Solutions to na couple of problems and a new problem for you to get your teeth into.
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:
Now here is an official solution, posted by Siva Budaraju:
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.
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:
A NEW PROBLEM
I finish by sharing another problem with you that I enjoyed solving:
A solution and two new problems by way of a quick post before setting off for Marxism 2018.
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 problem I posted earlier was this:
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:
The easier problem which generated some controversy is this one:
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:
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.
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.
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.
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:
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.
M IS FOR MATHEMATICS
Another lifelong interest, and something that I am very good at. Here is a frecnet problem from brilliant.orgthat took my fancy:
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.
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:
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.
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: A view from Strome Castle, Scotland
This 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 blogrun 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.
Notice of my imminent departure for a week’s holiday and solutions to Wednesday’s puzzles.
A couple of days ago I set some puzzlesfor 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
Here is the most elegant of the official solutions, posted by Callum Cassidy-Nolan:
SOLUTION TWO: 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):
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…
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.
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.
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:
A couple of easy problems on brilliant.org that have generated considerable controversy, and some butterfly pics of my own.
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:
PUZZLE TWO: FUEL TANK
Here it is:
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.
Some cricket related thoughts, photographs and solutions to my most recent set of problems.
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”.
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:
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.
J Bairstow, playing as a specialist batsman (he is plenty good enough to do so).
J Root (Captain)
D Malan – one of the few England test batsmen to be able to claim a successful tour of Australia and New Zealand.
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.
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.
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.
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.
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.
All of these problems were taken from brilliant. The first was:
First the answer:
Now here is Marvin Kalngan’s published solution:
SOLUTION 2: CLEAR ICE
Here is the problem:
And the answer:
SOLUTION 3: POLYOMINO
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:
SOLUTION 4: CONVERGENCE
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
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:
Some thoughts about the early stages of the English Cricket Season, some photographs and some puzzles.
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.
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:
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:
PUZZLE 2: 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:
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!):
PHOTOGRAPHS 4: SMALLER BIRDS
PUZZLE 4: CONVERGENCE
PHOTOGRAPHS 5: BUTTERFLIES
PUZZLE 5: CUBE
My own method for solving this one once again involved visualization, although other methods were also used.
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
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.