Answer to the Pythagorean Problem

The answer to a problem incliuded in yesterday’s post Monday Miscellany.

INTRODUCTION

You may remember that in yesterday’s post titled “Monday Miscellany” I included one of the problems I had solvced on the mathematical website Brilliant – this post presents the solution and also clears up a side issue raised on that website by disgruntled folk who had got it wrong.

THE PROBLEM

Here is the problem again from yesterday:

Pythag

THE SOLUTION

The answer is True. The formula for (x+1)^2 is X^2 + 2X + 1, and every odd number greater than 1 could serve as the 2X+1 part of that equation. 

THE SIDE ISSUE

Some people on Brilliant cavilled at this because there are some Primitive Pythagorean Triples whose smallest term is even (8,15, 17, 20,21,29 and 65,72,97 were all mentioned, although none of the complainers mentioned 12,35,37, 60,91,109 or 696,697,985). The question did not state that the triple of which the odd number is the lowest term was the lowest triple to feature that number, and indeed if one looks carefully at the triangles presented as part of the problem one can see clearly that the odd number is allowed to be in another triple where it is not the lowest term:

Pythagorean Triples Solution - a

Note that the number 5 features twice (ringed in the diagram above, once as the largest term in a triple and once as the smallest).

Thus that 15 features in 8,15,17 does not invalidate the claim of the question since it is the smallest term in 15, 112, 113. All the other odd numbers mentioned in triples of which they are not the smallest member likewise feature in triples in which they are the smallest member, the biggest being 985, 485112, 485113. 

Monday Miscellany

A mixed bag of an offering this Monday afternoon!

INTRODUCTION

This post will be every bit as varied as its title suggests, featuring a mix of politics, mathematics, music, nature and photography (and possibly more). 

SOME MATHS RELATED STUFF

I start with one of more recent followers, RobertLovesPi, and I have several pieces of his to share:

My next piece, courtesy of whyevolutionistrue is titled “The Coffer Illusion“, which concerns the picture below:

If the illusion defeats you, you can find out where the circles are by going to the original post. 

I finish this little section with a nod to the mathematical website Brilliant, which I am a regular visitor to (I am currently on a 64 day problem solving streak). As a sample here is a problem I solved today, rated at maximum difficulty by the site, pretty close to minimum by me:

Pythag

You can look at solutions to this problem on the website, and I will reveal the answer on this blog tomorrow. 

A FEW POLITICAL PIECES

There has been a lot of coverage from various people of events in Charlottesville. I choose to draw your attention to Richard Murphy’s excellent piece on Tax Research UK, titled “Charlottesville is a cradle of extremism: we should take note“. Below is a screenshot of the first few paragraphs:

charlottesville

My second link is to the petitionsite, regarding a young women in El Salvador who having been raped and then had a miscarriage has then been jailed for 30 years due to the Catholic church influence anti-abortion laws of that country. The screenshot below is formatted as a link to take you to this petition to sign and if possible share it:

Screenshot 2017-08-14 at 3.29.29 PM

I finish this section on a lighter note, courtesy of whyevolutionistrue. This little piece titled “Where is North Korea? Some Americans have no idea” reminds us how unacquainted USians are with that area known as the rest of the world! Here is a screenshot of the opening paragraph:
Screenshot 2017-08-14 at 3.39.40 PM

PHOTOGRAPHS

I usually end my blog posts with some of my own photographs, but this photograph section has an additional feature – as a nod to the principal subjects of many of the photos that follow I offer you a musica prelude – Ottorino Respighi’s “The Birds”:

cormorants and boatCormorants8Cormorant headsCormorants6Cormorants4Cormorants3Cormorants2Cormorants1Cormorants and gull 5Cormorant4Cormorants11Cormorants 10Cormorants and gull3TerngullCormorant3Cormorant2

Cormorant
I did not notice the white bird on the far side of the river until I was editing this one – I think from the shape and colour that is a Little Egret but the image is not clear enough to be sure.

Cormorants and swimming gullPollinator3Flybutterfly wingPollinator2white butterflyPollinator

Squirrel does Meerkat impression
This squirrel is clearly an impressionist – and his meerkat is very good!

Bird Pics on the #Inglorious12th

A post for the #Inglorious12th, featuring the right kind of bird shooting – that done with a camera.

INTRODUCTION

Today is August 12th, which is for well-heeled British hooligans the start of the grouse shooting season, known to them as “The Glorious 12th”. For folk like me, who view those who derive pleasure from taking pot shots at birds with utter contempt it is therefore the #Inglorious12th.

SHOOTING BIRDS THOMAS STYLE

I choose to mark today by posting pictures of birds shot the only acceptable way – with a camera. Most of these are from this morning, but I am also including some older pics.

Little Egret 1
The Little Egret that featured a couple of weeks back.
Magpie
A magpie on the path alongside the Great Ouse
wagtail
I was at the racecourse before my colleague arrived with a key, and this wagtail caught my eye while I was waiting.

small wadersmall wader2small wader3small wader4small wader5small wader6

CP1
The first of the new pictures – this and the final cormorant pic are in their correct positions, but some of the rest of the cormorant series are out of order.

Cormorant posing 1Cormorant posing 2Cormorant posing 3CormorantCP2CP3CP4CP5CP6two cormorantsCP and West Lynn churchFlying gullsRavenMoorhen in algaeJuvie moorhen

Jay1
I finish with this Jay – two live photographs and the relevant page from my bird book.
Jay2
The close-up shot.

Jay book page

 

 

Note of Warning to Autistics (via Nico Ian Nicholson)| Problematic Authors Seeking Interviews on Intersection of Queer/Trans and Autistic Identities

A warning about two people not to be co-operated with (nb I have a long record of assisting with autism related research, and as some of you will recall recently spoke to my local radio station – I am not remotely disinclined to co-operate with those who deserve it)…

American Badass Activists

ATTENTION AUTISTIC/ND FOLK! Read this carefully. This is important.

Eva Mendes, MA, LMHC and Meredith Maroney, MS are contacting Autistic people, particularly LGBTQ+ and racialized Autistics, and asking them to participate in interviews for an upcoming book project to be published by Jessica Kingsley Publishers.

The email will state that the book is about the intersection of queer/trans and autistic identities, and any accounts included will be anonymous.

Mendes is particularly problematic because of her book, “Marriage and Lasting Relationships with Asperger’s Syndrome.” It contains a great deal of pathology-based language, including “theory of mind” and “emotional intelligence” nonsense that suggests that Autistic people in relationships must be the ones to change.

RECOMMENDATION: Refuse to participate. And PLEASE SHARE WIDELY, especially with LGBTQ+, POC, and racialized Autistic people. — Nico Ian Nicholson


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Note of Warning To Autistic Community (via Nick Walker) | Anti-Autistic, Anti-Neurodiversity Documentary Alert

A warning about an upcoming documentary – please visit original post, read in full, comment an d share…

American Badass Activists

Dear Autistic Community & Allies: John Donvan & Caren Zucker, authors of the book “In a Different Key” – a book that promotes anti-autistic bigotry and abuse of autistic children, and deliberately smears and misrepresents the neurodiversity movement – are making a documentary film which they claim is “about the neurodiversity movement.”

Obviously, this is going to be another anti-autistic, anti-neurodiversity hit piece. If you’re pro-neurodiversity and you allow them to interview you for this film, you can bet the footage will be edited and presented in a way that misrepresents you and makes you look bad, just like they did to Ari Ne’emanand others in their book.

Best thing would probably be if all of us just completely refused to talk to them or be filmed by them.

Please share this information freely in the community – there are plenty of people in the neurodiversity movement with…

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Preparing for Two Auctions Simultaneously

A post about James and Sons’ August auctions.

INTRODUCTION

James and Sons usually have an auction near the end of each month. In August a combination of an intervening bank holiday and the fact that it takes a full day to set things up at Fakenham Racecourse means that we will be having two separate auctions near the end of August:

  • On Friday August 25th we will be having an auction at our shop in Fakenham.
  • On Wednesday August 30th we will be having an auction at Fakenham Racecourse.

THE FRIDAY SALE

This sale kicks off with a large number of coin lots. Here are some pictures:

33
This is a 17th century guinea weight (the S 21 that appears on each face refers to the monetary value of the guinea – 21 shillings)

33-a33-b

68
Lots 68-70 feature coins from the joint monarchy of William III and Mary II (1688-94)

68-a68-b6969-a69-b7070-a70-b

80
A farthing was a quarter of a penny, so these had a value of one eighth of a penny!

80-a80-b9191-b91-a

186
Lot 186 – a one punt piece.
186-a
This creature has been extinct for some considerable time – it is an Irish Elk and those antlers grew to have a span of up to 15 feet).

186-b

210
A commemorative medallion celebrating the acquittal of radical publisher John Horne Tooke.

210-a210-b

8
Lot 8
8-a
The two faces of the coin.

8-b8-c

199
Lot 199

199-a199-c199-b211211-b211-a

256
Lot 256 – this square coin is from 17th century Holland

256-b256-a272

273
Lot 273 – a two album collection.

273-a273-b

THE WEDNESDAY SALE

I included some of the pictures I have taken for this sale in my previous post, and although I will not ignore those items I will include only the main images and urge you to consult my previosu post for the rest.

 

1403
Lot 1403 – there is a little wallet incorporated in the inside back cover of the book to store the map when folded.
1415
Lot 1415 – the largest railway map I have ever seen – and it has stout front and back panels so that when folded it looks a bit like a book.
1422
Lot 1422 – A more modern and much smaller railway map, with promotional material on the reverse (four images)
1428
Lot 1428 – Some south Wales railway history.
1401
Lot 1401. Every lot from 1401 to 1428 is featured in this post

1401-b1401-a140214041404-a140514061406-a14071407-a1408140914101410-b1410-a1411141214131414-a141414161417141814191420142114231424142514261427

England 3 South Africa 1

My thoughts on the recently concluded series between England and South Africa mens teams, plus some photographs from work.

INTRODUCTION

On Monday I listened to what turned out to be the final day of the test series between England and South Africa (Tuesday would have been available had South Africa taken the game that far but they never really looked like doing so). In this post I look back at the match and the series.

THIS MATCH

England batted first and made at least 50 more than they should have done in the circumstances, getting to 360. When the ninth England wicket fell South Africa turned to the “clever ruse” of dropping the field back to allow the major batter (Jonny Bairstow on this occasion) to take singles so that they could bowl at the no11. This is a dubious tactic in any case, but South Africa’s execution of it was downright bad – on a number of occasions Bairstow took twos early in the over, which should never happen when this tactic is in play. I can think of no occasion on which it can be demonstrated that a side fared worse by attacking at both ends than they would be adopting this tactic, whereas I offer the following examples of times where adopting it caused problems:

  • Perth 1978 – Australia eight down for not many facing and England total of over 300, Mike Brearley gives Peter Toohey with 50 to his name singles so as to attack Geoff Dymock. The ninth wicket pair stage a very irritating partnership. In the end England’s superior skill and professionalism tell (Australia were depleted by the Packer affair and Graham Yallop proved to be a very poor captain). My source for this story is Brearley himself in “The Art of Captaincy”.
  • Melbourne 1982 – The ninth Australian wicket in their second innings falls with them still needing 74 for victory. England allow Border singles so they can attack Thomson, the no 11. Australia get to within a boundary hit of victory before Thomson flashes at a wide one from Botham and is caught by Miller with an assist from Tavare.
  • Sydney 2010 – Pakistan have bossed the game against Australia, leading by over 200 on first innings, and Australia are only 80 to the good with two second innings wickets standing going into the 4th morning. Pakistan decline to attack Hussey, and Siddle plays a straight bat the relatively few deliveries he has to face. In the end Pakistan need 176 to win, which is far more than they were expecting. The pressure is too much for an inexperienced batting line up, especially once Mohamed Yousuf has compounded his failure as captain by falling to a very poor shot to leave his side 57-4. Australia end up winning by almost 40 runs.

South Africa’s response, if it can be so described, was to scrape together 226 for a deficit of 136. A fine innings by Moeen Ali in the second England innings takes England to a lead of 379. Dean Elgar fell cheaply to start the South African second innings, and by the lunch interval Heino Kuhn and Temba Bavuma had also been accounted for. Amla and Duplessis resisted stoutly for a time, but the dismissal of Amla sparked a collapse, with no one else making a significant contribution as 163-3 at the high point of the innings subsided to 202 all out. Moeen Ali took five of the wickets to finish with 25 for the series alongside over 250 runs for the series (the first time this double has been achieved in a series of fewer than five matches). Moeen Ali was player of the match, and also player of the series for his all-round efforts.

THE SERIES AS A WHOLE

Barring the aberration at Trent Bridge this was a series that England dominated, and 3-1 is a fair reflection of that fact. Lord’s (it is named after Thomas Lord of Thirsk, so Lord’s is technically correct) saw the only really huge first innings tally of the series, and from that point on England were always going to win that match. I wrote in some detail about the Trent Bridge debacle at the time. At The Oval (these days there is always a sponsor’s name attached but I refuse to mention them whoever they may be) England made a respectable first innings total and South Africa crumbled, while this final match at Old Trafford went along similar lines. 

THE PLAYERS

I am going to finish the text element of this post by looking at both sets of players, starting with South Africa.

Dean Elgar – a tough competitor whose second innings 136 at The Oval when all around him were surrendering was a stand out performance. 

Heino Kuhn – resembles a test-class opener about as closely as Liam Dawson resembles a test-class all-rounder. The only surprise out his dismissal during the morning session fo what turned into the final day of the series was that it did not come sooner.

Hashim Amla – a magnificent batter now nearing the end of his illustrious career. This was not a great series for him but his fighting 83 in the final innings was a splendid effort.

Quinton De Kock – fine wicketkeeper and on his day a very destructive batter, but was miscast in the key number four role where was too often coming in with the team reeling from early blows. He was moved down for the final match of the series, but this was his equivalent of Adam Gilchrist’s 2005 in England – batting wise a series to forget.

Faf Du Plessis – it continues to be debatable whether he is worth a place as a batter, but the team play much better under his captaincy than when he is not present. 

Temba Bavuma – a very reliable batter. He needs to develop ways of keeping the scoreboard ticking – at the moment it takes him a very long time to score his runs.

Theunis De Bruyn – anonymous in this series, he did nothing significant with the bat and his bowling was not much used.

Chris Morris – occasional moments with his hard-hitting batting but his bowling was very expensive.

Vernon Philander – a great cricketer, but like Alan Davidson and Chris Old before him he is somewhat of a hypochondriac. He did not contribute fully to this series.

Keshav Maharaj – South Africa’s leading wicket taker of the series. 

Kagiso Rabada – A fine fast bowler who bowled well in this series and at times did enough with the bat to have embarrassed some of bhis supposed betters in that department.

Morne Morkel – A solid series – it was not South Africa’s bowlers who were chiefly responsible for their defeat in this series.

Duanne Olivier – more will certainly be seen of this young fast bowler.

Now for England…

Alastair Cook – continues to steadily ascend the test run scoring lists – in the course of this series he went past Allan Border’s aggregate. His effort on the truncated first day at The Oval put England in control of that game, a position consolidated by Ben Stokes’ century.

Keaton Jennings – surely he has run out chances after a series in which his highest individual score was 48 and during which he never looked convincing. 

Gary Ballance – given a chance to re-establish himself in the side because he scores so many in domestic cricket he failed, and looked out of place. He was deservedly one of the casualties of the Trent Bridge debacle.

Tom Westley – a solid start to his test career. He looks like he belongs in the test arena and I expect to see a lot more of him.

Joe Root – his first series as test captain, and with a 3-1 series win and himself being leading run scorer on either side for the series it was a splendid start. 

Dawid Malan – came in to the side after the loss at Trent Bridge and has not yet done much.

Jonny Bairstow – an excellent series with both bat and gloves.

Ben Stokes – regular contributor of runs, wickets and catches. Like the man I will be dealing with next he is that rarity, a genuine all-rounder.

Moeen Ali – deservedly named player of the series, he was outstanding with bat and ball. 

Liam Dawson – my comments about Heino Kuhn suggest that I do not rate Mr Dawson, and that impression is correct. He has neither the batting nor the bowling to be of use in test match cricket. If conditions warrant two spinners pick a real spinner, and if they don’t Moeen Ali will be the sole spinner.

Toby Roland-Jones – he started his test career firing with both barrels – a five-for including the top four in the opposition batting order, and has done well in both his matches so far. 

Stuart Broad – a good series for the big fast bowler. 

Mark Wood – two matches in the series, total figures 1-197 – ’nuff said.

James Anderson – 20 wickets in the series at 14 each. At the age of 35 he remains arguably the finest user of a new ball in world cricket. The authorities at his home ground of Old Trafford have recently paid him the compliment of naming one of their bowling ends in his honour – and he responded by taking four cheap wickets from that end at the first time of asking. I reckon he still has a couple of good years left in him which would enable him to sign off with a home world cup followed by a home Ashes series.

PHOTOGRAPHS

I always like to include photographs in my posts, and although I have none relating to cricket, here are a few from yesterday at work (these will be going under the hammer on August 30th, our second end of August auction, with a sale happening at our shop on Friday August 25th – more on this in a later post):

1403
Lot 1403 – there is a little wallet incorporated in the inside back cover of the book to store the map when folded.

1403-a1403-b1403-c

1415
Lot 1415 – the largest railway map I have ever seen – and it has stout front and back panels so that when folded it looks a bit like a book.

1415-a

1415-b
A stamp on the back of one the ordinary panels.
1415-c
The front panel
1415-d
The back panel
1422
Lot 1422 – A more modern and much smaller railway map, with promotional material on the reverse (four images)

1422-a1422-b1422-c

1428
Lot 1428 – Some south Wales railway history.