## 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

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

Here is the official “correct” answer:

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:

Here is Albert Lau’s published solution:

Second, the problem of Mr Mediocre’s Lawn:

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

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

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:

## 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…

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.

## 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).

## 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.

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:

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:~

I tackled this one this very morning:

# A BONUS PROBLEM

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

## 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…

## 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…

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

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

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…

Just beyond this I encountered a Red Admiral butterfly:

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

## 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.

# 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…