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…
.
No comments:
Post a Comment