**KNOW WHAT YOU NEED TO KNOW**

**Topic guide**

Either your school will have prepared one or the exam board have them on their website

Make sure you read it; anything you don’t understand you need to ask about

**Weak areas**

Find what you’re not good at and do something about it

Get help from your teacher or one of the websites listed below

**Strong areas**

Don’t ignore what you’re good at, it’s always nice to turn to a familiar topic

**PRACTISE WHAT YOU NEED TO KNOW**

**Best practice**

Work as if it is the real thing, without distractions; you can’t play Spotify in the exam room

Take care with your layout and give full working; method marks add up really quickly even if you don’t quite solve the questions

**Quality vs quantity**

Use of variety of sources so that you tackle a real range of question styles

Make sure that you are doing questions that are sufficiently challenging

**Use of support when stuck**

blind copying is not learning; help from friends and family is fine but make them explain why

If you are helping a friend, insist on showing them how to work it out

Highlight where you had help, help your teacher to help you; if they don’t know that you find the topic difficult they won’t do anything about it

**USE WHAT YOU KNOW YOU KNOW**

**If in doubt, do some maths**

Find the maths amongst the distractions

Look for cue words

Break the question down into simpler tasks

Don’t worry about tricky bits until you get to them

Look out for separate sub-questions that you can do without tackling an earlier tricky bit

**Use your time wisely**

Use the marks to tell you how long to take

Have a good go but don’t waste time

Do you understand/need to do the question?

Have you answered the actual question or have you just done some maths and need to do something with that value?

Checking: most pupils are rubbish at this!

Numbers – not just correct digits but correct info

Process – not just ‘add the add question’ but correct interpreting of the cue words

Maths – finally, you can check the actual maths. Look for common mistakes eg signs when multiplying with negatives, missing terms with brackets, plus c when integrating

**RESOURCES**

BBC Bitesize for GCSE – an oldy but goody

The Student Room – Forums and support for GCSE and A level

Khan Academy Math [US focus] – effectively GCSE and A level but with a US focus, lots of really good videos

SingingHedgehog – my website! Helppages and randomised worksheets with answers

Manga High – compete against classmates and the world up to GCSE but good for core skills at A level, best if school registers

nrich – weekly problems plus live puzzles to solve and explain up to A level.

Corbett Maths – GCSE and C1, daily 5 a day tasks plus many other good things

Plus Magazine – Plus magazine aimed at 16+ students with articles and puzzles

Mr Barton Maths – GCSE and A level, questions and videos and many other good things

Mr Carter Maths – GCSE comprehensive collection of differentiated tasks

Physics and Maths Tutor – collection of tiered A level papers

Twitter – questions from @ukmt, @corbettmaths; random stuff from @singinghedgehog

Geeky cartoons to enjoy:

XKCD – a webcomic of romance, sarcasm, math, and language

Spiked Math – a math comic dedicated to humor, educate and entertain the geek in you

SMBC – Saturday Morning Breakfast Club, cartoons about maths, science and life

]]>The first flowers were showing around the garden but primroses and the various bulbs are still some way off. The impending snow will do little to encourage them on!

**Jan 3rd**

Last year’s project, the Pollinators’ Bed, is well and truly established now, so this year’s blog will be a more general garden activity diary although there is an annual project too.

This year’s project is a large bed opposite the Pollinator and Black beds. The plan is to make a child-friendly zone with the colour, smell and structure of plants key attributes. There will also be different pathways, changes of levels and a water feature. Being attractive to pollinators will also be important.

The rather more seasonal weather this Christmas means that no flowers are out in either of the established beds. In the Pollinators’ Bed, last year’s Cerinthes do not seem to have survived or self-seeded and the Primroses are not yet ready however most of the plants seem well set and I have lost very little over the year; I am looking forward to the new mahonias flowering this spring. The Black garden is unsurprisingly sparce but again most plants look in good order. Team Dahlia and friends are well secured in the middle greenhouse as last winter although I will be looking to move most into larger pots for the coming year.

**Jan 10th**

I suffered a nervous breakdown in June 2015 and eighteen months later have not been able to return to teaching, a job that I love. I continue to struggle with both anxiety and low mood despite numerous changes of medication and hours of CBT therapy. The medication causes or exacerbates a variety of problems with most bodily functions impaired along with poor sleep, lucid dreaming and hallucinations. I have self harmed pretty regularly and had suicidal intentions more than once.

I found writing about my experiences of depression most distressing but have managed to distil some of my feelings and emotions through poetry and drawing. Although these are bleak pieces, I have found it helpful to recognise that other people suffer the same feelings and that you are not alone in the dark.

The author SJ Parris is in real life the journalist Stephanie Merritt. Her memoir The Devil Within, expresses with the considerable talent she possesses many similar events and feelings that I was not able to relate. It was a very moving, rewarding and ultimately uplifting read.

Zoë’s sketches bring to life the feelings that many have at dark times. The quickness of the drawings gives them an immediacy I do not have the skill to recreate. Showing maturity beyond her years, she uses her art as an outlet and release and inspired me to do likewise; I value her friendship very much.

I am very happy for people to use these to help or support in any appropriate way; I would ask that you credit me as author/artist and link back to the source ie here. Zoë’s work should be similarly credited with link to her DeviantArt page.

[All poems © Paul Harrison. Images © Paul Harrison, Zoë B]

**IMPOSTOR**

There’s an impostor living in my brain;

He seems the same but it isn’t me.

Ideas come so torpidly

And he can’t discern as lucidly, as I can,

The impostor in my brain.

There is an alien who has my voice;

He sounds the same but it isn’t me.

Vocabulary slips his grasp

And he doesn’t speak as fluently, as I do,

The alien with my voice.

There’s a charlatan dwelling in my head;

He appears the same but it isn’t me.

Tensions rise inexorably

And he cannot cope with complex tasks, as I can,

The charlatan in my head.

There is a masquerader, wears my face;

He looks the same but it isn’t me.

Organs do not work as well

And he cannot move as dextrously, as I can,

The masquerader with my face.

There’s a pretender looking through my eyes;

He scans the same but it isn’t me.

Hallucinations make him jump

And he cannot focus rapidly, as I can,

The pretender with my eyes.

There is a shyster lurking in my skull;

He reclines the same but it isn’t me.

Dreams recur remorselessly

And he cannot sleep as peacefully, as I can,

The shyster in my skull.

There is an interloper with my skin,

He moves the same but it isn’t me.

Pain is now his first resort

And he does not take care for himself, as I do,

The interloper with my skin.

**ME**

The me I knew is deep within.

My voice, my mind are chained and bound.

The stress, the strain I feel and taste,

Yet other senses are deceived,

As what I see can’t be believed.

And when at last I close my eyes

The endless torments then can start

With fiendish contests none can win,

Where waking brings but brief respite

From constant battle through the night.

Yet daytime does not always bring

Relief from terrors lurking deep.

The pleasures that I used to know

Have faded like a photograph

Of how life was when I would laugh.

And by my shoulder in the fight,

It’s hard to feel my loved ones there

Or friends who want to see me well;

For they can’t see and do not hear

The demons looming, crystal clear.

So inward looking I retreat,

Recede inside a silent shell.

And there, for future shame, dismay,

I sink to a perverse belief

That pain, chimeric, brings relief.

So who will search deep down inside

To find my essence in the dark,

Then coax it gently t’ward the light

And nurture back my very soul

To make me once more truly whole?

**SPARK**

Sitting on the precipice

Alone and all forlorn.

Hollow eyes stare nowhere through

The bleak and endless dark.

Brooding demons bubble up

And swamp you with their scorn.

Deep inside and far beyond

Look for that merest spark.

Let the darkness ebb and flow;

Suppressed, it comes reborn.

Pressure builds when you resist

The bite, allow the bark.

Midst those swirling stygian clouds

Amongst the gloom are borne

Moment’s friendship shared, not earned,

A touch, a kind remark.

Grant some good to coalesce

A face, a dew-drenched morn.

Beacons in the inky black

Faint on the canvas mark.

**KNOW WHAT YOU NEED TO KNOW**

**Topic guide **– issue it, refer to it, check understanding of it

**Weak areas **– individual/class, validity of homework vs classwork, support materials

**Strong areas **– individual/class, strengths give islands of confidence

**PRACTISE WHAT YOU NEED TO KNOW**

**Best practice **– use model answers, peer marking, test conditions, presentation

**Quality vs quantity **– variety of sources, breadth of tasks, multiple skills, options/extensions

**Use support of when stuck **– encourage good use of help, discourage copying, reward honesty, help your pupils to help themselves

**USE WHAT YOU KNOW YOU KNOW**

**Do some maths **– cue words, marks for follow-on errors, multi-task questions

**Make best use of your time **– mark schemes, ensure pupils have realistic expectations of topics and outcome, highlight standard errors, reward good exam technique: non-linear navigation, graphing/constructions for brain break, value of checking

**RESOURCES**

BBC Bitesize for GCSE – an oldy but goody

Khan Academy Math [US focus] – effectively GCSE and A level but with a US focus, lots of really good videos

SingingHedgehog – my website! Help pages and randomised worksheets with answers primarily KS3/4

The Student Room – Forums and support for GCSE and A level

Plus Magazine – Plus magazine aimed at 16+ students with articles and puzzles

nrich – weekly problems plus live puzzles to solve and explain upto A level. excellent scheme of work links for teachers

UKMT – Question of the day on twitter for GCSE and A level. Individual challenges, follow on rounds and team events. past papers full of good stuff can buy hard or electronic copies.

Manga High – compete against classmates and the world up to GCSE but good for core skills at A level, best if school registers

Hegarty Maths – online questions up to GCSE, needs the school to register and pay. Feedback to teacher looks really good.

My Maths – online questions for GCSE and A level, needs the school to register and pay.

Corbett Maths – GCSE and C1, daily 5 a day tasks plus many other good things

Mr Barton Maths – GCSE and A level, questions and videos and many other good things for students and teachers

Mr Carter Maths – GCSE comprehensive collection of differentiated tasks

Physics and Maths Tutor – collection of tiered A level papers

Tes Resources – GCSE and A level, a goldmine but needs time to find the nuggets

MEI – teacher resources and CPD for GCSE and A level, not just for their own schemes

FMSP Further Maths Support Programme – teacher resources and CPD for Further Maths

Underground Mathematics – brand new resource of rich tasks for A level from Cambridge University

Twitter – questions from @ukmt, @corbettmaths, teacher help from #mathschat, #mathscpdchat, @mathsjem, @nrich; random stuff from @singinghedgehog

Independent School 13+ 16+ Scholarships eg Oundle, KSC, Eton. Great source of sideways thinking problems

Geeky cartoons to enjoy:

XKCD – a webcomic of romance, sarcasm, math, and language

Spiked Math – a math comic dedicated to humor, educate and entertain the geek in you

SMBC – Saturday Morning Breakfast Club, cartoons about maths, science and life

]]>1st November 2016 or T minus 271: Track; Long Sprints – 2 sets of 180, 200, 220 walk back as rest, 5 mins between sets. Tight top left hamstring, sore calves.

T minus 270: Gym; Upper body circuit – 3 sets of 8 reps of 6 exercises. Back right lower intercostal sore.

T minus 269: Track; Running Through the Board part 3 – bounding from 9m board step-step-jump then hop-step-jump from 7m board short run up. Calves suing for divorce!

*need to work on knee parallel to ground in all phases*

T minus 266: Gym; 5x5x5 – 2 upper body and 1 core to failure, 2 unweighted leg.

T minus 265: Track(indoors); Bounds and Jumps – bounding drills; short run jumping exercises [focus on high knee] step – hop – jump, hop – hop – step – jump, hop – step – jump.

T minus 263: Gym; Complex training – drop and hold to low box to hurdle, box jumps, squats to 100kg plus upper body and core.

T minus 262: Track; Shorter sprints and bounding. Tight top left hamstring.

]]>I had the pleasure of watching Christian Taylor win gold in 2012; there was other stuff going on too, some Bolt chappy and a lot of fuss and nonsense about an 800m race but I had come to watch the jumpers.

I intend to keep a diary of my road to Aarhus and also give tips and ideas on training, diet and other things helpful to triple jumpers and athletes of all ages. To be strictly accurate, the title should say road back to Aarhus as I visited the town and area on holiday with my parents more than forty years ago. However, from a sporting perspective, I have never competed there so in that sense it will be a new experience.

I have just passed my 50th birthday and so move into a new age category. Veteran or Masters athletics works in 5 year age brackets so at the end of this season I could have been be competing against men nearly five years younger than me. Next season I will be one of the younger men in the 50-54 section.

Some of the team events have even wider age ranges and there too I have finally stepped out of the 35-49 section into the 50-59 group.

Here I am competing for my club in the regional finals; the two men in the background were both still in their thirties – hardly fair!

Targets for the year

Base targets: TJ 11.50m, win medals at British Masters Indoors and Outdoors, compete in European Master Outdoors.

Extension targets: TJ 11.83m [minimum distance for all time rankings aged 50-54] – 12m+, be champion at British Masters Indoors and Outdoors, win medal in European Master Outdoors.

]]>I was very pleased to discover recently that the shape formed from a net I like using for more advanced constructions has a proper name, a “Yangma”. I would normally require pupils to create the net using mathematical instruments then fold up to form a square-based pyramid with the apex above a right-angled corner. I also found out that this was first described in a Chinese book of mathematics called The Nine Chapters on the Mathematical Art or Jiuzhang Suanshu, an anonymous collection of writing from between two and three thousand years ago.

“Now that’s all very interesting,” you might say, “but what use is it?”

Well, if we were to use a set of nets, such as the one on my website, it is possible to make three congruent (identical in all aspects) yangma. They can then be arranged such that they form a cube.

“Okay you’ve made a cube,” you now say, “So what?”

Again, a fair response; time for an anecdote.

In my last job I had the pleasure and challenge of teaching children from various countries across Europe and beyond. Many of the children were a little ahead of their British counterparts at Arithmetic and Algebra but they had done little or no work on several topics, most relevant to this tale being constructions and shapes. A few years ago I had a lower ability set of 12/13 year old children which contained a Spaniard and a Russian. Their computational skills were much stronger than their peers but they struggled to apply their skills in context and had never seen a net before. Whilst the rest of the class were working on some algebraic tasks, I worked with the two boys to build some nets. We started with a cube and built up the difficulty until we each made a yangma. Dima, the Russian boy, was by now getting frustrated with his inability to make the shapes, as he prided himself on his mathematical prowess, with some justification. An ever growing mass of paper and glue was developing in the bin; it is a good job that my knowledge of Russian is limited to hello, goodbye and a few other random words as the crumpled shapes were being accompanied into the bin by a torrent of abuse that I suspect was not entirely appropriate! Finally we created the three yangma and put them together as in the picture above. Immediately his eyes lit up and through a growing smile of comprehension he said, “Ah, is why one third!”

“Very touching story,” you comment, “But **what **is one third?”

As any fule kno [younger readers who have not come across Molesworth need to enrich their lives immediately!], and Dima instantly recalled, the formula for the volume of a pyramid is, If we call the length of one side of our cube *a* then this gives us,

“Ah now that **is** interesting,” you concede, “But most pyramids are not like that, so how is the formula still the same?”

This is an excellent question and allows us to employ a valuable mathematical tool. Firstly, we consider our shape to be made of a series of increasing squares so as such a stepped yangma.

Next we start at the top and shift each layer diagonally across one unit, so it will end up like a stepped Egyptian pyramid.

Finally we repeat the exercise with an increasing number of squares with a decreasing difference in their size so that as the number of squares tends towards infinity, the difference between each one tends to zero and we get a pair of smooth-sided square-based pyramids of equal volume, one with the apex over a base corner the other with the apex over the centre of the base. Thus any square-based pyramid can be shown to have the same volume as the yangma and therefore a third of the volume of the appropriate cube.

“Actually, that was rather good but what if the base isn’t square, say it’s a rectangle. Now your three shapes thing won’t work.”

Ooh, good question but actually you are wrong! Here is a cuboid made from three different yangma. [again, the net is available on my website!]

Assume for a moment that the sides have length 3, 4 and 5. Our yangma will be as follows:

Base | Height | Base x Height | Volume |
---|---|---|---|

3 x 4 | 5 | 3 x 4 x 5 | 20 |

3 x 5 | 4 | 3 x 5 x 4 | 20 |

4 x 5 | 3 | 4 x 5 x 3 | 20 |

Although the shapes are now not congruent, they will still have the same volume which is one third of the whole cuboid so the formula holds true.

“Hmm”, you think for a moment.” What about triangular-based pyramids. They won’t make a cube.”

No indeed, but three of them will make a triangular prism which the Greek mathematician Euclid described in his Elements written about 300BC. Perhaps you might want to go away and work that one out for yourself?

[The Teachers’ Page on my website has links to all the nets and lesson notes as well as other interesting investigations to try]

]]>A quick rummage on the keyboard confirmed his find and that actually the basic tunes were very similar. A quick moment in trusty old Sibelius helped to make it clearer; I halved the note lengths in the second tune to get them to match better.

Although I was supposedly working on a maths problem for Rob, this appealed as a quick, fun challenge so I set to.

I first put together the words of both songs and sorted out a melange that fitted the Kurt Weill tune. I then found a decent karaoke version of the Bobby Darin arrangement on YouTube and ripped that to an mp3 file. Next I dropped that into Audacity, an excellent open source multi-track program, and recorded the song with the new words.

I use a Samson Q1U, a venerable but still excellent USB microphone. The pop screen is an invaluable addition and resting the mini tripod on some bubble wrap takes away the vibrations from the computer fan!

Finally, back to you tube for a collection of Postman Pat clips which I ripped to avi files. These were then arranged over the sound track in the old version of Windows Moviemaker, which is free and ideal for simple video editing projects.

All that was left to do was upload it to YouTube and let Rob know I was done.

Postman Pat … Postman Pat,

Postman Pat and … his black and white cat.

Early in the morning,

Just as day is dawning,

He picks up all the … postbags in his van

Postman Pat … Postman Pat

Postman Pat and … his black and white cat.

All the birds are singing,

And the day is just beginning,

Pat feels he’s a … a really happy man.

On the pavement … Monday morning, don’t ya know,

Lies a postbag … big and fat.

There’s someone driving … around the corner,

Could that be our boy … Postman Pat?

Ev’rybody knows his … bright red van.

All his friends will smile as … as he waves to greet ’em.

Maybe you can never … be sure there’ll be a ring

or a knock, knock … letters through your door?

Postman Pat, … Postman Pat,

Postman Pat and … and his fluffy little cat.

All the birds are singing,

And the day is just beginning,

Pat feels he’s a … a really happy man.

Farmer Thompson … Old Missus Goggins,

Reverend Timms and Ted Glen,

Yeah the parcels … will be right babe

Now that Pattie’s … back in town.

I said Farmer Thompson … Old Missus Goggins,

Look out Reverend Timms and old Ted Glen,

Yeah the parcels … they will be right babe

Now that Pattie’s______ back in town ______

Look out old Jessie is back!

As a lover of Lepidoptera there is nothing more exasperating than walking down a leafy lane with a group of friends and someone says, “Oh, what was that butterfly? It looked sort of purpley,” but by the time you have turned around, the hoped-for Emperor, if that is what it was, has fluttered away on the breeze. As you strain your eyes, scouring the canopy, surrounded by the sounds of the birds and the bees, you wish that you could have heard the thing, then you might have spotted it too.

Well your wish is about to come true. Here at Singing Hedgehog Industries, we are proud to announce Flutterby 1.0, an App for Apple^{®} or Android^{®} devices. We have harnessed the latest research into butterfly sounds and wing beat frequencies using results published at Bristol University ^{[1] }, University of Florida ^{[2] } and Carleton University, Ottawa ^{[3] }, and combined that with 3D sound technology from Surrey University ^{[4] }. Just pop on your favourite headphones, the app is Bluetooth^{®} compatible, and you will ‘hear’ the butterflies flutter by.

For those who cannot tell their Small White from their Green-Veined White, or if you are unable to see the underwing of a Brown Argus or female Common Blue, the next step is just for you. We are already working on Flutterby 2.0, with the latest imaging developments from the Biology department of Union College, Schenectady ^{[5] }. You will simply point the device’s camera at the insect and the software will analyse the flight characteristics of the butterfly and return the species name either on the screen or via a digital voice into your headphones.

UPDATE FOR 2017

As part of the development of Flutterby 2.0 we are pleased to confirm that you will be able to link directly with your iRecord account so that all of your findings are automatically logged using the device’s GPS to verify the location.

Richard Fox from Butterfly Conservation said, “I can’t believe the Flutterby App does all this,” and you won’t believe it either! Coming not very soon to an App Store near you.

References:

1. http://www.livescience.com/5814-butterfly-wing-ears-detect-birds.html

2. http://lee.ifas.ufl.edu/Hort/GardenPubsAZ/ButterfliesTalk.pdf

3. http://www.ncbi.nlm.nih.gov/pubmed/11076733

4. http://www.surrey.ac.uk/cvssp/research/3d_audio/index.htm

]]>It is party time *chez* Eastaway as one of the nation’s favourite mathematicians Rob and his wife have invited six people round for dinner. The desserts have been cleared and as the quests settle back with coffee and liqueurs, they turn to each other and embark on long conversations with the person to their left or right.

Lulled by the flickering candlelight, Rob idly muses that he has three possible outcomes: chat to the redhead on his right, be talked at by the stern brunette to his left or have no one to converse with all and end up a ‘gooseberry’ in the middle.

He snaps out of his reverie and, perceiving that he has been a little slow on the uptake, turns to his right to find the redhead is having a very earnest conversation with the nice maths teacher about the astonishing life cycle of Large Blue butterflies and their symbiotic relationship with ants.

Turning to his left, the brunette is lecturing her new acquaintance on the benefits of the Alexander technique to an athlete like him especially when competing. Realising that the lovely vase of flowers his wife put out as a centrepiece actually prevents conversation across the table, he puts his mind to what the chances of ending up as a ‘gooseberry’ would be.

He soon works out that the problem has too many variables for the napkin he is writing on and, encouraged by the loud tut from his other half who it appears has also ended up a ‘gooseberry’, stops defacing the linen and heads for the dishwasher.

The next day, Rob receives an email from Mr H, the nice maths teacher, asking if there is anything he can tackle while on a health-related sabbatical. Deciding that Mr H owes him a favour, he sets him to work on the problem.

**Solving the Problem**

I started by trying a few ideas in Excel and soon realised that 8 people was going to be a very complex task! I then used a key mathematical technique which was to reduce the problem to a simpler version, solve that, then extend that solution to the larger puzzle.

**Two, Three and Four People**

These simplest versions are trivial so I shall leave them for you the reader to deduce.

**Five People** [Yes, a pentagonal-ish Lego table; I was pleased!]

This is the first version where the problem gets interesting and where I soon hit an issue with the base logic I was using. Since there are five people and each person can turn left or right there are 2^{5} or 32 possible permutations with everyone having turned once. I used decimal to binary to create the five digit values; with 0 for left and 1 for right; note that the binary term is calculated on one less than the permutation number. The orientations of the dinner guests are then created by selecting the digits in order using the MID string function and setting left for zero and right for one. Here are the first eight rows:

Perm | Binary | 1 | 2 | 3 | 4 | 5 |

1 | 00000 | L | L | L | L | L |

2 | 00001 | L | L | L | L | R |

3 | 00010 | L | L | L | R | L |

4 | 00011 | L | L | L | R | R |

5 | 00100 | L | L | R | L | L |

6 | 00101 | L | L | R | L | R |

7 | 00110 | L | L | R | R | L |

8 | 00111 | L | L | R | R | R |

If we consider permutation 28 then 27_{10} = 11011_{2} becomes right, right, left, right, right; for the purposes of these examples I have taken the directions to be **as we see them** and set the people out in a line. I then tested for matching adjacent pairs so here Pilot Penn and Luscious Liz are facing each other and start conversing.

The others then have to change direction to see if they can get into a conversing pair. However, without additional strictures in place you end up with people being like windscreen wipers going left and right in sync, never coming face to face!` `

Putting the problem back into real life provided the solution to this issue. If it is obvious the pair on the left are stuck in deep conversation you would look right and stay looking right. In our example, Lady Alexander does not need to look left again so will very soon talk at Mr H, leaving Rob as the ‘gooseberry’. This solved most of the issue but there was one group of arrangements that needed further thinking.

**Three in a row**

If we now take as an example permutation 8 so that we get 7_{10} = 00111_{2} giving left, left, right, right, right. We have to remember that persons 1 and 5 are also sat next to each other at the round table so Rob and Mr H can talk about a common interest, like the inability of English batsmen to build an innings. Pilot Penn now has a converser to the left so he will turn right. Similarly, Lady Alexander will now turn to the left. Luscious Liz now has the casting vote as to who become the ‘gooseberry’. This means in the simulation we have to introduce a random variable so that each permutation does not necessarily derive the some outcome each time. This will apply for every combination that starts with one pair and three potential ‘gooseberries’. Our original thirty-two possibilities will clearly escalate but this is still a manageable problem to account for every possible outcome, if rather time consuming even using Excel, and confirm the not unexpected probability of being a ‘gooseberry’ is a fifth or:

**p(g _{5})=0.2**

Evidently, [Rob will no doubt enjoy the Fronted Adverbial here!] as we increase the number of people round the table, the potential for groups of undecided guests increases. Thus with the original problem, the worst case scenario has just one conversing pair, two adjacent guests who will turn away and four in the middle who have to be set randomly. At this point the large number of possible outcomes means that a more flexible approach is needed.

**The Monte Carlo Method**

For problems when it is difficult to calculate all the possible outcomes we can compute a large number of randomised solutions and average them. The Monte Carlo method was first formalised by Stanislaw Ulam and John von Neumann when working on behaviour of sub atomic particles. Being secret work, they used a code name to describe the process, which referred to the casino in Monaco. The implementation in this instance works as follows: I used a simple spreadsheet in Excel to test for converser or ‘gooseberry’ and a short VBA program to run the simulation. The seed for the binary code is generated randomly and the number of guests set for that worksheet. The number of loops required and a seed value for the probability are input. After testing for the initial arrangement there is a stack of move and status pairs so that however unusual the sequence of initial positions and changes, the guests will settle to a stable configuration. The step number indicates which pair that is, so that the probability for the arrangement can be stored and used for calculation.

Here is the VBA program which is started by pressing the large button, obviously!

Sub MonteCarlo()

‘*declare the variables*

Dim TheLoop As Long

Dim LoopNumber As Long

Dim ProbNumber As Double

Dim ProbAverage As Double

Dim ProbRow As Integer

Dim ProbColumn As Integer

Dim StepNumber As Integer

Dim Guests As Integer

Dim ResultsAverage() As Double

Dim ResultsNumber() As Double

*‘set initial values of required loops, probability seed and number of guests*

LoopNumber = Range(“D2”).Value

ProbAverage = Range(“B6”).Value

Guests = Range(“C2”).Value

*‘clear storage area and set up arrays for the running average and separate probabilities*

Range(“Values!A:C”) = “”

ReDim ResultsAverage(1 To LoopNumber) As Double

ReDim ResultsNumber(1 To LoopNumber) As Double

*‘main program loop*

For TheLoop = 1 To LoopNumber

*‘display current values and generate new binary seed*

Range(“A6”).Value = TheLoop

Range(“C6”).Value = ProbAverage

Calculate

*‘see where all guests are in fixed position and set cell values for required p(g)*

StepNumber = Cells(4, 11 + Guests).Value

ProbRow = 4 + 2 * StepNumber *‘moves down to first fixed position*

ProbColumn = 10 + Guests *‘allows code to run for different numbers of guests*

*‘find p(g) for that arrangement and calculate new average probability*

ProbNumber = Cells(ProbRow, ProbColumn).Value

ProbAverage = (ProbAverage * (TheLoop) + ProbNumber) / (TheLoop + 1)

*‘store values in the storage array*

ResultsNumber(TheLoop) = ProbNumber

ResultsAverage(TheLoop) = ProbAverage

*‘go again!*

Next TheLoop

*‘fill the storage area with the results*

For TheLoop = 1 To LoopNumber

Sheets(“Values”).Cells(TheLoop, “A”) = TheLoop

Sheets(“Values”).Cells(TheLoop, “B”) = ResultsNumber(TheLoop)

Sheets(“Values”).Cells(TheLoop, “C”) = ResultsAverage(TheLoop)

Next TheLoop

End Sub

The process on my old XP powered PC was rather slow due mainly to the refreshing of the guests each time. However, it did allow me to watch the change in the figures diminish over time as they approached the true value. Reducing the visible window to just the output values helped increase the speed significantly.

**Six People**

There are two possible outcomes: all six in conversation or two pairs and two ‘gooseberries’ the latter in several permutations. We should then expect the value to be some way below one third as the groups of three depreciate the figure from the sets of two and two. I was slightly surprised at how much more often the set of three conversations appeared than the two and two arrangement.

**p(g _{6})=0.0998**

**Seven People [Yes, a heptagonal table!]**

This was a good test of the logic built into the spreadsheet and the code. There can only be one value for p(g) each time as the guests can not be arranged other than three pairs and one ‘gooseberry’. Running simulations with seeds of 0.144 and 0.142 showed the value settle to one seventh to six decimal places within a little over a thousand steps.

**p(g _{7})=0.142857**

**Eight People**

Finally I felt ready to tackle the actual problem set! There are two possible stable configurations: four conversations or three pairs and two ‘gooseberries’ of which there are a number of permutations. We would thus expect a value some way below one quarter as the sets of four diminish the value from the sets of three and two, as with six guests.

**p(g _{8})=0.1283**

Even with a million loops the final value is only stable to three decimal places. However I am reasonably confident in the value expressed above. This has been an enjoyable challenge; I hope that I have shown that something as esoteric to most people as the Monte Carlo method can be applied to a simple model of a real life situation and be implemented in a straightforward manner with Excel, plus a touch of VBA.

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