## 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:

## Autism Related Finds

Three autism related links including one featuring a seriously good infographic and one in which I get an honourable mention, and some photographs.

# INTRODUCTION

I was planning to do a major sharing post today, but other factors have intervened, so I am just going to share three autism related pieces.

# AUTISTIC ADULTS WANTED FOR ART PROJECT

This was posted on messymiscreation and is exactly what you might expect a piece titled “Autistic Adults, I’m seeking your input on an art project” to be. I hope it gets plenty fo responses.

# AN INFORGAPHIC ON DIVERSITY

This is an example of why Erin Human is a firm favourite of mine. This post has wonderful title: “Diversity is Beautiful“. Below is a screenshot of the feature infographic. I urge all of you to visit the original and read the accompanying text.

# NAS WEST NORFOLK OFFICIAL POST ABOUT THE TENTH ANNIVERSARY

Our vice-chair Rachel Meerwald with some input from the rest of the committee created a post for our website about this event, which you can see here. Below is a screenshot of the end of the piece (my reasons for choosing this section will be instantly apparent!).

# A FEW FINAL PHOTOGRAPHS

## Cars, Buses and Trains

My gloss on an excellent little fact sheet produced by George Monbiot.

# INTRODUCTION

This post was inspired by a fact sheet created by environmental campaigner George Monbiot which you can read in full by clicking the screenshot below:

# CARS

This short piece outlines some very valid objections to the over-use of cars. However, the pollution aspect of the problems caused by the over-use of cars (which in this country has reached scandalous proportions) is more properly a criticism of the fact that the vast majority of cars continued to be powered by the infernal combustion engine. There are many non-polluting means of powering vehicles available these days. Addressing the pollution issue however does not address the problem of congestion. To avoid misunderstandings: Monbiot’s fact sheet is bang on the money, and everyone should read it in full.

As an example of my own approach as a non-driver, here courtesy of google maps is a suggested walking route from my home to the scout hut on Beulah Street, which I quite often have cause to visit:

My usual choices of walking routes are actually longer than those recommended above because I prefer routes that spend less time around roads even if they take longer (see this post from yesterday for examples of two routes that I used on Saturday). There is a bus route that I could use if so inclined – there is a stop close to the Wootton Road end of Beulah Street but for a journey of this distance I positively prefer Shanks’ pony.

However, I freely acknowledge that while cars are over-used for short journeys there is another reason why there are far too many cars on British roads, and that leads to the next section of this post…

# BUSES AND TRAINS

British public transport is in a shocking state. There are many people, particularly in rural areas, who have no public transport options available to them, and even where there are public transport options they are overpriced and unreliable. It is only by creating a public transport system that works for those who use it that we can seriously reduce car usage.

I always like to include photographs in my posts, so to conclude this little post here is a shot of the front of King’s Lynn railway station: