Solutions to Good and Bad Mathematical Problems

Solutions to the problems I put up here on saturday and some new problems.

INTRODUCTION

On Saturday I produced a mixed bag of offerings from brilliant.org, promising to provide solutions in a later post, and now that time has arrived.

THE WORST SET PROBLEM I HAVE EVER ENCOUNTERED ON BRILLIANT

bad problem

I got nowhere close to solving this for reasons which will soon become obvious.

Here is the official “correct” answer:

solution to dreadful problem

My beef is with that diagram. The answer given works on the red section being the largest part of the parallelogram whereas the diagram shows it as the smallest, which was the basis on which I worked. This is way beyond a diagram being “not to scale”, which I have no great issue (the most famous of all such schematic diagrams, variations of which can now be seen representing transport systems in cities everywhere in the world is of course H C Beck’s London Underground diagram), but showing what is actually the lagrest single area of the diagram as the smallest is a bridge too far (Beck enlarged the central area so the stations were easier to see, but he did not actually make it cover a larger area than the surrounding suburbs, merely a less small area than was actaully the case, which to me is what if the approach is to have any validity is what “not to scale” should mean). The diagram in this question was literally worse than useless – wkith no diagram at all it would be have been a better question than it was with the actual diagram.

ANSWERS TO THE TWO FABULOUS PROBLEMS

First, what I now call the “Mendrin Circles Problem” after its creator:

Mendrin Circles

MSC.jpg

Here is Albert Lau’s published solution:

Lau solution

Second, the problem of Mr Mediocre’s Lawn:

First the answer:

Mr Mediocre

Jeremy Galvagni’s published solution was worthy of this splendid problem:

JeremyG

The ‘chessboard shading’ in this diagram is the key to the excellence of this solution – it rules out A and D, while B’s location rules it out, leaving only C as an option.

THE BONUS PROBLEM

Honey the ant

The answer:

Honey solution

Attaching the leash to an edge or a corner introduces restrictions which are not there if it is attached to the centre of a face. At full extent Honey can be diametrically opposite her starting position, which means that the entire surface of the cube is available to her.

A NEW PROBLEM

This is a very old problem indeed:

Fermat Challenge

 

A Grockle’s Eye View of Cornwall 12: Homeward Bound

Bringing my series on my visit to Cornwall to a close.

INTRODUCTION

All good things come to an end, even this series of blog posts, and with this post we do indeed reach the end of my account of my visit to Cornwall. 

THE JOURNEY HOME

With my train due to leave St Germans at 9:27AM on Monday July 16th we had decided that a departure from Fort Picklecombe at 8:30AM was warranted due to the fact that I had a reserved seat on that train, which was going all the way to London, and would have been in the region of £100 down had I missed it. We arrived early at the station, and once the train arrived I found my seat easily enough, although as had happened on the journey down they had reversed the running order of the train and I was facing against the direction of travel instead of with it (trust me, for a photographer this is quite significant). Here are the last of the Cornish pictures…

sea birdSt GermansScheduleRiverMarshy riverRiver with boardsYacht and buoyWaterside housesSaltash stationCrossing the Tamardetail from suspension bridgesuspension bridge and boatsTamar Bridge

The rest of this post takes place outside Cornwall. The train ran a trifle slow, arriving into Paddington sufficiently late to ensure that I would not make my intended connection at Kings Cross. However, the failings of Great Western were as nothing compared to those of Great Northern. The train arrived at Cambridge where it was supposed to divide in two as usual with four coaches going on to King’s Lynn and the rest going back to London, then a good couple of minutes later we got an announcement telling us that the split was not happening and that we needed to go to Platform 7. As a direct consequence of this mess up we then hit two red signals, at Waterbeach and again at Watlington, arriving into King’s Lynn much later than we should. This made it four successive journeys on Great Northern where the schedule had not been adhered to. Their failures appear to have got worse – just a few days ago I saw a special bus service running between King’s Lynn and Ely.

DevonportSouth Devon RailwayStationStation IIView from the trainNewton Abbot stationNewton AbbotNewton Abbot canopy and big buildingRiver near Newton AbbotRacecourselong bridgeLone boatAcross the waterAcross the water IIChurch towerAcross the water IIIBoats and buildingsBoats and hillsidesGrand HouseChurchButterflyriverside walksrural riverExeter St Davidstree and housesTiverton ParkwayTiverton Parkway IITauntonTrain, ExeterTrain, Bristol IIMetal curlicuesTrains, BristolBristol with trainsBristol Temple MeadsCleaning Britain's Railwaysbridge and pipebridge and pipe IIBath SpaBath Spa IIChurch tower IIBarge and churchBig churchcricket fieldRailway buildingGWR MuseumGWR Museum IISTEAM building

Tunnel trucks
These trucks look like tunnel segments on wheels.

SwindonPlatform detailGoods trainsingle trainReadingReading Stationtrains near ReadingLondon Underground train

Royal Oak
Westbourne Park, the penultimate Hammersmith & City line station west of Paddington – until quite recently local mainline train services stopped here as well. When the Metropolitan Railway opened in 1863 it used GWR rolling stock (broad guage in those days), and there were track connections between the two railways. This line to Hammersmith became part of the Metropolitan Railway in 1864, and between 1877 and 1910 services ran through to Richmond by way of a viaduct from Goldhawk Road the far end of which can still be obeserved at Ravenscourt Park. The last legacy of these connections is that today the Hammersmith and City platforms at Paddington are nos 15 and 16 of the main station.

Hammersmith and City line trainPaddington platforms

Interchange, Baker Street
The interchange from Circle and Hammersmith and City through to the four platforms that serve the Metropolitan line at this station (and escalators down another four platforms which serve the Bakerloo and Jubilee lines)

Screen

 

A Grockle’s Eye View of Cornwall 11: The End of the St Michaels Mount Day

Concluding my account of the day at St Michael’s Mount as we near the end of my series about my visit to Cornwall.

INTRODUCTION

Welcome to the latest installment in my series of posts about my recent visit to Cornwall. This post completes the day at St Michael’s Mount, leaving me with a post to do about the journey home and finally a page from which all the posts about this trip can be accessed. The fact that this will mean (including the page) 13 pieces relating to the trip bothers me not a jot – I have no more time for triskaidekaphobia than I do for any other ridiculous superstition.

WRAPPING UP ST MICHAEL’S MOUNT

Having finished our exploration of the mount itself it was time for lunch, which was excellent. The establishment at which we ate our lunch has a rule that alcoholic drinks can only be served if food is ordered at the same time, and according to their interpretation cream teas do not count as food, so on two occasions in the course of that meal we ordered portions of chips to go with drinks. Mention of cream teas (a speciality of the far west of England) brings me to a debate that rages unchecked: which goes on the scone first, the cream or the jam? The cream advocates argue that cream in this context is the equivalent of butter (and if it is Cornish clotted cream it is so thick that one can pretty much slice it like butter!), and that if you put the cream on first you do not get jam in it. I am not sure what the jam advocates base their case on.

Lunch consumed it was time to head back to our parking place on the edge of Penzance. The tide was just starting to turn but was still a long way out, and unlike the Mont St Michel, on which the current setup of St Michael’s Mount is modelled the tide here comes in slowly (no danger of galloping horses being swallowed by an inrushing tide, as allegedly happened at Mont St Michel on one famous occasion), so we were still able to walk back across a vast expanse of beach to rejoin the official footpath just west of Marazion. I omitted to remove my socks and shoes for this part of the journey, and they ended up thoroughly soaked, although by the end of the walk they had dried out again (without the sea breeze the heat would have been fiendish).

Mathematical Pic
The first few photos here, including these mathematical pictures are from lunch

Mathematical picsMathematical Pic IIMetal pictureThrough the windowCornish beer ICornish Beer IICornish Beer IIIBoilersdetail from bottleDetail from bottle IIPaintingLooking towards the mainland

Sea birds
Just before recrossing the causeway to the mainland, we saw these birds – the one with the red bill used for digging would have been having a bonanza – there were huge numbers of worm casts in the sand.
Causeway close-up II
The causeway from close up

Worm castReturning to MarazionLooking WestGWR works, near PenzanceBeach viewShell

Farewell to St Michaels Mount
A farewell shot of the mount

St Michael's Mount panorama

Chough
A shore dwelling corvid, the Chough

Good and Bad Mathematical Problems

Two splendid problems, one terrible one and one intermediate one, all courtesy of brilliant.org.

INTRODUCTION

This week on brilliant.org I have encountered two of the finest mathematical problems that I have seen on that site, and also unequivocally the worst. In the rest of this post I will share these problems with you – solutions and explanations will appear in a later post.

FIRST THE BAD NEWS

This is the terrible problem referred to in the introduction. In almost 15 months og being a brilliant.org solver I have complained about precisely one problem – this one:

bad problem

The “correct” answer when I give it will make it clear precisely why I was so annoyed by the setting of this problem.

TWO FABULOUS PROBLEMS

These two problems were beautifully set, very fun to solve, and one has a published solution that I consider to be quite superb. First:~
Mendrin Circles

I tackled this one this very morning:

Mr Mediocre question II

A BONUS PROBLEM

Can you give Honey the ant maximum possible access to the surface of the cube?

Honey the ant

A Grockle’s Eye View of Cornwall 10: The Map Room at St Michael’s Mount

Continuing my account of my visit to Cornwall. Today we look at the map room at St Michael’s Mount.

INTRODUCTION

Welcome to the latest post in my series about my visit to Cornwall. As part of my coverage of a day at St Michael’s Mount I am devoting a whole post to the map collection there.

A Collection of Antique Maps

For a cartophile such as me this section of the visit was particularly good fun…

Plymouth to Senan
In the days before googlemaps this is what a walking route looked like!

Exeter to TruroeWilliam Holes - Devon and Cornwall

Jan Janssons Cornwall
At one time the Dutch were trailblazers when it came to cartography.

two antique maps of CornwallJames Pigot's CornwallModelFour antique maps of CornwallNicolas Sansons West of England MapThe Isles of ScillyChristopher Saxton's CornwallMounts BayMap room mantelshelfMounts Bay IIThree views of cornish buildingsduoDuo IICottage PicSextetSextet II

Falmouth
Falmouth – one of the people who have noticed my posts about cornwall is thecornishbird, whose offerings include this about Falmouth: https://cornishbirdblog.com/2018/07/14/things-to-do-in-falmouth/

sextet IIIThree maps in one frameCornwall and Devonshiresix maps, two pictures

A Grockle’s Eye View of Cornwall 9: The Top of St Michael’s Mount and the Descent

Continuing my account of a visit to St Michael’s Mount.

INTRODUCTION

This is the third to last post about the day at St Michael’s Mount, and the ninth in my series about my summer visit to Cornwall. The next post will be all about the vast collection of antique maps that are on show here, and then a final post about the last stages of the day. I then have the journey home to cover to complete the series.

THE REMAINDER OF THE SUMMIT

At the end of the previous post in this series were about to head indoors for the second time in our exploration of St Michael’s Mount…

The mainlandStained glass VIIRoseCarving and stone inscriptionTrio of miniaturesShepherdEagle and windowangelthree in one frameSeptetRose II

Organ
This organ looks splendid.

Organ close-up 1Organ close-up 2DedicationAngel and windowMemorial and windowMemorials and windowPainting and windowTrio and window panelsText and windowThe second Lord St LevanArchitectOrgan and windowLiving roomPortrait XPortrait XIMantel clockbusts and windowCupid

At this point we entered the map room. As an appetiser for the next post I offer one picture from there…

Christopher Saxton's Cornwall

After the maps came a display featuring large amounts of weaponry…

remains of spiral staircase
The remains of a spiral staircase

hornDollArmour IgunsTruncheonstruncheons and gunsMore gunsWeapons VSpearsWeaponryWeaponry IIShield IIHorses Head

Cornish map with extras
A final map, separated from all the others.

girls armourbreastplates and helmetsGiant clock IClockfaceClock case18th Century mugMug provenanceUniformCapShoulder plate

That ended the indoor stuff until lunch time.

THE DESCENT

The descent takes past an emplacement of mini-cannons which are of French Revolutionary origin…

mini cannonsMini cannoncannon markings ICannon markings IICannon markings IIIMini cannon emplacementCannon markings IVCannon markings VCornish flagCannon markings VICauseway from the cannon emplacementFarewell to the cannons

Just beyond this I encountered a Red Admiral butterfly:

Red Admiral I
At first it had its wings folded making ID difficult…

Red Admiral II

Red Admiral III
…but when it opened them out there was little room for doubt.

Red Admiral IV

Not long after this we could see our next destination, where we would be having lunch.

Nearly back at sea levelsea level buildings

A Grockle’s Eye View of Cornwall 8: Ascending St Michael’s Mount

Continuing my account of my visit to Cornwall, with the ascent of St Michael’s Mount.

INTRODUCTION

Welcome to the latest post in my series about my stay in Cornwall. This post takes us up St Michael’s Mount and covers some of the stuff at the top. There will be at least two and possibly three more posts about the day.

BASE CAMP (!)

Among the places at ground level, before the ascent begins are the restaurant where we would be having lunch and a visitor’s centre which provides a comprehensive introduction. After these one passes through a field that contains a dairy cottage before the ascent begins.

Ships in the distanceSea ViewFind Your WayMapLooking back at the mainlandMuralIllyriaFootprintsMural PlaqueModel of the mountPlan the dayWelcome to St Michael's MountPicture of the mountTL1

TL2
No sops for creationists here! The straight truth about the formation of the granite of which the mount consists – it was formed longer before the dinosaurs roamed the earth than we are after the last dinosaurs disappeared.

TL3The castleThe GardensTL4Boat 1Boat 2Boat info boardThe DrakeAmphibious vehiclesDUKWBoat on the islandThe St MichaelOur venue for lunchMarazion from the IslandThe abbey from belowCreelsOutdoor artworkHutConservation notice

THE ASCENT BEGINS

The climb up to the buildings on top of the mount begins by way of The Pilgrims Steps, continues past the Giant’s Well and the Giant’s Heart and a cannon emplacement. Then comes the first indoor section and a roof terrace where we pause until the next post in this series…

The pilgrims stepsView across the sea

Giants Well
The Giant’s Well – from an old folk tale

Giants Well sign

The vicinity of the Giants Heart
And about 50 yards further along The Giant’s Heart.

The giants heart sign

The Giants Heart
The actual heart!

FrontageArcher's alcoveSea view from aboveSea view from above IISea view from above IIICannonsCannonMarking on cannonView from the cannon emplacementSea View from above IVThe main buildingDistant view of PenzanceYachts and a headlandPenzance from St Michael's Mounttwo yachtsThree yachts

Wall mounted militaria I
The first weapons display – in the next post you will see another one.

Swordsthree swordsGunsGunCrossbowJewellery displayCannon and ammogiant crestCrossed swords ICrossed swords IIDrummers kitPortrait 1Portrait 2Trio IMantelpiece and trioClockCentral UnitGrandfather clockCentral Unit IIBoyPortrait IVCorner unitChest IChest IITrio IILadyArtists representation of the mountMiniaturesquartetLady IIQuartet IIMiniatures IIMiniatures close-upFancy CabinetLiving spaceLibrary and game zoneChess boardChess board IIHeraldry display wallRoof patterning and friezechapelShieldsshields and ornamentsLong tablefrieze workstained glass Istained glass IIstained glass IIIsingle panelShields anf friezeshields and friezeRoof beamsAlcovewooden bas-reliefShieldSilverwareSilverware IISilverware IIIStained glass VStained glass VIShip panelstained glass womanTriple panelstained glass heraldic lionStained glass jester panelCircular stained glass panelsSilverware IVdecorated chestQuintetOval picturePicture of a Cornish gentFireplaceTrio IVbay window

Garden from above I
Us hoi polloi can only view these gardens from above – we never actually get into them.

A study in blueGarden from above IIGarden from above IIIGarden from above IV

Sundial
A very elaborate sundial.

Detail from sundial

Roof terrace
This where our next post will start from.