The hind legs of a panto horse?

Photographic Quiz

Photo no. ??:- What is the relevance of this medal from another club to Coventry Godiva Harriers? Why did it take nearly a decade for Godiva to realise that they had missed the boat?

 Dear Blog,

            Perhaps we could trudge the Gert North Run together? 

                                                                             Colin

Temptation

Dear Blog,
               5 places ...... I'm tempted.
                                                     Colin

BUPA GREAT NORTH RUN 2012 /entry

Decision time for you Blog. Big decision. Twelve months ago plus a week and a few days, we started corresponding because you wanted to become my friend and help me raise money for the two charities for disable children, Newlife and Tiny Tims Children’s Centre. And a good friend you have been. I suffered for four hours in the London Marathon to raise a few quids for the two charities for disable children, Newlife and Tiny Tims Children’s Centre. We have discussed the possibility of you starting to run / hobble/ trudge, haven’t we Blog? Well here is an incentive. An incentive which is irresistible! Tiny Tims Children’s Centre has five places in the GREAT NORTH RUN in the Autumn. Five. F  I  V  E. Get on your shoes Blog and get cracking. If you want a place to raise some quids for Tiny Tims Children’s Centre, get in touch with me like yesterday. If you have another friend besides me who wants a place to raise some quids for Tiny Tims Children’s Centre, get in touch with me like yesterday. If you have a friend of a friend who wants a place to raise some quids for Tiny Tims Children’s Centre, get in touch with me like yesterday. If you have a friend of a friend of a friendⁿ (where n>1) who wants a place to raise some quids for Tiny Tims Children’s Centre, get in touch with me like yesterday. Spread the word, preach the gospel, go forth into the country and let the people know. I am serious Blog, GET IN TOUCH!!!!!!!!!!!!!!!!!!!!!! FIVE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Question 1:- What is the very most annoying aspect of trudging? Answers on the back of a cheque made out to ‘Tiny Tims Children’s Centre’.

Here’s one for starters….. the most annoying thing about trudging is seeing one end of the waist chord of your tracksters disappear into the waist band as you put them on just before going out for a trudge.

Answer 1:-Do you search for another pair if you have enough quids in the bank to own two pairs?

Answer 2:-Do you hunt for a safety pin to rethread the chord before you go out?

Answer 3:-Do you use the said safety pin to put a tuck in your tracksters and look a right prat as you trudge along?

Answer 4:-Do you go out without tracksters and get your goolies frozen so they snap off in the cold?

Answer 5:-Do you write to me for a place on The Great North Run to raise money for the charity for disable children, Tiny Tims Children’s Centre?

Question 2:- Where do the companies who make trackster bottoms put such short tie cords in the waist band? What lengths should they go to, to hold the tracksters up? Answers on the back of a cheque made out to ‘Tiny Tims Children’s Centre’.

I’ll tell you in my next letter to you Blog, what happened on this morning’s trudge. Haven’t time now, I have got to rethread these blessed tracksters before I do my afternoon weights session.

                                         Colin

PS If you want some help with coaching Blog, let me know at the same time.

National Cross Country Championships at parliament Hill Field, London 2012. Result?

Photograph Quiz:

Photo no. 84:-Where is this pavilion? WHO is the bloke WHO is sixth from the left WHO won the Inter Counties Cross Country Championships and WHO ran in more than one International Cross Country Championship. One of Coventry’s finest, nay priceless, runners. Because you are so hopeless at these quizzes Blog, I will give you a clue   … WHO wrote the rock opera named after him?

 Dear Blog,

The National Cross Country Championships at Parliament Hill Fields, London are on tomorrow,  Saturday; are you going to watch Blog, or are you going to run and get your commemorative pin? I think the first senior National I ran was at Norwich. I was at the Pink Panther University at the time … and it was a long long way to Norwich. It was decided to do the journey in two stages on the way down South and East. So we set off on Friday night, stopped over in a guest house on the Friday evening in Doncaster, completing the trip on the Saturday morning. I cannot remember too much about the race or the weekend; if you are slightly interested, Blog, I will look it up for you. I do remember the leisurely breakfast and leaving about 10am to get to Norwich in good time for the senior race. We continued down the main A1 road from Doncaster. For some reason, after an three quarters of an hour or so, some bright sparked asked where Merlin was; he wasn’t on the coach!!! He shared a room with *** and he was still in bed when *** went down for breakfast. The penny didn’t take too long to drop, and we soon worked out that we must have left him at the guest house in Doncaster; we were university students, weren’t we??? We had left him at the guest house, believe it or not Blog. We had left him behind. We needed him for the team. We did not have the time to travel back for him. What to do? We needed him for the team. No mobile phones in those days Blog!! Problem. The phone number of the guest house was produced, so all we needed was a phone box. A phone box on the A1?? We did find a phone box near the main road. A phone box Blog, is like a mobile phone but about the size of a small cupboard, three sides of which are panes of glass, the whole contraption being painted pillar box red. A pillar box Blog, is a red painted cylinder with a hole near the top; they were planted at regular intervals at the road side. They were used to post letters in. A letter Blog, is an e-mail written on paper and sealed inside another piece of paper so no one else can read it until the seal is broken open. Now this may disgust you Blog, but remember it was a long time ago; to seal the piece of paper Blog, you had to lick it. Disgusting I know, but nevertheless true!  So this phone box Blog, which is like a mobile phone about the size of a small cupboard, three sides of which are panes of glass, the whole contraption being painted pillar box red, is used for sending talking messages, but you have to go inside the box to talk the message and pay for the talking message by inserting quids into a slot. So we send this talking message back to Doncaster. The owner of the guest house kindly agreed to bring the missing Merlin to us on the A1, at the side of which our mini bus was parked up.

So what did we do while we waited? The A1 was the main arterial north / south highway in England, Blog. No hard shoulders in those days Blog. No central reservation in those days Blog. Not much traffic in those days Blog. What a brilliant idea …. the next 40 odd minutes was spent waving and applauding every third car that passed us while we stood at the side of the A1 which was the main North / South route. Blog, I kid you not. Well actually I might have told you a tinsey winsey white lie … it might have been every fourth car; come on Blog, it WAS a long time ago.

Marathon training tip:- if you get a sniffle these days, take a pill, go to bed for a couple of weeks and recover. Alternative remedy, tried and tested and used by countless runners years ago:- get a plastic dustbin bag, cut holes in same for head and arms. Wear next to skin, apply plenty of other running kit including hat and gloves. Go for a long slow run and   S    W    E    A    T        I    T        O    U    T!!! Come back, have a shower, go to bed. Get up next day as right as rain or as wright as reign. Blog, I am not joking.

Tuesday’s trudge was to cure a chesty cold. (cf above) It was very dark, no moon? It was very still and very black. To accommodate my chest, I went for a long slow trudge over the fields in the pitch dark. I’ve said to you before that going for a relaxed trudge can subconsciously solve problems or produce new ideas, haven’t I Blog?? Well ………..a poem was coming together in my mind; the dark, the quiet, the stillness all mixing with the trickling sweat, so a another poem by Colin …………. my reverie was suddenly disturbed by a dog wearing a head torch. Blog I kid you not. A dog wearing a head torch. Barking mad if you ask me? Well back to this poem; Coleridge may have been distracted by a Royal Mail delivery postman, but the dog did not have the same effect on me; I remembered my poem. Having suffered at the hands of Billy Boy while at school, having to learn a sonnet or two by him by heart, I thought I would compose a sonnet in his style. So Blog, here it is in true fourteen lined iambic pentominoes. You have SILENCE by Colin.

,

,

,

.

,

.

,

.

,

,

,

.

,

.

Even if I say it myself, I think those last two lines are quite brilliant, don’t you Blog?? They took a bit of recalling after I was startled by the dog, I must admit, but there was no way was I going to do a Kubla.

See you on the hill above the swimming baths tomorrow Blog.

                           Colin               

Washday flues.

Photograph Quiz:

Photo no. 83:- What is this object? What is special about it athletically and commercially?

Monday morning, freezing cold, I get back from my trudge, wanting a quick shower and my breakfast AND WHAT DO I GET? Would I just hang the washing out, it won’t take a minute. Normally, I would not be asked to hang out the washing on the washing line because I allegedly, cannot do it correctly. How can you hang out towels on a washing line incorrectly? That does take some doing!! Apparently I can do it wrong so in normal circumstances I am not allowed to hang the washing out on the washing line. Sweating from my trudge, sweat trickling down my face, wet through with sweat, the strong possibility of catching man flu from catching a chill in the freezing cold hanging out washing; but I have to hang out the washing. Monday is Monday is wash day. Monday is washday now, has always been so and will be so forever more. If you took careful note of my last message to you Blog, you will have immediately recognised the inductive reasoning therein. Well done. The sun rises in the East and sets in the West, has always done so and will do so forever more. If you took careful note of my last message to you Blog, you will have immediately recognised the inductive reasoning therein. Now because I was trusted with this important task, I have to confess to you that I do enjoy the satisfaction it gives. Red items are pegged with red pegs, white items with white pegs, green coloured clothing with green pegs and so on and so forth. It is a little indulgence I allow myself. The finished line of clothes gives a deep feeling of a job well done with all the colour coordination. But today was just too much. It may have been the cooling effect of the sweat on my tired body from my trudge or it may have been the first signs of the onset of man flu from not having a warm shower after my trudge, but when I saw all the yellow items that were in the good-for-life wash bag, I knew that my wife had over stepped the mark. Never let it be said that I am not a very tolerant, caring husband, but too may yellow items in the good-for-life wash bag and not enough yellow pegs in the peg bag is really taking the biscuit. I tried to control myself, but there is only so much a man can take, and not enough coloured pegs to coordinate my wash line on a Monday morning when I was all hot and sweaty from my early morning trudge was enough to tip me over the edge. I stormed up the garden path, through the orchard, across the vegetable patch, passed the large greenhouse, around the polytunnel, across the paddock into the kitchen to give my wife a piece of my mind. Normally I am calm and collected, but not today! Not enough yellow pegs indeed. Doesn’t she plan her shopping list and make a note of when a shortage might arise? I always make sure the wine cellar is stocked and I cannot remember the last time we ran short of a can of beer. Organisation. Simple. So where was the required number of yellow pegs? The yellow pillow cases hanging on the clothes line with red pegs simply looked ridiculous. Stupid. Same with my yellow socks. What would the neighbours have to say about that? I would be the laughing stock of the village if it got out. So I was at boiling point as I stormed into the kitchen. I had all my facts ready. I was going to tell her that she had to budget her housekeeping allowance more carefully and buy more yellow pegs or she simply had to be more careful about which coloured items she washed together to accommodate the various coloured pegs we had. I was going to pull no punches. I was going to tell her straight. And, what is more, if she tried to retaliate like the last time I had to chastise her for her lack of household budgeting by putting all my white underclothes in the wash with her red running top, the one that was not colour fast and the dye leeches out, making my vests and underpants pink, I will be very upset. It took many weeks of bleaching to get rid of the pink; the clothes never became fully white again but ended up a mucky grey. I was too embarrassed to go to the juju lady when I was ill in the fear that she might ask me to strip and I would be all exposed, resplendent in my pink undies. Or grey undies. What would the neighbours have to say about that? I would be the laughing stock of the village if it got out. Pink undies!!! (Or grey undies!!!) So equipped with all my arguments I threw open the kitchen door, stormed inside without taking off my outdoor shoes and putting them on the shoe rack behind the kitchen cabinet which is on the left hand side of the sink next to the Aga cooker but in front of the waste bin, and what do I find? She wasn’t there. I cannot remember giving her leave of absence, not on a Monday morning with all the dishes from the previous night still to be washed up. She’d gone. GONE. She had brazenly left me a note to say that she had popped down to the Village Hall for a coffee morning and I would have to make my own breakfast. Can you believe that Blog, or can you believe that Blog? And I don’t even know where she keeps the paracetamol to treat my man flu.

                                                   Colin

I've gottabe jokin?

I’m back Blog. I’m back!!! I have just spent a week in Yorkshire and my batteries have been full recharged. The weather was fine and warm, ideal for a trudge or two or three. What more can one ask from life, than to trudge in the pure Yorkshire free air. Free air. Free. FREE. The ideal of which every Yorkshire man dreams. The only thing a Yorkshire man dreams of!! Trudging the moors or the canals, it costs nowt with a capital ‘N’.

And when I got back to the Manse, there was a pleasant surprise awaiting me. What fantastic news Blog. Not only am I a Colossus of the athletic world, but I am now receiving just recognition for my ground pioneering work in the Land of Sums. I had a very special letter waiting on the silver platter by the door in the entrance hall where it had been left. A letter conferring an Honour upon me; quite deservedly so, even if I say so mesen. Fleetingly I thought it might be a wind up; I thought someone was taking the Mickey. But no. No. NO. The letter was from the C.E. of 4M. The C.E. no less. The C.E. in person. Hand written, I kid you not Blog. 4M as you may well know Blog, is a multi-NATIONAL GROUP of companies with wide ranging interests including the Media and Entertainment. The particular honour they wished to bestow upon mesen was for ‘Services to Sums’. 2321. The letter was from the 4M was for me discovering the new number 2321, the number I told you about last week Blog. As you may know Blog, the 4M (the Mickey Mouse Mathematical Mausoleum) is the sole repository for all wrong calculations. Every error made in sums is kept here, all are collected, collated and stored. It makes no difference if the mistake if the sum is committed accidentally by a school child or on purpose by a highly paid Banker. All errors, the lot, go to 4M, without exception. Because my ingenious self-correcting sum means many errors in the future will be eliminated, 4M will have huge amounts of unused storage space available. This storage space can be sold on the open commodities market to anyone wanting storage facilities. Because the amount of space I have made available to the Mickey Mouse Mathematical Mausoleum was unexpected, 4M can undercut the market price charged by their competitors and so make a killing. 4M shares have sky rocketed on the stock markets around the world. The income that the spare capacity has generated will part finance the generous award they have given to me.

So, hence forth and forever more they have decreed that 2321 will be known as Kirkham’s Konstant (KK) and the self-correcting procedure I detailed in our recent communication Blog, will be referred to as Colin’s Conjecture (CC). Kirkham’s Konstant will stand astride the mathematical community in the same way as Planck’s  Constant supports the Construction Industry, Einstein’s Uncertainty Principle underpins the confidence of all Physics’ Research and Portnoy’s Complain  the Medical World.  I must confess that I feel as Newton did; as Newton said when he entered his orchard to inspect his apples ‘I have only been able to see further because I have been able to stand on the shoulders of others’. Interesting quote from 1666. Step ladders weren’t invented until 1727, the year of Newton’s death so he was unable to use them to help him look into the future, nor to help him avoid getting his fingers burned in the Great Fire of London by climbing above the conflagration. Of course, once the step ladder was invented, builders could utilise them along with Planck’s Constant to speed up the rebuilding work of London after the devastating fire. And this lead to the industrial revolution beginning in earnest; Eisenberg’s invention of the air ship being one of the first ideas to take off; Kundt’s Tube, being the forerunner of the London Underground, was another novel idea born out of Colin’s Conjecture and last, but by no means least, the hand held battery powered drill being the revolutionary tool which enabled Sir Thomas Fairfax to drill efficiency into Cromwell’s Model Army. Once he had the bit between his teeth he was able to beat the Royals without having to play extra time. Oh yes, Blog. Part of the condition of the award of KK and CC is to publish a short synopsis of the investigation into how 2321 came to be so important….. So here is a flavour …. Added as an addendum to study at your leisure. Got to get a few miles in now, before it gets too late. Given any thought to my offer of coaching help in your future athletic plans. I was going to charge you for my services and give the money to Newlife and Tiny Tims Children’s Centre but seeing as how 4M are so handsomely rewarding me, I will waive the fees.

                                                              Colin

P.S. Newlife and Tiny Tims Children’s Centre will miss out on my fund raising this year because I will not be able to trudge in the London Marathon as I do not have a place.

P.P.S. One more year over and I again was most successful in persuading all my admirers not to waste their money on sending me a Valentine’s Card. So that they all didn’t feel too embarrassed about not posting me a card, I used the excuse that I would be away in Yorkshire and if my letter box was stuffed full of cards, it would be an open advertisement to burglars to burgle. I am pleased to say that they all appreciated my concern for my own property and did not send a single card. Not one. NONE. NIL. Zilch!!!

                                                          2321

Mathematically we are used to seeing order emerge from chaos.

If a scientific research is undertaken using a given set of instruments, the interpretation of results of the investigation will be by means of the investigative tools used. Prosaically, the infinite set of monkeys with their infinite set of type writers may produce vast quantities of gibberish and the occasional recognisable word, perhaps, but the print-out will all be of the type face of the type writer keys.

 If a simple rule is repeatedly applied to a set of digits, should one expect a simple pattern to emerge?

Apply the rule ‘count the number of each digit of a random number. List the count in ascending order on the next line. Repeat the procedure.’

Although any number could be selected, it is logical to start simply with the first digit, ‘1’. [for the purpose of this article, ‘0’ is regarded as the last digit … purely for convenience. Proceed to numbers with more than one digit.

1 … we have one ‘1’, therefore, applying the stated rule, we write ‘11’ on the next line viz:-

11 … we now have two ones, therefore on the next line we will list ‘21’

21 … we now have one ‘1’ and one ‘2’ etc..

1112

3112

211213

312213

212223

114213

31121314

41122314       at this stage, do we expect this to continue and spiral out of control??

But to continue

31221324

21322314

21322314    clearly this will now continue to repeat.

If we extract the actual digits, ‘1’, ‘2’, ‘3’, ‘4’, we are left with 2321 [which will be referred to as the base root]

If now we continue to repeat this procedure with the base root

2321

112213

312213

212223

114213

31121314

41122314

31221324

21322314        once again this will now continue to repeat

21322314 ==> 2321   This will be the definition of a basic root. A basic root is one which when continually simplified gives the same root.

Perhaps at a later stage, we might like to speculate on why the first digit should produce the base number 2321.

Note at this early stage, we would end with the same result if our chosen random number had been eleven ‘11’ or twenty one ‘21’ etc..

Note the number of steps to repetition form ‘1’ is 12, for ‘11’ it would have been 11 steps, for ‘21’ it would have been ‘10’ steps etc.. The step relationship will be considered at a later stage.

If our start number had been 21, the next line would give 1112, which is line four of start digit ‘1’, so the digit line display for ‘1’ could be tidied to ...

1

11

21

1112 <== 12

3112

211213

312213

212223

114213

31121314

41122314

31221324

21322314

21322314 ==> 2321

If we now try the same procedure for ‘2’

2

12 …. Which we have just met! Therefore ….

1

11

21             2

1112 <== 12

3112

211213

312213

212223

114213

31121314

41122314

31221324

21322314

21322314 ==> 2321

‘3’ and ‘4’ can be similarly computed and imposed on the grid, noting that ‘13’ and ‘14’ come free.

1

11                                               3

21             2                                13

1112 <== 12                              1113

3112                                           3113

211213                                       2123            4

312213 <================112213        14

212223                                                           1114

114213                                                           3114

31121314 <=======================211314

41122314

31221324

21322314

21322314 ==> 2321

From our observation of ‘12’ and ‘21’, we can now impose ‘31’ and ‘41’

1

11                                               3

21             2                                13

1112 <== 12                              1113 <== 31

3112                                           3113

211213                                       2123            4

312213 <================112213        14

212223                                                           1114 <== 41

114213                                                           3114

31121314 <=======================211314

41122314

31221324

21322314

21322314 ==> 2321

This cascade effect will be seen to become important at a later stage.

At a later stage, it can be seen that other numbers will link in also. ‘32’, ‘23’, ‘33’,‘34’, ‘43’,

1

11                                             3

21           2                                13

1112 <= 12                              1113 <= 31

3112                 32          33      3113

211213 <====1213 <= 23      2123       4

312213 <===============112213   14

212223                                                    1114 <= 41

114213                                                    3114            34

31121314 <====================211314 <==1314 <=43

41122314

31221324

21322314

21322314 ==> 2321

It may soon be worth tabulating the steps each of our ‘random’ numbers takes to reach 2321

                                                      *********

‘5’ introduces a variant derivative

5

15

1115

3115

211315

31121315

41122315

3122131415

4122231415

3132132415

3122331415 which appears to give a new base root of 32311

3122331415

But if the modelling continues in the same way, viz:-

32311

211223

213213

212223

114213

31121314

41122314

31221324

21322314 ==> 2321, our original base root!

21322314

32311 may be considered to be an intermediate root, as it does not comply to the definition of a ‘basic root’. . 

At this stage Blog, step numbers were considered and a method of simple notation devised.

Before considering other derivatives, it is worth looking at the other five digits, ‘6’, ‘7’, ‘8’, ‘9’, and ‘0’

6

16                                                                                  and ‘61’

1116

3116

211316

31121316

41122316

3122131416

4122231416

3132132416

3122331416 ==> 32311

3122331416

7

17                                                                                 and ‘71’

1117

3117

211317

31121317

41122317

3122131417

4122231417

3132132417

3122331417 ==> 32311

3122331417

8

18                                                                                 and 81 

1118

3118

211318

31121318

41122318

3122131418

4122231418

3132132418

3122331418 ==> 32311

3122331418

9                                                                                   and ‘91’

19

1119

3119

211319

31121319

41122319

3122131419

4122231419

3132132419

3122331419 ==> 32311

3122331419

0

10

1110

3110

211310

31121310

41122310

3122131410

4122231410

3132132410

3122331410

3122331410 =>32311

For the last five digits, the same pattern can be seen which translates into the digit => ‘*’

                                                                                                                                           *

                                                                                                                            * 1 ==>  1*

                                                                                                                                           111*

                                                                                                                                           311*

                                                                                                                                           21131*

1                                                                                                                                         3112131*

11                                             3                                                                                        4112231*

21           2                                13                                                                                      312213141*

1112 <= 12                              1113 <= 31                                                                        412223141*

3112                 32          33      3113                                                                                  313213241*

211213 <====1213 <= 23      2123       4                                                32311 <=====312233141*

312213 <===============112213   14                                              211223

212223                                                    1114 <= 41                                213213

114213                                                    3114            34                          212223

31121314 <====================211314 <==1314 <=43             114213

41122314                                                                                                   31121314

31221324                                                                                                   41122314

21322314                                                                                                   31221324

21322314 ==> 2321 <===================================21322314

Other derivatives do exist, but before their consideration, uniquely we have ‘22’ !!!

‘22’ obeys our definition for a ‘basic root’.

The method of simple notation I have adopted makes the following so much simpler but I will slog through the long way, just for you Blog. With my notation, the same sequences can be seen quickly without the slog. It then becomes self-evident.

However …..

‘20’ is our next number in sequence, with ‘30’, ‘40’, ‘50’ initially suggesting a similar pattern … but

20                                         30                               40                            50

1210                                     1310                           1410                        1510

211210                                 211310                       211410                    211510

312210                                 31121310                   31121410                31121510

21221310                             41122310                   4112131410            4112131510

31321310                             3122131410               5112132410            511213141510

31123310 ==> 3131            4122231410               412213141510        611213142510 

31123310                             3132132410               512213241510        51221314151610

                                             3122321410               413213142510        61221314251610

                                             3132231410               412223241510*      51321314152610

                          32311 <== 3122331410               314213241510#      51222314251610

                                                            422211<== 412223241510*      41421314251610

                                                                                         4213111 <== 41221334151610*

                                                                                                                41222324151610##

                                                                                                                41421324151610##

                                                                                                                41221334151610*  

We now have a new derivative from ‘20’ which if we continue

3131             ….. to distinguish derivatives call this

2123

112213

312213

212223

114213

31121314

41122314

31221324

21322314 ==> 2321 … or more fully ‘20’ is S9E6

21322314

 But with ‘40’ and ‘50’ we introduce the idea of a loop, ‘40’ has one element hence [L9,1] and ‘50’ has two elements hence [L12,2]. But continuing

422211                                 4213111     both could be considered to be ‘intermediate roots’.

213214                                 41121314

21221314                             41121324

31321314                             31221324

31123314                             21322314

31123314 ==> 3131            21322314 ==> 2321

Therefore ‘30’ takes on D10S8, ‘40’ takes L9,1E5S9 and ‘50’ L12,2S5

Def:- an ‘intermediate root’ is one which comes about when the number sequence starts to repeat. Distinguish between intermediate roots with and without loops.

A program would be useful .. my programming ability is not up to the task. (Remember Algol 60 … )

60                                     70                                     80                                        90

1610                                1710                                 1810                                     1910

211610                            211710                             211810                                 211910

31121610                         31121710                         31121810                            31121910

4112131610                     4112131710                     4112131810                        4112131910

511213141610                 511213141710                 511213141810                    511213141910

61121314151610             61121314151710             61121314151810                61121314151910

71121314152610             7112131415161710         7112131415161810            7112131415161910

6122131415161710         8112131415162710         811213141516171810        811213141516171910

7122131415261710         712213141516171810     911213141516172810        91121314151617181910

6132131415162710         812213141516271810     81221314151617181910    101121314151617182910

6122231415261710         713213141516172810     91221314151617281910    91221314151617181920

5142131415261710*       712223141516271810     81321314151617182910    81321314151617182910

6122132425161710#       614213141516271810*   81222314151617281910    81222314151617281910

5142131415261710*       712213241526171810#   71421314151617281910*  71421314151617281910*

54111211                         614213141516271810*   81221324151627181910#  81221324151627181910#

51121415                         641111211                       71421314151617281910*  71421314151617281910*

41121425                         61121416                         7411111211            <===   7411111211

31222415                         41121426                         71121417

2132131415                     31122416                         41121427

4122231415                     3122131416                     31222417

3132132415                     4122231416                     2132131417

3122231415                     3132132416                     4122231417                                                                                                                                                                                   3132231415                     3122331416 *                  3132132417

3122331415*                   3122331416                     3122331417*

3122331415                     32311                               32311

32311

By now it should be self evident that if all numbers investigated come down to one of the intermediate  roots, a mistake in the computation will follow through and still give a common intermediate root ( mostly not the correct one ??)

The state of play so far :-

** ** ** ** ** ** ** ** ** ** **                     If we slowly work through, trying to spot family patterns.…

     ** ** ** ** ** ** ** ** ** **                      24 <== 44                         25        similar pattern for 26, 27, 28, 29, 62, 72, 82, 92. 

     ** ** ** 24 25 26 27 28 29 **                      1214 <== 42                     1215 <== 52

     ** ** ** 34 35 36 37 38 39 **                      211214                              211215

     ** 42 43 44 45 46 47 48 49 **                      312214                              312215

     ** 52 53 54 55 56 57 58 59 **                      21221314                          21221315

     ** 62 63 64 65 66 67 68 69 **                      31321314                          31321315

     ** 72 73 74 75 76 77 78 79 **                      31123314 ==> 3131 <== 31123315

     ** 82 83 84 85 86 87 88 89 **                      31123314                          31123315 and therefore the  rest of the twenties and 

     ** 92 93 94 95 96 97 98 99 **                      their palindromes. (and 44 plus all the other multiples of 11!)                                                                                                                                                    )

** ** ** ** ** ** ** ** ** ** **                 Then surprisingly…   

     ** ** ** ** ** ** ** ** ** **                  34                                         

     ** ** ** ** ** ** ** ** ** **                  1314 <==43                        

     ** ** ** 34 35 36 37 38 39 **                  211314                                 

     ** ** 43 ** 45 46 47 48 49 **                  1121314                                 

     ** ** 53 54 ** 56 57 58 59 **                  41121324                             

     ** ** 63 64 65 ** 67 68 69 **                  31221324                              

     ** ** 73 74 75 76 ** 78 79 **                  21322314    

     ** ** 83 84 85 86 87 ** 89 **                  21322314 == > 2321

     ** ** 93 94 95 96 97 98 ** **

The rest of the thirties follow the pattern 3$ and because as previous patterns show, $ is too big to

35                                                  contribute as not enough digits exist to reach $.

1315 <== 53

211315

31121315

41122315

3122131415

4122231415

3132132415

3122331415 ==> 32311

3122331415

** ** ** ** ** ** ** ** ** ** **

     ** ** ** ** ** ** ** ** ** **

     ** ** ** ** ** ** ** ** ** **

     ** ** ** ** ** ** ** ** ** **

     ** ** ** ** 45 46 47 48 49 **

     ** ** ** 54 ** 56 57 58 59 **

     ** ** ** 64 65 ** 67 68 69 **

     ** ** ** 74 75 76 ** 78 79 **

     ** ** ** 84 85 86 87 ** 89 **                 

     ** ** ** 94 95 96 97 98 ** **

45                                  whereas 46 (and 47, 48, 49 and 74, 84, 94 )

1415 <== 54                               1416  <== 64     as well as 56                same pattern  57 with 58, 59 and 85, 95)  

211415                                        211416                               1516  <==  65                    1517

311211415                                  31121416                           211516

4112131415                                4112131416                       31121516

5112132415                                5112132416                       4112131516

4122131425                                412213141516                   511213141516

3132132415                                512213241516                   611213142516

3122331415 ==> 32311             413213142516 <=======512213141526

3122331415                                412223241516*

                                                    314213241516#

                                                    412223241516*

                                                    422211

                                                    213214

                                                    21221314

                                                    31321314

                                                    31123314

                                                    31123314 ==> 3131

67                                                68                                        86, 69, 96 following on

1617 <== 76                               1618 <== 86

211617                                        211618

31121617                                    31121618

4112131617                                4112131618

511213141617                            511213141618

61121314151617                        61121314151618

71121314152617                        71121314152618

61221314151627                        6122131415161718

51321314152617                        7122131415261718

51222314251617*                      6132131415162718

41421314251617##                    6122231415261718*

51221334151617##                    5142131415261718###

51222314251617*                      6122131425161718###

4411211                                      6132131415261718###

411224                                        6122231415261718*

212224                                        62211211

124214                                        413216

212224                                        21121314

114214                                        41221314

311224                                        31221324

21221314                                    21322314 ==> 2321

31321314                                    21322314

31123314 ==> 3131

31123314

78                        similarly      79                                              and     89

1718 <== 87                           1719 <== 97                                        1819 <== 98

211718                                    211719                                                 211819

31121718                                31121719                                             31121819

4112131718                            4112131719                                         4112131819

511213141718                        511213141719                                     511213141819

61121314151718                    61121314151719                                 61121314151819

7112131415161718                7112131415161719                             7112131415161819

8112131415162718                8112131415162719                             811213141516171819

7122131415161728                712213141516171819                         911213141516172819

6132131415162718                812213141516271819                         812213141516171829

6122231415261718                713213141516172819    <====          713213141516172819

5142131415261718*              712223141516271819*                      

6122132425161718#              614213141516271819###                  

5142131415261718*              712213141526171819###                  

54111211                                713213141516271819###                  

51121415                                712223141516271819*                      

41121425                                722111211

31222415                                513217

2132131415                            2112131517

4122231415                            5122131517

3132132415                            4122132517

3122331415 ==> 32311         313213141517

3122331415                            511233141517

                                                511223142517

                                                412213142517

                                                413213241517

                                                412223241517*

                                                314213241517#

                                                412223241517*

                                                422211

                                                213214

                                                21221314

                                                31321314

                                                31123314

                                                31123314 ==> 3131

Now let 2321 = +            3131 = #        and                32311 = *

       0              1               2               3              4              5                6               7                8                9    

       *              +               +               +             +              *                *               *                *                *

     10            11             12             13            14            15              16             17              18              19

      *              +               +               +             +              *                *                *                *                * 

     20            21             22             23            24            25              26             27              28              29    

      #              +                                +             #               #                #               #                #                #

     30            31             32             33            34            35              36             37              38              39

      *              +               +               +              +             *                 *              *                *                *

     40            41             42             43            44            45              46             47              48              49    

      #              +               #               +             #              *                #               #                #                #

     50            51             52             53            54            55              56             57              58              59

      +              *               #               *              *              #                #              #                 #                #

     60            61             62             63            64            65              66             67              68              69    

      *              *               #               *              #              #                #              #                 +                +       

     70           71              72             73            74            75              76             77              78              79

      *             *                #               *              #              #                #              #                 *                #

     80            81             82             83            84            85              86             87              88              89    

      *              *               #               *              #              #                +              *                 #                # 

     90            91             92             93            94            95              96             97              98              99

      *              *               #               *              #              #                +              #                 #                #  

      *      +      +      +      +     *      *       *      *     *   

      *      +      +      +      +     *      *       *      *     * 

      #      +              +      #      #      #      #       #     #    

      *      +      +      +      +     *      *       *      *     * 

      #      +      #      +      #      *      #       #      #     #  

      +      *      #      *      *      #      #       #      #     #

      *      *      #      *      #      #      #       #      +     +         

      *      *      #      *      #      #      #       #      *     #

      *      *      #      *      #      #      +       *      #     #

      *      *      #      *      #      #      +       #      #     #  

It is a great deal prettier if a colour coding is adopted, Blog

At this stage you may take a breather, Blog, before the heavy stuff starts and we do away with numbers…

.