[Before reading this, the reader must note that this
article is purely autobiographical, and readers of my blog articles on history
should not expect anything of that nature in this article.
I had posted this article on the blog yesterday
morning, but when I checked it on the blog in the evening, I found that the
mathematical symbols such as fractions (from "insert"
"equation" in Microsoft Word) could not be read in the article. Thus,
the formula given by me here as "Ap = A÷1 x
(A+1)÷2 x (A+2)÷3
x (A+3)÷4 ….. x (A+p)÷(1+p)" was visible only as "
x x
x x … x". So I deleted the
article and changed all the symbols to ordinary symbols, e.g. using a division
sign "÷" to express fraction-relations, and I am re-uploading it
again].
When I was in the tenth standard in school (in
1973-74), I "invented" or "discovered" some part of the
concept of permutations and combinations. I know how ridiculous this sounds
(and probably is), but I really did. While sitting on the second-last bench in
class (either during some boring part, or in a free period) I sat scribbling
things on a paper (a childhood habit: my fingers used to itch to write
something or the other), and I idly wrote out some long word, and its
three-letter abbreviated form, on the paper. Then, equally idly, I wondered how
many three-letter combinations I could form from the word. To avoid confusion,
I abandoned the long word, and just took the first 10 alphabets, abcdefghij,
and started to find out. I realized that keeping the first two letters
constant, I got the following 8 combinations: abc, abd, abe, abf, abg, abh,
abi, abj. Shifting the second one forward, I got 7 combinations: acd, ace, acf,
acg, ach, aci, acj. And so on. I noticed that to get the sum total of the
number of possible combinations, this required the addition of 8, 7, 6, 5, 4,
3, 2 and 1. I do not remember the full story of my investigations, but I ended
up inventing a series of formulae for adding a series of numbers, a process
which, (in my schoolboy lingo) I named "down-addition".
Two points before I give the formulae:
1. My tryst with mathematics ended after the
following sequence of events: pleased with my own inventiveness, I approached
the assistant principal of our school (a Jesuit priest named Father Bulchand,
who was the Assistant Principal of my school, St. Xavier's High School, Mumbai,
at that time, in 1973-74, and for a long time after), who was also suitably
impressed by my work and told me to show it to another fellow-priest teaching
at the neighboring St. Xavier's College. This priest was also impressed and
sent me to a former student who was then a researcher at the TIFR (the Tata
Institute of Fundamental Research) in the Navy-Nagar area at the extreme south
end of Mumbai. This man (unfortunately I do not remember the name either of
this gentleman or the priest-professor at St. Xavier's College) took my notes,
told me that he would look into the formulae and place it on record in the
appropriate place (probably some file or folder). But he also gave me his
advice on how, after completing my SSC and joining college, I could become a
researcher in mathematics and try to join the TIFR where I could make a career
of research in Mathematics. But, for some reason, the extremely serious and
dedicated atmosphere at the TIFR that I saw, with serious-looking young people
engaged in perpetual research, actually put me off rather than inspired me: frankly,
a life of sitting in the clinical atmosphere of an academic institution (after
following long academic procedures of selection and admission) and churning out
research papers on a regular basis somehow put me off. My work is based on more
sporadic, unplanned and internally-inspired (whimsical?) sources of impetus,
although once I really get into the subject I go as deep in my analysis as (I
find) necessary. So my interest in research in mathematics just sort of fizzled
out quickly.
However, I wrote down the formulae I had invented in
a notebook, and kept carrying them forward into new books as the passage of
time made it necessary to dispose of increasingly yellowing and tattered
earlier notebooks.
Now I am setting down these formulae in this blog
article, so that they are placed on record once and for all.
[An important side-effect of all this was that I turned to
my childhood ambition of becoming a writer of books (English story-books
modeled on my favourite English writers), and then started wondering whether I
should not write a book in my mother-tongue Konkani. I started out to write a joint
family play bhɑṅgrɑ̄ muddi ("gold ring"), and
realized that the existing Devanagari alphabet was not adequate to represent
Konkani sounds. I did not know anything about phonetic Roman, so I actually
invented a new alphabet for Konkani. But obviously this was a totally
impractical idea, and today my alphabet functions as a personal secret code in
which I write things I may not want others to read. But this triggered off my
interest in two things: in Konkani linguistics (see my forthcoming blog
article, "The Konkani language"), and in different alphabets of the
world. This led to my study of most of the alphabets (now actually in use) and
then to the actual study of languages of the world (learning by heart the
numbers 1-100 in as many languages as I could was my hobby in my college days.
See my blog article "India's Unique Place in the World of Numbers and
Numerals") and general linguistics. This led me, in turn, to the concept
of the Indo-European language family, to the Aryan-Invasion-Theory, and finally
to the historical analysis of the Rigveda (as the oldest Indo-European and
Indo-Aryan text)].
2. To the non-mathematician, these formulae may
appear obscure and uninteresting. A mathematician may find that all these formulae
are quite elementary and already exist, expressed maybe in some different way.
The Assistant Principal of my school, after he went through my formulae and
notes, did point out to me that the subject with which I started out was a
branch of Mathematics taught in colleges called "Permutations and
Combinations" (as a school-boy in Bombay in 1973-74, I was not aware of
any of this) and showed me a college text-book with a chapter on this. The
first of my formulae, a little differently expressed, was known to him, but he
was not aware of the others. But actual mathematicians may know all of these
formulae, similarly or differently expressed. But it makes no difference to me:
I am merely setting out my formulae on the internet to place them on record,
and indirectly also to be able to dispose of my old and yellowing notebooks, as
well as to close my connection ("R.I.P.") with this subject once and
for all.
I.
The Formulae
Formula 1:
Ap = A÷1
x (A+1)÷2 x
(A+2)÷3 x (A+3)÷4 ….. x
(A+p)÷(1+p)
Where A is the number to be "down-added",
p is the degree of "down-addition".
Formula 2:
Ap(to L) = Ap
-
(Up + Up-1W + Up-2W1 + Up-3W2 + Up-4W3 + Up-5W4 …… + U1Wp-2)
Where A is the number to be "down-added",
p is the degree of "down-addition", L is the lowest number till which
the addition should go, U = L - 1, and W = A - U.
Formula 3:
Ap(dif. Y) = Y(Mp) + N x
(M+1)p-1
Where A is the number to be
"down-added", p is the degree of "down-addition", Y is the
difference between two consecutive numbers
in the addition list, A÷Y = M with or without a remainder, and the remainder if any (of A÷Y) is N.
II.
Formula 1
The "degree" in this is as follows (I give
below the first three degrees of "down-addition" for 10):
100:
10. (i.e. no "down-addition").
101
= 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
102
= 101 + 91 + 81 + 71 + 61
+ 51 + 41 + 31 + 21 + 11
That is:
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 6 + 5 + 4 + 3 + 2 + 1
+ 5 + 4 + 3 + 2 + 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
103
= 102 + 92 + 82 + 72 + 62
+ 52 + 42 + 32 + 22 + 12:
That is:
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 6 + 5 + 4 + 3 + 2 + 1
+ 5 + 4 + 3 + 2 + 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 6 + 5 + 4 + 3 + 2 + 1
+ 5 + 4 + 3 + 2 + 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 6 + 5 + 4 + 3 + 2 + 1
+ 5 + 4 + 3 + 2 + 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 7 + 6 + 5 + 4 + 3 + 2 + 1
+ 6 + 5 + 4 + 3 + 2 + 1
+ 5 + 4 + 3 + 2 + 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 6 + 5 + 4 + 3 + 2 + 1
+ 5 + 4 + 3 + 2 + 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 5 + 4 + 3 + 2 + 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 4 + 3 + 2 + 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 3 + 2 + 1
+ 2 + 1
+ 1
+ 2 + 1
+ 1
+ 1
Applying the formula
Ap
= A÷1 x
(A+1)÷2 x (A+2)÷3
x (A+3)÷4 ….. x (A+p)÷(1+p):
101 =
10÷1 x 11÷2
= 55.
102 =
10÷1 x 11÷2
x 12÷3 = 220.
103 =
10÷1 x 11÷2
x 12÷3 x
13÷4 = 715.
III.
Formula 2
The "degree" in this is as follows [I give
below the first three degrees of "down-addition" for
10p(to 4), the 4 being for the purpose of
this example: the lowest number could be anything below 7]:
101(to 4)
= 10 + 9 + 8 + 7 + 6 + 5 + 4
102(to 4)
= 101(to 4) + 91(to 4) + 81(to 4) + 71(to
4) + 61(to 4) + 51(to 4) + 41(to 4)
That is:
10 + 9 + 8 + 7 + 6 + 5 + 4
+ 9 + 8 + 7 + 6 + 5 + 4
+ 8 + 7 + 6 + 5 + 4
+ 7 + 6 + 5 + 4
+ 6 + 5 + 4
+ 5 + 4
+ 4
103(to 4)
= 102(to 4) + 92(to 4) + 82(to 4) + 72(to
4) + 62(to 4) + 52(to 4) + 42(to 4)
That is:
10 + 9 + 8 + 7 + 6 + 5 + 4
+ 9 + 8 + 7 + 6 + 5 + 4
+ 8 + 7 + 6 + 5 + 4
+ 7 + 6 + 5 + 4
+ 6 + 5 + 4
+ 5 + 4
+ 4
+ 9 + 8 + 7 + 6 + 5 + 4
+ 8 + 7 + 6 + 5 + 4
+ 7 + 6 + 5 + 4
+ 6 + 5 + 4
+ 5 + 4
+ 4
+ 8 + 7 + 6 + 5 + 4
+ 7 + 6 + 5 + 4
+ 6 + 5 + 4
+ 5 + 4
+ 4
+ 7 + 6 + 5 + 4
+ 6 + 5 + 4
+ 5 + 4
+ 4
+ 6 + 5 + 4
+ 5 + 4
+ 4
+ 5 + 4
+ 4
+ 4
Applying the formula
Ap (to L) = Ap - (Up + Up-1W + Up-2W1 + Up-3W2 + Up-4W3 + Up-5W4 …… + U1Wp-2)
Where A is the number to be "down-added",
p is the degree of "down-addition" [Note:
The number of factors to be added within the brackets will also be p],
L is the lowest number till which the addition
should go,
U = L - 1, and W = A - U
[Here A is 10, L is 4, U=L-1 =3,
W=A-U =7]:
101(to 4) = Ap
- (Up)
= (10 x 11÷2) - (3 x
4÷2) = 55 - 6
= 49.
102(to 4) = Ap - (Up + Up-1W)
= (10 x 11÷2
x 12÷3) - [(3 x 4÷2 x 5÷3) + (3 x 4÷2 x 7)]
= 220 -
(10 +
42) = 168.
103(to 4) = Ap - (Up + Up-1W + Up-2W1)
= (10 x 11÷2
x 12÷3 x 13÷4) - [(3 x 4÷2 x 5÷3 x 6÷4) + (3 x 4÷2 x 7 x 8÷2)]
= 715 - (15 + 70 + 168) = 462.
IV.
Formula 3
The "degree" in this is as follows [I give
below the first three degrees of "down-addition" for Ap(dif.2)
the 2 being for the purpose of this example; the difference number could be
anything below 7]:
101 (dif.2)
= 10 + 8 + 6 + 4 + 2
102 (dif.2)
= 101 (dif.2) + 81 (dif.2) + 61 (dif.2) + 41
(dif.2) + 21 (dif.2)
That is:
10 + 8 + 6 + 4 + 2
+ 8 + 6 + 4 + 2
+ 6 + 4 + 2
+ 4 + 2
+ 2
103 (dif.2)
= 102 (dif.2) + 82 (dif.2) + 62 (dif.2) + 42
(dif.2) + 22 (dif.2)
That is:
10 + 8 + 6 + 4 + 2
+ 8 + 6 + 4 + 2
+ 6 + 4 + 2
+ 4 + 2
+ 2
+ 8 + 6 + 4 + 2
+ 6 + 4 + 2
+ 4 + 2
+ 2
+ 6 + 4 + 2
+ 4 + 2
+ 2
+ 4 + 2
+ 2
+ 2
Applying the formula
Ap(dif. Y) = Y(Mp) + N x
(M+1)p-1
Where A is the number to be
"down-added", p is the degree of "down-addition", Y is the
difference between two consecutive numbers
in the addition list, A÷Y = M with or without a remainder, and the remainder if any (of A÷Y) is N.
[Here, A is 10, Y is 2, M= 5,
N= 0]
101 (dif.2)
= Y(Mp) = 2 x (5 x 6÷2) = 30.
102 (dif.2)
= Y(Mp) = 2 x (5 x 6÷2 x 7÷3)
= 70.
103 (dif.2)
= Y(Mp) = 2 x (5 x 6÷2 x 7÷3 x 8÷4)
= 140.
As the above example gave no remainder, we will
examine another example where there is a remainder:
Ap(dif.3)
101 (dif.3)
= 10 + 7 + 4 + 1
102 (dif.3)
= 101(dif.3) + 71(dif.3) + 41(dif.3) + 11(dif.3)
That is:
10 + 7 + 4 + 1
+ 7 + 4 + 1
+ 4 + 1
+ 1
103 (dif.3)
= 102(dif.3) + 72(dif.3) + 42(dif.3) + 12(dif.3)
That is:
10 + 7 + 4 + 1
+ 7 + 4 + 1
+ 4 + 1
+ 1
+ 7 + 4 + 1
+ 4 + 1
+ 1
+ 4 + 1
+ 1
+ 1
Applying the formula Ap(dif.Y)
Y(Mp) + N x (M+1)p-1
Where A is the number to be
"down-added", p is the degree of "down-addition", Y is the
difference between two consecutive numbers
in the addition list, A÷Y = M with or without a remainder, and the remainder if any (of A÷Y) is N.
[Here, A is 10, Y is 3, M= 3, N=
1]
101 (dif.3)
= Y(Mp) + N x
(M+1)p-1
= 3 x (3 x 4÷2) + 1 x (4) = 18 +
4 = 22.
102 (dif.3)
= Y(Mp) + N x
(M+1)p-1
= 3 x (3 x
4÷2 x 5÷3) + 1 x (4 x
5÷2) = 30 + 10 = 40.
103 (dif.3)
= Y(Mp) + N x
(M+1)p-1
= 3 x (3 x
4÷2 x 5÷3 x 6÷4) + 1 x (4 x 5÷2 x 6÷3) = 45 + 20 = 65.
To round off the "research", I should have
gone on to try to derive a fourth formula for "down-addition" for a
number with a lowest number + a difference between consecutive numbers, eg. Ap
(to L; dif. Y). However, I didn't, and these are the only three formulae
invented by me, after which I did not venture into mathematical
"research" of any kind. As I said earlier, these formulae will
probably be already known to mathematicians (perhaps expressed rather differently,
and in a more sophisticated and developed way). The whole purpose of writing
this blog was to get these formulae off my chest.
V.
A Failed Attempt at Mathematical "Research"
It was not only the dullness of the prospect of
sitting in the clinical atmosphere of an academic institution, mechanically churning
out mathematical "research" (in the form of formulae), all my life, as
already related, that put me off "research" in mathematics. It was another
later, and failed, attempt at such "research" that finally set
the seal on my disillusionment, and loss of interest in, such research. In my
college days, I suddenly became interested in (or obsessed with) finding out a
regular mechanical method for preparing magic squares with even numbers.
A magic square is a square in which the numbers add
to the same total horizontally, vertically, as well as diagonally (i.e. for the
two full diagonals only). As most people know, there is a regular mechanical
method for preparing magic squares with odd numbers. For those who don't I will
explain that method. See for example the following squares with 1, 3, 5 and 7
for example:
|
1
|
|
8
|
1
|
6
|
|
3
|
5
|
7
|
|
4
|
9
|
2
|
|
17
|
24
|
1
|
8
|
15
|
|
23
|
5
|
7
|
14
|
16
|
|
4
|
6
|
13
|
20
|
22
|
|
10
|
12
|
19
|
21
|
3
|
|
11
|
18
|
25
|
2
|
9
|
|
30
|
39
|
48
|
1
|
10
|
19
|
28
|
|
38
|
47
|
7
|
9
|
18
|
27
|
29
|
|
46
|
6
|
8
|
17
|
26
|
35
|
37
|
|
5
|
14
|
16
|
25
|
34
|
36
|
45
|
|
13
|
15
|
24
|
33
|
42
|
44
|
4
|
|
21
|
23
|
32
|
41
|
43
|
3
|
12
|
|
22
|
31
|
40
|
49
|
2
|
11
|
20
|
It will be noticed that the totals from every
direction in the case of the four above squares is, respectively: 1, 15, 65,
175.
The method of forming magic squares with odd numbers
is simple:
1. Start with the number 1 in the middle square on
the top-most horizontal row.
2. After that, keep going forwards and upwards, i.e.
in the north-east direction:
a) when you
reach the top horizontal row (as after no. 1 itself to begin with), you cannot
move upwards, so go to the bottom-most horizontal square in the next vertical
row, and continue moving north-eastwards.
b) when you
reach the last vertical row, you cannot move forward, so go to the first square
in the vertical row just above, and continue moving north-eastwards.
c) When you cannot do either (the concerned square
already being filled), just go to the square directly below, and continue
moving north-eastwards.
When all the squares get filled up, you have your
magic square!
While there is this perfectly simple and mechanical
method of forming magic squares with odd numbers, there did not seem to be any
simple and mechanical method of forming magic squares with even numbers. So,
when in college, I decided I would try to find such a method. I spent many
obsessed months.
But I was forced to give up the quest for two
reasons:
1. I finally realized that it was actually impossible
to get a simple and mechanical method of forming magic squares with even
numbers. Actually this should have become immediately obvious simply on the
mere examination of the lowest odd number square (1) vis-à-vis the lowest even
number square (2):
|
1
|
|
1
|
2
|
|
3
|
4
|
It can be seen that while the single number 1
obviously adds up to 1 from any direction, it is impossible to
arrange the numbers from 1 to 4 on a 2x2 square in such a way that the total
will be the same from every direction.
2. At the same time I became conscious that my
obsessive preoccupation with trying to find this method (of forming even number
magic squares) was actually driving me crazy: I was literally beginning to view
everything around me in terms of numbers and squares! This is hard to explain -
only someone who has experienced similar things will understand it - but I
realized that I would land up in a lunatic asylum if I continued with this kind
of pointless cerebration. So I decided to give the whole thing a permanent
break.
However, there was a further bit of failed, or
rather fated-to-be-obsolete, research in store for me:
I joined Central Bank of India in 1978 as a clerk,
when I was still in college. At that time, all banking in India was manual
banking: there were no computers, there were manual books called
"journals", "ledgers", etc. which contained all the banking
transactions. All the savings accounts were entered on paper sheets which were
fitted together into savings ledgers (likewise current accounts into current
ledgers). Depending on the number of accounts, there were many different
ledgers serially numbered (1,2,3… as per the account numbers, e.g. ledger 1
contained account numbers 1 to 523, ledger 2 contained account numbers 524 to
1241, and so on). At the end of the day, all the entries (credits, debits) in
the accounts were written into books called "supplementaries", a
separate supplementary for each ledger, and the clerk concerned sat with the
officer to check whether all the entries were correctly entered in the ledgers.
The next day, a separate clerk handling what was called a "cash-book"
(though nothing to do directly or solely with "cash"), totaled all
the previous day's transactions in the branch to see whether the sum totals of
all the cash, clearing and transfer transactions tallied with the figures shown
in what was called the "general ledger". This was the daily routine.
Once a month (usually on the last day), there was
what was called "jotting": every clerk had to take the total of the
figures of all the accounts on that day, in each individual ledger allotted to
him, and tally the figures with the ledger-wise figures shown in the general
ledger. If there was a difference (i.e. in ledger 4, the general ledger showed
that the figure should be 54,37,411/- but the "jotting" total of that
ledger in the jotting-book showed 54,37,296/-), then there was a difference of
Rs.115/, and the concerned clerk and officer had to sit and search out the
difference in that ledger by checking all the transactions in that ledger
throughout the month (i.e. since the previous "jotting") to find out
where the mistakes occurred. This monthly tallying of the ledgers was a big
task, and the staff used to spend several days searching out the differences in
the different ledgers.
At that time I devised a special short-cut method by
which I could find out the difference simply by noting down the relevant
figures on a sheet of paper and doing some special calculations. It was my
challenge that any difference in any ledger, which remained untraced even after
several days or weeks of diligent checking by the clerk and officer concerned,
could be traced by me within one hour at the maximum. Sometimes, the difference
was actually a carry-over from the previous month which remained untraced, but
no-one could fathom this because the previous month's ledger was ostensibly
"tallied" (wrongly); but this method could immediately tell (within
one hour) that this was the case: normally, the searchers would continue to
search and search hopelessly without realizing this. Then I would proceed to
search out the difference from the previous month - within one hour.
My method became so well-known in neighboring
branches that I started to be called there to search out untraceable differences.
Everyone was mystified by what I did on a sheet of paper to find the
difference: some jokingly commented that it seemed I was writing matka
figures on the sheet (matka was an illegal form of lottery run by the
underworld in Mumbai, very popular with the gambling masses, and gamblers could
often be seen writing mystic figures on paper to forecast the "lucky"
number for the day!). Finally, someone suggested to me that I should set out
the exact procedure in a booklet which could then be learnt by other bank
employees. I enthusiastically set out and prepared a booklet setting out the
entire method: it was really a very easy and logical method.
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