequences.
18	Bridges Math Art http://blog.andreahawksley.com/bridges-math-art-2015/ Notes from the 2015 conference.

About me

A little about me. I'm a self confessed geek who grew up consuming everything that Martin Gardner wrote. I taught myself to program, first on a TRS-80, then onto a BBC Micro where I learned 6502 Assembly language. After that, I've pretty much used PC's ever since. I was educated in the UK, where I obtained a Master's Degree in Aeronautical and Astronautical engineering. Then, through sale of a software company, I moved to the USA. I worked for Microsoft for almost 14 years (over ten of which were in their games division), then a couple of years of consulting, and I now, proudly, work for Facebook as a Data Scientist out of our Seattle office.

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