Solutions to Guards and Clock Problem

Solutions to na couple of problems and a new problem for you to get your teeth into.

INTRODUCTION

A couple of aeons ago in the post I put up immediately before setting off for Marxism 2018 I presented two problems from brilliant.org, one easy and one hard. Now at long last I offer solutions to them.

THE GUARDS PROBLEM

Here is the answer:

Guards answer

Now here is an official solution, posted by Siva Budaraju:

Guards solution

Yes, Anna, you were right about this one, as you are about many things. 

THE CLOCK PROBLEM

For this one I shall present the confirmation that my answer was correct, my sneaky way of solving the problem and then an official solution.

Clock answer

I got the solution by realising that if there was an arrangement of the hands that enabled this to happen it would not be unique – as with problems involving two hands on a clockface there would be a number of possibilities, which would mean that finding such an arrangement would not be very difficult, and this was supposed to be a dificult problem, which led me to the conclusion that there could not be a time when the three hands divided the clockface into equal segments. Now here are two official solutions:

clock solution

A NEW PROBLEM

I finish by sharing another problem with you that I enjoyed solving:

3 and 2

 

 

 

A Solution and A Couple of New Problems

A solution and two new problems by way of a quick post before setting off for Marxism 2018.

INTRODUCTION

I am writing this immediately before setting off for Marxism 2018, a four-day political festival taking place in London. I should be able to do some posting at the event and at my accommodation. In my recent “An A-Z of Me” I included a mathematical problem from brilliant.org, and I now post the solution to that problem and offer you a couple of others.

THE SOLUTION

The problem I posted earlier was this:

plus-plus

The answer is that there is not, because it turns out that the perimeter of a plus-plus is always a multiple of four, which 2018 is not.

TWO NEW PROBLEMS

These are also both from brilliant.org. One is easy but generated a bit of controversy, while the other is very hard to solve properly, but by applying logic it can be easily worked out what the correct answer is. First, the hard problem:

clock problem

The easier problem which generated some controversy is this one:

Guards

An A-Z of Me

A personal A-Z, inspired by a post on The Cornish Maid.

INTRODUCTION

This post was inspired by The Cornish Maid’s post “A-Z about Me!!!“. Like the post I have just referenced there will be one entry for each letter…

MY PERSONAL A-Z

A FOR AUTISM

I am autistic myself, and also branch secretary of NAS West Norfolk. This entry is a very appropriate starting point because it was my diagnosis and the role I then had running a support group for Asperger East Anglia that led me to create this blog. 

B IS FOR BOOKS

I have always loved books, and am a very voracious reader. In addition to my own collection I am a regular user of several of Norfolk’s libraries, and yes I do use them to borrow books. 

C IS FOR CRICKET

I have been an enthusiastic follower of cricket for over 3o years (my attempts at playing the game foundered on a chronic lack of talent). The fact that my employers had an auction yesterday and have another on Saturday means that I am off work today, and therefore able to listen both installments of the Women’s T20 double header. Here is the feature image from Saturday’s upcoming auction:

2412-b

D IS FOR DETECTIVE STORIES

This is an extension of my love of books as a whole. I regularly borrow large quantities of detective ficition from thbe libraries. Among my very favourites are Edward Marston’s Railway Detective stories.

E IS FOR EAST RUDHAM

The village in West Norfolk where I began to rebuild my life after mental health issues had nearly destroyed me. I lived there for just over five years and was a regular visitor until my parents recently moved to Cornwall.

F IS FOR FERRY

I have travelled on many ferries in my lifetime, but the one I particularly think of nowadays is the Lynn Ferry which has been running for over 800 years.

Ferry 2

G IS FOR GREECE

I first visited Greece for a family holiday about 35 years ago and have been back mnay times. It remains a favourite holiday location. I have produced a number of posts about my most recent visit.

Tseria

H IS FOR HISTORY

One of the many subjects I enjoy reading about. One of the reasons I enjoy going to Greece so much is the presence of so many historic sites.

I IS FOR IRRELIGIOUS

I have been a staunch atheist for my entire adult life. For those who take the approach that the Northern Ireland census form used to I am a “catholic atheist” – that being the specific religion that I rejected. To paraphrase Richard Dawkins most people are as atheist as me about almost every god who has ever been believed in – I just go one god further than they do.

J IS FOR JOURNEYS

I love travelling, and being a lifelong non-driver am able to make good use of almost all my journeys – if the route is not familiar to me I will be observing the scenery and taking photographs, and if it is it represents reading time.
Major bridge from 37,000 feet up
K IS FOR KERNOW

Kernow is the Cornish name for Cornwall (this is the only entry in my A-Z that overlaps with The Cornish Maid’s), and although unlike the person who inspired this post I do not live there I have been there a number of times over the years and my parents have recently moved to that part of the world. It is a Cornish picture that appears on the reverse of my personal cards:

framed lighthouse

L IS FOR LONDON

I grew up in London, and still visit the place on occasion. Also, I run a London Transport themed website, www.londontu.be. I will be back in London during the latter part of next week, for Marxism 2018 which runs from Thursday to Sunday. 

Crossing the the Thames

M IS FOR MATHEMATICS

Another lifelong interest, and something that I am very good at. Here is a frecnet problem from brilliant.org that took my fancy:

plus-plus

N IS FOR NATURE

Nature has always been very important to me, and I love being out and about in nature with my camera for company. My name is often to be found among those supporting campaigns to protect nature, and as a thoroughgoing internationalist I take pride in having been the first non-Swede to sign the online petition to save Trosa nature.

Dragonfly
A spectacular creature, presumably some form of dragonfly.

O IS FOR OVAL

Because of their shape many cricket grounds have Oval in their name. The two with which I am most familiar are The Oval, in South London not very far from where I grew up, and served by two stations, Oval and Vauxhall; and the Adelaide Oval, which owes its name to a transplanted Surreyite who suggested it because he wanted to be reminded of home.  Of the innings I have seen live at the ground the most memorable at either of these two venues was played by David Gower in 1990. England could do no better than draw the game, which as it happened was enough to give them the series. Gower made 157 in that innings, and by the time he was out the draw had long since been secured.

P IS FOR PHOTOGRAPHY

This is a hobby of mine, and also something I do at work. Here are some recent pictures:

Black Headed Gull III
Three recent bird pictures

Moorhen IIITwo moorhens

840
This was lot 840 at yesterday’s auction – some good tools but the box being solid iron makes for an extremely heavy item.

840-a

2188
Lot 2188 in Saturday’s upcoming auction.

Q IS FOR QUIZ

With my eclectic interests and retentive memory I am pretty good at quizzes (unless they are overloaded with questions about pop music), and generally enjoy taking part.

R IS FOR RAILWAYS

Railways are one of my special interests. I have travelled on railways in many different countries and have also built up a decent collection of railwayana. I may add to my collection on Saturday. 

20502050-a2050-b2050-c2050-d2050-e

 S IS FOR SCOTLAND AND SWEDEN

These are two of my favourite countries to visit, both very scenic. I could find no way to split them so I have decided to honour both places.I have produced a number of posts about both Sweden and Scotland. Here are a couple of pics: through the window 2A view from Strome Castle, Scotland

DSCN9318This river is in Northern Sweden.

T IS FOR T20

Yes – another cricket related entry. T20 (where each side bats for 20 overs) has been a great success since its introductiuon in 2003. However the new 100-balls per side competition is being too clever by half (and consigning the County Championship to the start and end of the season when conditions are least suitable for long form cricket). 

U IS FOR UNIVERSE

I find it fascination reading theories about our universe, its possible origins and its possible place in a wider cosmos. I also find the history of how we moved from considering our planet to be at the centre of a fixed universe to recognising it as pale blue dot (hat tip to Carl Sagan who wrote a book of that title) in the immensity of the cosmos to be fascinating.

V IS FOR VARIETY

One of the things I enjoy about my current job is that there is plenty of variety there. I am firmly in the camp of those who say that variety is the spice of life.

WHY EVOLUTION IS TRUE

Jerry Coyne’s 2009 book with that title remains a firm favourite (along with his more recent Faith versus Fact), and it is also the title of a blog run by Professor Coyne that I follow. 

X IS FOR EXHORT

As I near the end of this post I exhort you to produce your own version – it is time consuming but fun. You have seen my version, and if you followed the opening link you have seen the version that inspired me to take on this challenge – now go and do likewise!

Y IS FOR YARBOROUGH

This is a bit of a cheat – it is my way of mentioning the game of Bridge which is a firm favourite of mine. A yarborough is a hand with no card higher than a nine and 4-3-3-3 distribution, and is named in honour of Lord Yarborough who had all bridge players at his house contribute a guinea to a kitty, while if someone had the misfortune to be dealt the hand that now bears his name they got 1,000 guineas (he was on to a winner – the actual odds against the hand coming up are 1827 to 1). I do not get to play very often but I am a pretty good player of the game.

Z IS FOR ZOOM

A zoom lens can be a real boon for a photographer (my current camera has a zoom capacity of up to 60X) – a little tip from experience is to not stretch the zoom lens right to its limits – leave a bit of space around whatever you are photographing (you can always crop it out during the editing process). This post was inspired by a Cornish blogger, so I end with a Cornish picture.

St Michael's Mount and Flying Gull

 

Answers to Puzzles and Notice of Absence

Notice of my imminent departure for a week’s holiday and solutions to Wednesday’s puzzles.

INTRODUCTION

A couple of days ago I set some puzzles for you, and in this post I will be answering them. Also, I am off to Greece for a week’s holiday later tonight (I fly out of Gatwick at 5:40AM, so am envisaging taking the 21:36 train out of Lynn, connecting to a Thameslink service at St Pancras and arriving the airport just after 1AM – the second latest set of connections available to me, and I know British public transport too well and trust it too little to rely on the last possible connections) and although I will take any opportunities that arise to go online I will still have comparatively little access to the internet, so you will hear little from me between now and a week tomorrow evening when I shall be back home.

SOLUTION 1: DR FRANKENINE

Dr Frankenine

Here is the most elegant of the official solutions, posted by Callum Cassidy-Nolan:

CCN

SOLUTION TWO: FUEL TANK

Fuel Tank

As you can see from the above graphic, almost half of those who attempted solutions on brilliant.org got it wrong. A perusal of the comments section revealed a degree of reluctance on the part of some of the errant solvers to admit to being wrong (never mind arguing with the umpire, some of these folk were metaphorically following that up by arguing with TV replay umpire) so I am to explain the whole process of getting the right answer (though it took me milliseconds to work out and not much longer to enter the correct answer):

  1. After stage 2) half of the original fuel remains and is then topped up with new fuel, and we are told to assume that perfect mixing occurs…
  2. After stage four one quarter of the tank of perfectly mixed fuel has gone, and as it is perfectly mixed one half of that fuel is original, meaning that a further one eighth of total tank full of original fuel has been used.
  3. One half plus one eighth = five eighths of the original tank full of fuel has been used, leaving three eighths of the original tank full still there.
  4. As a percentage three eighths is 300/8 = 37.5 per cent, and the question has asked for that figure. 

One particularly offensive complainant attempted to use the fact that the question had said decimals allowed to claim that an answer of 0.375 should have been permitted. This is wrong – the question specifically asked for a percentage, and the reason for mentioning the a decimal figure was OK was so that people did not think they needed to round to the nearest whole number, which in correct mathematical notation would have been 38 (the figure being rounded away is five or greater so you round up, had it been four or less it would have been correct to round down). 

Another complaint people made was being marked wrong for including the percentage symbol – I am less unsympathetic to this than I am to the indefensible claim that 0.375 should have been marked as right, as this latter has missed out turning the answer into a percentage, but including the symbol is still a mistake as the way the question is asked renders it needless to do so. 

Please note that I did not create this problem and had nothing to do with deciding what answer should be marked as right – I have treated it at length because I was annoyed on the composers behalf that so many solvers rather than attempting to see by looking at correct solutions, of which plenty of good ones were posted, why their chosen answer was wrong instead opted for the ‘argue with the umpire’ approach, in some cases being very unpleasant about it. Here to end this little post is David Vreken’s published solution:

DV

A Couple of Puzzles and Some Butterflies

A couple of easy problems on brilliant.org that have generated considerable controversy, and some butterfly pics of my own.

INTRODUCTION

A couple of this weeks puzzles on brilliant.org have generated considerable controversy, so I am going to share them with you as a prelude to the butterfly pics. Both puzzles are actually very simple, and I will provide solutions and explanations some time on Friday.

PUZZLE 1: DR FRANKENINE

Here is a screenshot of the first puzzle:

Dr Frankenine

PUZZLE TWO: FUEL TANK

Here it is:

Fuel Tank

BUTTERFLIES

These are from Saturday and Monday. In addition the ones I have photographs of I have seen one other species, mainly white but with flame coloured wingtips but not yet been able to photograph it.

Small Tortoiseshell IIISmall Tortoiseshell IIWhite butterflySmall Tortoiseshellgreen and white butterflyveined white butterfly

Cricket, Solutions and Photographs

Some cricket related thoughts, photographs and solutions to my most recent set of problems.

INTRODUCTION

I have a couple of cricket related things to share, a few photographs, and solutions to the puzzles I set in the post “Cricket, Photographs and Puzzles”. 

CRICKET

The third round of Championship matches were scheduled to start yesterday, but most did not get underway due to the weather, and those that did get underway were heavily affected by the weather. I have two other things to mention in this section:

WORLD CUP 2019 SCHEDULE

The schedule for the 2019 World Cup is now available to the public (see here for full details). The tournament is as usual spread out over far too long (starting at the end of May but not finishing until mid July) because the organisers will not stand up to the TV people and schedule multiple matches for the same day. In 2015 I put up a post demonstrating how a 16-team tournament could be scheduled to last no more than three and a half weeks, and I reproduce the text from that post below:

THE SUTCLIFFE FORMULA FOR ORGANISING A CRICKET WORLD CUP

There has been much talk at the Cricket World Cup about how the tournament should be formatted, especially given that there are those who would reduce it to a ten team tournament (so utterly harebrained a notion that I do no more than mention it). Several of the associate nations at this world cup have given good accounts of themselves, with Ireland having a strong chance of progressing to the quarter finals.

My formula for a Cricket World Cup would be as follows:

16 teams to play in the tournament. Stage one would involve two groups of eight teams, the top four from each group progressing. Each group would play its matches in sets of four (hence two groups of eight), making seven rounds of matches for each group, to played on alternate days (i.e. this stage would span two weeks, with each side having a day off between matches.

After the group stage would be a three day break before the quarter-finals, which would be played all on one day. After a two day break the semi–finals would take place. Then following another two day break the final would take place. This would mean that the tournament would be played in a period of three and a half weeks (a sensible length for a global tournament).

As for the TV people: If they don’t like it they can lump it.

A SUGGESTED ENGLAND TEAM FOR 1ST TEST MATCH

The early stages of this cricket season have been less than satisfactory, but I have some thoughts about an England team for the first test match nonetheless. In batting order:

  1. A Cook
  2. H Hameed – Mark Stoneman has had ten test matches without producing a serious score, and the fact that he has reached 50 five times but not gone beyond 60 is enough for me to call time on him. Hameed is restored to full fitness (it was injury that ciost him his place after an encouraging start to his test career) and should be given another chance.
  3. J Bairstow, playing as a specialist batsman (he is plenty good enough to do so).
  4. J Root (Captain)
  5. D Malan – one of the few England test batsmen to be able to claim a successful tour of Australia and New Zealand.
  6. B Stokes 
  7. M Burgess (wk) – a fine wicketkeeper, and in what is currently a very exclusive club of batsmen who have produced two major scores this season. 
  8. S Curran – a left-arm pace bowler who has been knocking on the door. The fact that he bowls with his left-arm will lend variation to the seam attack.
  9. J Leach – finally given a chance in the last test of the New Zealand leg of the tour, he bowled well and must surely be persevered with.
  10. S Broad
  11. J Anderson
  12. D Bess – stranger things have happened than an English pitch in May warranting the selection of two spinners, so Bess gets the nod as 12th man. Incidentally, controversial as it would be, the person who would have miss out were I going the two spinners route is Broad, going for a new-ball combo of Curran and Anderson.

Of the three players who went into the Ashes tour as England players and who do not feature above James Vince may yet redeem himself by producing some huge scores for Hampshire, while Stoneman and Ali are as far as I am concerned firmly in the category of ex-England players.

PHOTOGRAPHIC INTERLUDE

We now change focus, and I am marking this with some photographs. There will be more after I have presented solutions to the problems I left you to tackle.

Gull on chimneytwo gullsBird on The Granaries roof

OSSA1
During the fine weather last week I used the ‘outside study area’.

OSSA2White Butterfly in flightFlowers 1Flowers 2Red FlowersFlowers 3

SOLUTIONS

All of these problems were taken from brilliantThe first was:

MATCHSTICKS

matchsticks

First the answer:

matchstick answer

Now here is Marvin Kalngan’s published solution:

Kalngan

SOLUTION 2: CLEAR ICE

Here is the problem:

Clear Ice

And the answer:

clear ice

SOLUTION 3: POLYOMINO

The problem:

Polyomino

The answer:

polyominoa

I solved this one the lazy way – I noted that shape B very easily forms a rectangle, and after visualising various assemblages of shape A and noting that none were rectangular I opted for B only given that this is a Basic level problem. Stefan van der Waal published this solution:

van der Waal

SOLUTION 4: CONVERGENCE

The problem:

Convergence

The answer:

CVergence

Since the sequence involves numbers between 0 and 1 being multiplied together, and such numbers multiply to produce smaller numbers, the series actually converges on 0.

SOLUTION 5: CUBE

The problem:

Cube

The answer: 

Cubesol

I solved this as follows:

1)Because you are specifically allowed to rotate the cube you can see every individual block that appears on the outside…
2)This means that the only blocks that can you cannot see are those wholly inside the cube…
3)…Which since the surface layer is 1 block thick, and occurs twice in each direction amounts to a 4 x 4 x 4 cube…
4)…Therefore 64 blocks are invisible, which means that (216-64) = 152 blocks are visible.

To end this section here is Aaa-Laura Gao Gao’s solution:

ALGG

FINAL PHOTOGRAPHS

For those of you who have made it to the end:

White butterflyRiver view IRiver view IIWBNar Meets OuseSmall Tortoiseshell XVSmall Tortoiseshell XVIOuse Rowers IOuse Rowers IIBaden PowellBaden Powell IIMallard pair on Great OuseMariners cornerFriar NicholasMariners corner IILower Purfleet

 

Cricket, Photographs and Puzzles

Some thoughts about the early stages of the English Cricket Season, some photographs and some puzzles.

INTRODUCTION

The second round of County Championship matches in season 2018 are now on their second day. Additionally the fact that here in England we seem to have skipped spring, going dorectly from a long, unpleasant winter into summer means I have a particularly fine selection of photographs for you, and there will be puzzles. 

THE COUNTY CHAMPIONSHIP

Scoring is low everywhere. At Chester-le-Street it is looking a first innings tally of 169 will be sufficient for Kent to record an innings victory (Durham, shot out for 91 yesterday morning are 39-7 in their second innings, needing their last three wickets to double that paltry tally just to avoid the innings defeat). Essex and Lancashire are already into their third innings as well, Essex having scraped together 150 first up and Lancashire replying with 144. Essex are 39-0 in their second innings. Somerset, having actually claimed a batting bonus point by reaching 202 are poised for a handy first innings lead, Worcestershire being 153-8 in response. Surrey also topped 200 – making 211, and Hampshire are 79-6 in response. Yorkshire made 256 in their first innings, and Nottinghamshire are 110-6 in response. Derbyshire made 265, and Middlesex have also reached three figures, being 101-5 in response. Gloucestershire are 47-0 in response to Glamorgan’s 236. Northamptonshire were all out for 147 and Warwickshire are about to overhaul them, with wickets in hand. Finally, Sussex batting first are a comparatively monumental 304-7 (three batting bonus points, although they will not get a fourth as they have had 108.3 overs, and bonus points are only awarded in the first 110 overs of a team’s first innngs) against Leicestershire.

Every match is in progress, which beats last week, when Yorkshire failed to produce a playing surface on which the game could be played, resulting in their match against Essex being abandoned without a ball being bowled. 

The low scoring is a major problem – the batters will gave little confidence since they are not making runs, and as soon as they face conditions in which the ball does not get up to mischief most of the wicket-taking bowlers will revert to being their workaday selves (we saw, unforgettably for all the wrong reasons, over the winter how seamers who bowl accurately but not especially fast are cannon fodder for international class batsmen on good pitches). 

From the point of view of England possibles these two rounds of championship matches have been largely valueless – the 75 from James Vince on the opening day was the usual Vince fare – excellent while it lasted, but did not last long enough to be satisfactory and given the conditions no bowling figures can be taken with anything other than a substantial helping of salt.

PHOTOGRAPHS 1: AN ASPI.BLOG FIRST

The Muscovy ducks first saw a few months back are still in residence, and they have been joined by an unusual visitor, the second largest bird species I have seen in King’s Lynn – Canada Geese.

Muscovy ducks and Canada Geese
The white patch at the top front of the otherwise pure black neck (the head and bill are also pure black) is, along with the colossal size, the key identifier of these birds as Canada Geese.

Canada Goose and Muscovy ducksCGICGIICGIII

PUZZLE 1: MATCHSTICKS

My first offering from brilliant (the source of all of today’s puzzles – note also that all can be solved without even using pen and paper, never mind mechanical assistance – I did) is an exercise in visualization:

matchsticks

PHOTOGRAPHS 2: MUNTJAC

This muntjac was nibbling the grass on the playing field of the Lynn Academy, and I was taking pictures through a screen of plants:

Muntjac IMuntjac IIMuntjac IIIMuntjac IV

PUZZLE 2: CLEAR ICE

Clear Ice

PHOTOGRAPHS 3: SQUIRREL

I got two shots of this squirrel, one om the ground, and one as it swarmed up a tree trunk:

Squirrelsquirrel swarming up tree

PUZZLE 3: POLYOMINO

Another exercise in visualization (my own success with this one enabled me to celebrate what I call my brilliant.org Pi Day – 314 successive days on which I had solved at least one of their problems!):

Polyomino

PHOTOGRAPHS 4: SMALLER BIRDS

BlackbirdMoorhen on branchMagpieperching blackbird

PUZZLE 4: CONVERGENCE

Convergence

PHOTOGRAPHS 5: BUTTERFLIES

Small Tortoiseshell XIITwo Small TortoiseshellsTwo Small Tortoiseshells IISmall Tortoiseshell XIIITwo butterfliesSmall Tortoiseshell XIVPeacock Butterfly with closed wings

PUZZLE 5: CUBE

My own method for solving this one once again involved visualization, although other methods were also used.

Cube

In view of some of the moans that appeared on brilliant in relation to this problem please note the crucial words “by rotating” in the question – they are absolutely key.

PHOTOGRAPHS 6: THE REST

PollinatorSmall birdShy guinea pig

AFTERWORD

While I have been completing this post Durham have succeeded in making Kent bat again, though it is still massive odds against that game even making it onto the third of the scheduled four days.