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Eunoia
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--> Most recent Blog ![]() Comments Policy Impressum Maths trivia Search this site YouTube Videos Eunoia, who is a grumpy, overeducated, facetious, multilingual ex-pat Scot, blatantly opinionated, old (1944-vintage), amateur cryptologist, computer consultant, atheist, flying instructor, bulldog-lover, Beetle-driver, textbook-writer, long-distance biker, geocacher and blogger living in the foothills south of the northern German plains. Not too shy to reveal his true name or even whereabouts, he blogs his opinions, and humour and rants irregularly. Stubbornly he clings to his beliefs, e.g. that Faith does not give answers, it only prevents you doing any goddamn questioning. You are as atheist as he is. When you understand why you don't believe in all the other gods, you will know why he does not believe in yours :-) Oh, and he also has a neat English Bulldog bitch 'Frieda'. And her big son 'Kosmo'.
Geocaching Stats
Some of my bikes
My Crypto Pages ![]()
My Maths Pages ![]() Nota bene : Cuius rei demonstrationem mirabelem sane detexi hanc marginis exiguitas non caparet. Nearby village Totem Pole (Meerhof Woods)
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Wednesday, August 31, 2011
Notting Hill Carnival ;-)![]()
Y ou can say what you like, but it looks to me like these two UK coppers are getting a stand-up[sic!] lap dance during the Notting Hill Carnival last weekend! Talk about job benefits, eh, Inspector Gadget ? ;-)
Tuesday, August 30, 2011
So nearly nuked! (on 9/11 too!) :-(G ermany's date of fate is 9/11 too, but in our case that means 9th of November. On Friday morning, 9 November 1923, Adolf Hitler and his thugs marched on the Feldherrnhalle. As a result, 20 people died (Göring was injured and Hitler sent to prison). On November 9-10, 1938, was the Reichskristallnacht, a pogrom on Jewish homes, shops, towns and villages. Around 1,668 synagogues were ransacked, and 267 set on fire. On a more positive note, on 9 November 1989 the Berlin Wall fell; the end of the Iron Curtain :-) But the one very few people seem to know about was November 8-9 1983, the day when sabre-rattling brinkmanship took us to the brink of WW3 :-( A ten-day NATO exercise called Able Archer 83, starting November 2nd 1983, centred on the Supreme Headquarters Allied Powers Europe (SHAPE) HQ. The exercises simulated a period of conflict escalation, culminating in a coordinated nuclear release. The 1983 exercise simulated all 5 steps up to DEFCON 1, included heads of government for (nuclear) realism and used radio silences broken only by use of our new codes. It was so realistic that the Soviet 'Politburo' believed it was preparations for a genuine nuclear first strike by NATO/US on East Germany :-( After all, US nukes in Germany were being moved around and the US and UK were exchanging newly unbreakable coded messages. So the Soviet block armed their short range bombers in East Germany with a nuke each and escalated them to sitting on the runways with engines running. Flight time from Magdeberg (East Germany) to Hamburg (West Germany) at Mach 1 would be only 12 minutes; from the Iron Curtain to nuke RAF Gutersloh is only 9 minutes at Mach 1. Military targets would have included RAF Gutersloh, Frankfurt Airport (south side), the US air base at Ramstein, the US nuke stockpile at Buren, etc etc. Civilian targets would have included the cities of Hamburg, Frankfurt, Bonn, Munich, Düsseldorf etc etc. Even the standard 550 kiloton nukes had 30 times the power of the Hiroshima bomb. Around this time sabre-rattling brinkmanship by the US had American bombers flying directly towards Soviet airspace, peeling off at the last moment, occasionally several times per week. This led to the Soviets shooting down KAL007 as some of you may remember. The US was stationing Pershing II missiles in Europe and Ronald 'Rayguns' was raving rhetorically about SDI, so the Soviets were understandably nervous. When our spies in East Germany told us about the nukes-armed bombers sitting with engines running on the runways in East Germany, and we saw the KGB spies over here reporting things as 'real' preparations for a first strike, the US realised it needed to defuse the situation sharpish. So Reagan was sent to his ranch on AF1 and this act was publicly televised. Heads of state like Thatcher(UK) and Kohl(Germany) were withdrawn publicly from the exercise and Able Archer was much reduced in scale and much was reported about the scale-back. Reagan subsequently changed his policies to rapprochement having taken us to the brink of WW3, every bit as scary as the Cuban Missile crisis of the Kennedy/Kruschev era! We have never been so ![]()
Comments (2) : Sunday, August 28, 2011
One Time Pads : Cold War Coding.N o, this is NOT an article about those new-fangled convenience coffee machines ;-) On friday I'd promised to tell you about cold war communications between East Germany (Markus Wolf's Ministry for State Security) and their 221 spies in West Germany. Many people may remember that prior to 1990 (reunification of Germany) you could tune your radio to the top of the FM band and hear a very clearly enunciated woman's voice reading out groups of five digit (random?) numbers. If your transistor radio couldn't reach that high up, you just opened it up and unscrewed the ferrite core from the inductance coil outwards a little bit (less, Henry ;-). Then you wrote down the numbers you heard. Example :- "95529 53598 24002 37816 79973 76503 66505 79556 78849 71657 44943 26137". Counterespionage organisations (e.g. BND, NSA, GCHQ etc) would run Hadamard Transforms on these number sequences, only to confirm that they were indeed random and thus not susceptible to frequency analysis codebreaking. The fact that they were random indicated that the code was a one-time-pad. The East German spies would have each had a well-hidden one time pad of sheets of random numbers. Mere possession of such a pad made you highly suspicious! When the spy's time-slot came around, he would have listened to the radio and written down the number sequence intended for him, crossing off a group for each group received, so each 5-digit group was only used once (whence one-time-pad). In the example OTP shown below, three groups have been crossed off from a previous session, leaving 95457ff.
The spy would then subtract each of these 5-digit numbers in turn from the 5-digit numbers he had received on the radio (see example above). Please do that exercise, you should get 72 101 108 108 111 32 87 111 114 108 100 33. Converting these decimal numbers into ASCII characters gives you the plaintext of the secret message :-) Check: the last number now crossed out on your OTP should be 26104. In an ideal situation, these OTP codes are unbreakable even if intercepted. However, that's assuming there were only two copies of the OTP, one at the sending end and one at the receiving end. Here, I think we can safely assume that the KGB in Moscow would have had a third copy :-) The major problem of OTPs being key-distribution; how do you get the OTPs to your spies in the first place? Postal service? No, the BND could intercept that. So dead-letter boxes were used. These were found by our side either by chance or nosy neighbours or tracking embassy staff (contact agents) etc etc. Then we were able to copy (NOT steal!) the OTP so that when the radio transmissions came, we were able to decode them too. Of course both sides use dead-drops, nowadays these are contactless too. Example : On January 23, 2006, the Russian FSB accused Britain of using wireless dead drops concealed inside hollowed-out rocks to collect espionage information from agents in Russia. According to the Russian authorities, the agent delivering information would approach the rock and transmit data wirelessly into it from a hand-held device, and later his British handlers would pick up the stored data by similar means. The spy's radio I showed you on friday had an instructional manual (VVS MfS o059-A35/86) which limited the 'telegrams' (from West to East) to 100 5-digit groups, and preferably below 50. This could be interpreted as implying that the number groups of the OTP coming from a pseudo-random number generator, some of which are cryptographically weak. But the real reason for the length restriction was to keep the D/F (direction finding) window as short as possible, even using the squirt compression I mentioned on friday. Even using these OTP codes, the East Germans always used cover names for their agents. But they also made procedural mistakes. Example: Markus Wolf (East German spymaster) liked to congratulate his agents on their birthdays which gave us only a 1 in 365 chance of identifying them. But in the case of Gunter Guillaume he sent a message asking GG to congratulate his wife on her birthday too. That narrowed the search to a married couple of high-up civil servants with both birthdays known. Bingo! PS: Further reading for budding mathematicians, cryptographers and IT nerds interested in reading about Cryptographically secure pseudorandom number generators. Comments (3) : Friday, August 26, 2011
Secret Spy Stash found nearby!J ust last week a treasure hunter was combing an area near Bielefeld (a town not 40 miles north of here) with a metal detector. I guess he was looking for coins, swords etc left over from the Battle of Varus (during the Roman occupation, 9th century AD). Suddenly he got a strong signal and so started digging. About 2 feet down he found a wooden box, only about 40-50 years old. This is what he found (spade shown for scale):-
It's a DDR spy's short wave radio transmitter! On the left you can see the morse key in its bakelite case (so the morse key is WW2 army issue, 70 years old). On the right a couple of spare valves (US issue, Sylvania). Parts like these, and the batteries, are all West German, US, UK so there would be no problem getting replacements; there are no labels on any of the kit indicating it was of DDR (=East German) origin. The wire cards in the centre are aerials. The East German secret service had 221 spies in West Germany, but only 15 of them had such transmitters for use in emergencies and warning the DDR if we (NATO) were on the brink of WW3 (such as on 9/11/83). Usually the spies sent documents they had stolen/copied back east to the DDR via secret couriers, because obviously they couldn't trust the postal service ;-) We can assume that the spy lived nearby so that he could dig up his radio in the woods quickly in an emergency. Why would he live here? Well the UK's Senne tank training grounds are there (where (UK?) tactical nukes were likely stored), RAF Guterloh is just 10 miles west (Harriers, Phantom F4 fighter/bombers etc), the US stockpiled their nukes in the woods near Buren just a few miles west of here, so in the case of WW3 preparations by Nato a local spy could observe if nukes were being hastily moved, e.g. to Gutersloh. BTW, the little cover on the left side of the transmitter contains a little magnetic recorder and a wind-on handle to position the tape and replay quickly. Thus the spy could tap in his morse offline at usual keying speeds, then replay it very quickly when transmitting (squirt mode), thus spending less time on the air and so giving us less time for a direction finding fix :-( Eastern block spies received their instructions via coded signals sent just at the edge of normal FM bands, I'll tell you about that coding next week. I'll also tell you about the stupid US brinkmanship of 9/11/83 which almost provoked WW3. We have never all been so nearly nuked in our lives! :-( Comments (4) : Wednesday, August 24, 2011
US East Coast Earthquake Cause Found ;-)
A pparently there was a minor earthquake along the US east coast yesterday. Some reporters claim it was major, seeing as how it woke some members of Congress. Now people are tweeting/blogging their photos of the 'tremendous' destruction not caused. So, not to be outdone, I add my contribution which is an artist's (Salvador Dali, actually) impression of the cause of the quake : Washington was hatching something evil again ;-) Monday, August 22, 2011
Sonofabitch! ;-)
His Lordship (Kosmo) actually sat still for 30 seconds, a rare photo :-)
Sunday, August 21, 2011
Red Arrow Down :-(
D iamond Nine Outside Right* pilot (Red 4) Flt.Lt. Egging was killed in a crash of his Hawk yesterday near Bournemouth, UK :-( He managed to steer the stricken Hawk T2 away from people and houses before it went down in a field, sliding into the river. He did not have time to eject it seems :-( He is survived by his wife Dr Emma Egging, to whom my condolences. The UK press has more details. Meanwhile, we are racing in the sodding rain at Schotten, Germany. Vintage racing motorcycles around the short town track, as every year in mid-August.![]()
Friday, August 19, 2011
Writing Styles AnalysisS o I write like Cory Doctorow? And like Dan Brown, and David Foster Wallace, and Arthur C. Clarke, and William Gibson and, and, and...
I should be so Thanks to an article in the FAZ, a German national newspaper which I read daily, I have become aware of a robotic website called I write like ... You can submit any text you wrote for analysis, blog, website, short story, poem, book extract, whatever. The site purports to analyse your writing style and tell you which famous author writes in a similar style. Unfortunately, it does not explain how this is done, it could just be a Bayesian vocabulary differentiator or even a random choice :-( I decided to try it out. My first test was to see if it was consistent (i.e. NOT a random choice). Repeatedly submitting the same text but with the sentences in a different order gave consistent results, which is a good sign. My next test was to take five books from my shelves at random and see if it could identify the authors from a single paragraph taken from each. It correctly identified Dan Brown, William Gibson and H.P.Lovecraft. It failed on two less well known authors, Thomas Landauer and Kathleen Meyer. From this I deduce it has only a limited number of well-known authors in its comparison pool. Unfortunately the website does not say who these are, nor even how many there are :-( My next test was to take a blog article of mine where I had made a deliberate attempt to write in the style of Ian Fleming (author of the James Bond books). I was pleased to see that the program thought I had achieved this. Finally, I went the whole hog and analysed 20 recent blog articles of mine to see if there was statistical evidence of a particular style. There is :-) Five of my twenty articles are in the style of blogger Cory Doctorow, the tool tells me. Some common themes of his work include digital rights management, file sharing, and post-scarcity economics none of which I have ever written about. I have however read his first (2003) novel Down and Out in the Magic Kingdom, but none of his other books. He is not on my blogroll, and I make no effort to copy his style. I don't know if we use similar vocabularies. Judge for yourselves. My five articles allegedly in Cory Doctorow style are :-
Three of my twenty articles are in the style of thriller author Dan Brown. Brown uses recurring themes such as cryptography, keys, symbols, codes etc. and so the vocabulary may be similar. Brown's prose style has been criticized as clumsy, so that we have in common ;-)Another three of my twenty articles are in the style of David Foster Wallace. I must confess to never having heard of the good but troubled man, let alone read any of his works. Wallace's fiction is often concerned with irony, Wikipedia tells us, which might explain the third article Slut Walk being regarded as similar, but the other two?Two of my twenty articles are in the style of SF author Arthur C.Clarke. These two articles of mine are about space travel and jet fighters, and so the vocabulary may be similar?Another two of my twenty articles are in the style of William Gibson. I am a great fan of Gibson and have read all of his works including the Steampunk one with Bruce Sterling. However, I fail to see any similarities :-(My last 5 articles are allegedly in the styles of Ursula K. Le Guin (see On going deaf...), Ray Bradbury (see Crypto Grille for kids :-)), H.P.Lovecraft (see Whence the Anacreontic Song?), Ian Fleming (see British Understatement ;-)), and, surprisingly, Margaret Atwood (see Trojan T-Shirts :-)). WTF? BTW, FWIW, the robot thinks THIS article is in the style of Cory Doctorow too. I wonder if he wrote the SW ;-) Why don't you try having this robot analyse YOUR writing style(s)?
Comments (5) : Wednesday, August 17, 2011
21 Dumb things Bikers do :-)T hose of you who have ever owned a motorcycle should be able to identify with this list; the rest of you may just smirk ;-)
) a link to your list or mail a comment :-)
![]() Comments (5) : Moonday, August 15, 2011
Slut-Walk, Germany, 2011Firstly, let me outline what the weekend's SlutWalk was :
Participants (mostly young women) protest against excusing rape by referring to any aspect of a woman's appearance.
They dress like whores and sluts, marching through the streets in their underwear or otherwise scantily clad (if at all),
protesting that men should not grope or molest or even rape them just because
they (un)dress like common whores. Often they The SlutWalks began in april 2011 when Constable Michael Sanguinetti, a Toronto (Canada) Police officer, suggested - not unreasonably IMHO - that to remain safe, "women should avoid dressing like sluts." Dutifully, the yellow press sends its Meanwhile, are red-blooded males supposed to visit the local scrap-metal sculpture parks and molest the statues? Or grope robot angels instead? ( Attention please : Sarcasm alert, but NSFW! ;-) While I agree that there is no excuse for rape, I am old-fashioned enough to appreciate that Constable Michael Sanguinetti was making a sensible contribution.
The clubbing scenes I saw on a previous visit to the UK (Bristol city, as it happens ;-), was full of scantily clad - even knickerless - binge-drinking girls
falling about drunkenly on the pavement/sidewalks
and generally looking for a Seriously, IMHO, we need a little bit more restraint on both sides if the pheromenal molestation-rate is to be decreased! Meanwhile, the up-and-coming(sic!) Christian party politician Christian von Boetticher (40) has resigned after his affair with a 16-year-old girl in 2010 became pub(l)ic knowledge. It is not reported how she dressed ;-) Sunday, August 14, 2011
NPR's Top 100 Science-Fiction, Fantasy BooksTo follow the NPR (US National Public Radio) meme, copy this list, putting in Bold those you have read. Link your list in Dana's comments.
1. The Lord Of The Rings Trilogy, by J.R.R. Tolkien So I've read 75% of them already :-) Thanks for the Meme, Dana :-) Saturday, August 13, 2011
50th anniversary of the Berlin Wall :-(For those who think "We don't need no education...." ; a timely reminder :-(
Walter Ulbricht made this speech on the 15th June 1961, closing with the infamous lie "No one has the intention of erecting a wall!". Barely 2 months later, exactly 50 years ago today, the Berlin Wall was built :-( Well, on the positive side, it inspired this Pink Floyd number :-)
Friday, August 12, 2011
Trojan T-Shirts :-)Trojan T-Shirts : what a great idea :-) The far right neo-fascist NPD party organised a rock festival in Gera (in eastern Germany) with the intent of getting the (~600) attending youth to join their political party. To this end they hand out propaganda material, CDs and also T-shirts. The T-shirts were black and displayed a skull and the right wing slogan "Hardcore rebels - national and free". However, the NPD were given these 250 T-shirts for free, as it turns out from an anti-fascist organisation called Exit. And the great trick is that the Exit T-shirts are Trojan T-shirts :-) Because when they are washed, the skull and the fascist slogan disappear and a new slogan appears "You too can EXIT the fascist scene - call 0177-240 45 92, - and visit www.exit-deutschland.de" Is that a neat idea, or what? Of course, some wearers haven't noticed yet ;-) Wednesday, August 10, 2011
Wall Street : take a tip from a Biker :-)If I - as a mere Biker - may be so presumptious . . .
The bikers' secret for a winning life :-) Monday, August 8, 2011
The Golden Ratio (φ) number system
L et me share this mathematical curiosity with you. The rectangle shown here is approximately a golden rectangle. The ratio of the height of the upper red section to the lower green section is the same as the ratio of the total height is to the red section. Coincidentally, I drew the ratio of the height to the width to be the same too. This ratio is called the Golden Ratio and is much used by artists (e.g. Leonardo da Vinci), it being aesthetically pleasing. So we can solve R/G= (R+G)/R to get R/G = approximately 1.6180339887, usually designated as φ (the Greek letter Phi). Note φ - 1 = 1/φ :-) φ is an irrational number, meaning it cannot be represented as a fraction. And so it is somewhat surprising (to me at least) that ALL non-negative integers can be represented as the sum of a few powers of φ but no fraction has a terminating series of powers of φ. Does that blow your mind, or what? ![]() If you wanted to make a φ-based calender, the powers are not all that large. For example 31 uses the powers [7,1,-3,-5, and -8] and even 2011 only has the powers [15,13,10,2,-2,-11, and -16]. Wikipedia has more details as usual :-) This page is for blogreader Wendy Templeton (Oz), who can now go ahead and sing "φ men went to mow, went to mow a meadow" ;-) Sunday, August 7, 2011
How dumb is CNN?Sometimes I have my doubts about the media and their ability to draw logical conclusions from their own sloppy reporting :-(Example? Just look at this statistic - produced without any interpretation on the CNN website - tabulating where people ostensibly live longest :-
Presumably their dumber readers will all want to move to Monaco, because that way they will get to live longer ;-) May I suggest another possibility? Due to the tax-free status of Monaco, many rich people live there. Rich people can afford better medical care. Therefore they live longer. But the real reason is different. Look down your own table, CNN. Seven of the top eight states in the longevity stakes are tiny. So when people there get REALLY sick, they are taken to hospitals in the neighbouring large countries, which have larger (and more specialised) hospitals. Where they ultimately die! This gets counted towards the mortality stats in the neighbouring large countries. The healthy people stay in the tiny states, increasing the nominal longevity in the tiny states. And look at the third column for Macau, CNN. Only a mere 3.72 deaths/year per 1000 residents? So in Macau the longest life expectancy (100% of residents dead) is 269 years? Really, CNN? Don't you think you should add some text explaining 1) How these stats were measured and 2) How the results are to be interpreted ? And BTW, correlation is not causality. Goddamn stupid and sloppy reporting, CNN. Shame on you! Comments (1) :
Saturday, August 6, 2011
Mr. Bean's Nemesis ;-)We learn from the UK national press that Rowan Atkinson, the renowned actor and mediocre car driver, spun and crashed his McLaren F1 supersports car on a village road in Haddon (UK) on thursday. Despite police claims that no other vehicle was involved, I have my suspicions about a Reliant Robin 3-wheeler on the wrong side of the road ;-)
Thursday, August 4, 2011
British Understatement ;-)Skip to 4:02 though 4:34 to watch the background to today's anecdote :-) Overheard in the officers' mess one evening at RAF Valley (Wales) : Excited F-15 Strike Eagle american fighter jock : "Man, we went through the Mach Loop at Cad West at 400 knots at only 250 feet! Wow! Hedgehopping!" Cool RAF type, very British : "I know, I saw you there :-)" American fighter jock, in a supercilious tone : "Oh, were you on the hillside watching me? Did you get a photo of me down that low?" Cool RAF type, very British : "No, I was in the Hawk at your 6 o'clock, doing 500 knots and 200 feet underneath you. Sadly no, I don't have a single photo of you, but I DO have a movie; it'll still be in the gun camera ;-)" Comments (3) : Wednesday, August 3, 2011
Couch Potato Calories :-(As we all know, you get fat by eating more Calories than your body burns. Also, Couch Potatoes are generally fat, so I got to wondering how many Calories do you burn when doing nothing [sleeping, sitting on the couch, watching TV, playing PC games, yes, even blogging ;-) ?].![]() Turns out there is a term which dieticians call the Resting Metabolic Rate which can be calculated quite easily, as I'll show you. It is a function of your weight, height, age and gender.
10 * weight in kilos, plus So for me this would be 88*10 (=880) + 6.25*182 (=1137.5) - 5*67 (335) +5 = 1687.5 Calories. If I consume more I need to exercise more to avoid putting on weight. Put another weigh(sic!), if my regular eating & drinking habits have me inputting 2000 Calories, then if I did not exercise (i.e. were a Couch Potato) then my weight would increase to 120 kilos! FATTY! :-( Now put down that plate and go do the calculations on yourself! Then skip the pizza and evening beers and go for a jog(hurt?) instead :-)
Monday, August 1, 2011
Dollar Debt Deadline Due?
What will happen with the US dollar this weak(sic!)? Comments (4) : |
23 Recent Writings
FWIW : 23 is the number of the Illuminati, folks ;-) Notting Hill Carnival ;-) So nearly nuked :-( One Time Pads Spy Stash found! East Coast Earthquake Sonofabitch ;-) Red Arrow down :-( Writing Styles Analysis Dumb things Bikers do Slut Walk Top 100 SF Books 50 years of The Wall :-( Trojan T-Shirts :-) Wall Street : Bikers' tip Golden ratio numbering How dumb is CNN? Mr. Bean's Nemesis ;-) Brit Understatement ;-) Couch Potato Calories Dollar Deadline... US budget crisis... An old friend lost :-( My Right Royal Pun ;-) No hybrids forseen Norwegian Terrorism :-( Four Colour Theorem 42 & no longer counting On going deaf :-( Food Porn Anecdote ;-) Crypto Grille for kids Finding NEMA Scrap metal sculptures PETling hurling prize Archive 2011: Jan Feb Mar Apr May Jun Jul Aug Archive 2010: Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Archive 2009: Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Archives 2002-2008 offline to save server file-space. Blogroll Ain Bulldog Blog Badtux... Balloon Juice Cocktail Party Physics Cosmic Variance Curmudgeonly... Decrepit Old Fool Demeur Dilligaf II Doug Alder Dr Grumpy Earth-Bound Misfit En Tequila Es Verdad Fail Blog Finding life hard? Greg Laden HaggisChorizo Inspector Gadget Kees Kennis Making Light Monkey Muck Mostly Cajun Noded (JR) Not Always Right Observing Hermann One Good Move Pergelator Pharyngula Rants from t'Rookery Stupid Evil Bastard The Magistrate's Blog Too many tribbles Xtreme English Yellowdog Grannie Link Disclaimer ENGLISH : I am not responsible for the contents or form of any external page to which this website links. I specifically do not adopt their content, nor do I make it mine. DEUTSCH : Für alle Seiten, die auf dieser Website verlinkt sind, möchte ich betonen, daß ich keinerlei Einfluß auf deren Gestaltung und Inhalte habe. Deshalb distanziere ich mich ausdrücklich von allen Inhalten aller gelinkten Seiten und mache mich ihrem Inhalt nicht zu eigen. This Blog's Status is
Blog Dewey Decimal Classification : 153FWIW, 153 is a triangular number, meaning that you can arrange 153 items into an equilateral triangle (with 17 items on a side). It is also one of the six known truncated triangular numbers, because 1 and 15 are triangular numbers as well. It is a hexagonal number, meaning that you can distribute 153 points evenly at the corners and along the sides of a hexagon. It is the smallest 3-narcissistic number. This means it’s the sum of the cubes of its digits. It is the sum of the first five positive factorials. Yup, this is a 153-type blog. QED ;-) ![]() Books I have written
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