Tag: Solutions

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

 

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

 

 

 

Solutions (And New Problems)

Solutions to my .last set of problems and a new set. Also some photographs.

INTRODUCTION

It has been a few days since my last post, and the principal reason for this will made clear in my next post. Meantime I am starting proceedings for today by answering the questions I included in my previous post, and then setting a couple of new ones.

SOLUTION 1 – TRIANGLE CENTERS

Here is the original problem, from brilliant:

triprob

Here is the answer:

TriAnswer

To explain, here is Alex Warneke’s published solution (one of a number, but the one I like best)

Draw a circle around each figure. The circle drawn around each polygon has a larger circumference than the polygon and therefore a larger radius than the circle. If we consider the ratio of circumference of a circle to perimeter of inscribed regular n-gon we see it is bigger for smaller n and smallest for n = 3.

SOLUTION 2: EBBINGHAUS DISILLUSION

This was the problem:

Ebbinghaus

The two blue circles are exactly the same size. Here is an edited version of the above, deliberately clumsy so that my method of editing it can be seen by all:

Ebbinghaus Disillusion

PROBLEM 1: DECK OF CARDS

The first of two new problems from brilliant that I am sharing in this post:

Card problem

This is a multi-choice question, the possible answers being:

a) Less than 50%
b) More than 50%
c) Exactly 50%

PROBLEM 2: GROYNE

Again from brilliant:

Groyne Q

As the title of this section indicates I have identified a clear-cut mistake in the wording of the question – there may be room for doubt as to whether the indicated structure is a groyne or a jetty, but it is most definitely not a ‘breakwater’ – such would be entirely out at sea, not stretching from the land into the sea. 

PHOTOGRAPHS

MinsterMajestic Clock TurretGreyfriars Tower

Puzzles and Solutions

Some solutions and some new problems/ questions. Also details a thank you card for TFL and some photos.

INTRODUCTION

I have solutions/ answers to some problems and a few new problems for you. I was going to be blogging about my activities on Monday, Tuesday and Wednesday but that will have to wait until tomorrow now.

SOLUTIONS 1:
OPENING THE “DOOR OF DEATH”

In my post “Coming Up – A Trip To Cornwall“ I set this juicy little puzzle:

DoD

Each of the three filled in columns contains one large number and several smaller numbers. In each case the big number is the sum of all the smaller numbers – 6 = 3+2+1, 28 = 14 + 7 + 4 + 2 + 1 and 496 = 248 + 124 + 62 + 31 + 16 +8 + 4 + 2 + 1. In each case the smaller numbers listed below the large number turn out to be all of that number’s factors. A number that is equal to the sum of all its factors is called a perfect number. Looking closer still we that 6 = 3 x 2, 28 = 7 x 4 and 496 = 31 x 16. In each case these multiplications consist of a larger number that can be expressed in the form (2 ^ n)-1 and the smaller number is equal to 2 ^ (n-1). Further, the larger multiplier is in each case a prime number. Investigation reveals that the next prime number of the form (2 ^ n)-1 is 127, and the other multiplier must therefore by 64. Multiplying these two numbers gives 8,128. Thus the final panel will consist of 8128, with vertically below it the numbers 4064, 2032, 1016, 508, 254, 127, 64, 32, 16, 8, 4, 2 and 1. This was a tough puzzle but it would have been downright vicious had I made the door’s mechanism consist of five panels, these four and the fifth for you to work out – for a bonus can you explain why?

I also asked if anyone could identify to the real life door that I had used as the basis for the “Door of Death”. It is one of the doors to King’s Lynn Town Hall and in reality of course it is not remotely deadly (indeed barring dying of boredom during a council meeting there I can think of no risk of death anywhere in that building).

SOLUTIONS 2: GAYWOOD RIVER QUIZ

This one appeared in my post “The Gaywood River”. The answers are below:

Quiz - answers

NEW PROBLEMS

I have three problems for you. The first comes from Trivia Hive. Unfortunately I cannot present it to you in their format without giving away the answer, so instead I present in plain text:

In which country is Europe’s only desert located?

a)Italy
b)Greece
c)Poland
d)Spain

Puzzle number two comes by way of the twitter feed of estate agents Abbot Fox:

Puzzle

Can you reveal the street name?

My third and final problem comes courtesy of the mathematical website brilliant and sends you on a treasure hunt:

TH

A THANK YOU CARD FOR TFL

I was delighted to receive an email from campaigning organisation sumofus inviting me to sign a thank you card to TFL for having given Uber the boot. I have already shared this invitation on facebook and twitter and ‘pressed’ it to my London transport themed website. I now invite my followers here to add their names to this thank you card: 

TYTFL

PHOTOGRAPHS

These photographs were taken in Norwich…

Window

Mural 1
This is on the ‘ceiling’ portion of a covered passage near…
Castle
…the castle
Haart Map
This map is in the office of Haart in Norwich, and I could not resist trying to photograph it from the street.