It is said:
God took the strength of a mountain,
The majesty of a tree,
The warmth of a winter sun,
The calm of a quiet sea,
The generous soul of nature,
The comforting arm of night,
The wisdom of the ages,
The power of the eagle's flight,
The joy of a morning in spring,
The faith of a mustard seed,
The patience of eternity,
The depth of a family need,
Then God combined these qualities,
When there was nothing more to add,
He knew His masterpiece was complete,
And so, He called it ... Dad.
Tuesday, August 21, 2007
Sunday, August 19, 2007
P12. As a lover, I wonder if you have ever thought of Goodnight this way?
GOOD-NIGHT
Goodnight! Ah! No, the hour is ill,
Which severs those it should unite.
Let us remain together still,
Then it will be good night.
How can I call the lone night good!
Though thy wishes wing its flight,
Be it not said, thought or understood,
Then it will be 'good' night.
To hearts which near each other move,
From evening close to morning light,
The night is good; because my love,
They never say GOODNIGHT.
I THINK YOU WILL AGREE WITH THIS!
Goodnight! Ah! No, the hour is ill,
Which severs those it should unite.
Let us remain together still,
Then it will be good night.
How can I call the lone night good!
Though thy wishes wing its flight,
Be it not said, thought or understood,
Then it will be 'good' night.
To hearts which near each other move,
From evening close to morning light,
The night is good; because my love,
They never say GOODNIGHT.
I THINK YOU WILL AGREE WITH THIS!
Monday, July 30, 2007
S5. FORGET AND FORGIVE
A kindergarten(KG school) teacher has decided to let
her class play a game. The teacher told each child in
the class to bring along a plastic bag containing a
few potatoes. Each potato will be given a name of a
person that the child hates, so the number of potatoes
that a child will put in his/her plastic bag will
depend on the number of people he/she hates. So when
the day came, every child brought some potatoes with
the name of the people he/she hated. Some had 2
potatoes; some 3 while some up to 5 potatoes.
The teacher then told the children to carry with them
the potatoes in the plastic bag wherever they go (even
to the toilet) for 1 week. Days
after days passed by, and the children started to
complain due to the
unpleasant smell let out by the rotten potatoes.
Besides, those having 5 potatoes also had to carry
heavier bags. After 1 week, the children were relieved
because the game had finally ended.
The teacher asked: "How did you feel while carrying
the potatoes with you for 1 week?" The children let
out their frustrations and started
complaining of the trouble that they had to go through
having to
carry the heavy and smelly potatoes wherever they go.
Then the teacher told them the hidden meaning behind
the game. The teacher said: "This is exactly the
situation when you carry your hatred for somebody
inside your heart. The stench of hatred will
contaminate your heart and you will carry it with you
wherever you go. If you cannot tolerate the smell of
rotten potatoes for just 1 week, can you imagine what
is it like to have the stench of hatred in your heart
for your lifetime?"
Moral of the Story:
Throw away any hatred for anyone from your heart so
that you will not carry sins for a lifetime. Forgiving
others is the best attitude to take. "Learn to
Forgive and Forget."
her class play a game. The teacher told each child in
the class to bring along a plastic bag containing a
few potatoes. Each potato will be given a name of a
person that the child hates, so the number of potatoes
that a child will put in his/her plastic bag will
depend on the number of people he/she hates. So when
the day came, every child brought some potatoes with
the name of the people he/she hated. Some had 2
potatoes; some 3 while some up to 5 potatoes.
The teacher then told the children to carry with them
the potatoes in the plastic bag wherever they go (even
to the toilet) for 1 week. Days
after days passed by, and the children started to
complain due to the
unpleasant smell let out by the rotten potatoes.
Besides, those having 5 potatoes also had to carry
heavier bags. After 1 week, the children were relieved
because the game had finally ended.
The teacher asked: "How did you feel while carrying
the potatoes with you for 1 week?" The children let
out their frustrations and started
complaining of the trouble that they had to go through
having to
carry the heavy and smelly potatoes wherever they go.
Then the teacher told them the hidden meaning behind
the game. The teacher said: "This is exactly the
situation when you carry your hatred for somebody
inside your heart. The stench of hatred will
contaminate your heart and you will carry it with you
wherever you go. If you cannot tolerate the smell of
rotten potatoes for just 1 week, can you imagine what
is it like to have the stench of hatred in your heart
for your lifetime?"
Moral of the Story:
Throw away any hatred for anyone from your heart so
that you will not carry sins for a lifetime. Forgiving
others is the best attitude to take. "Learn to
Forgive and Forget."
Sunday, July 29, 2007
P10. I MISS YOU
I MISS YOU
Fate had only ordained that
We be part of one family,
But our mutual love and care has
Blended us in a MAJESTIC BOND
The sparks of life in me
Maybe slowly, but surely, fading
Yet the purity of my love
Still remains like it were just born.
When miles came between us
And our lives were drawn apart
Our thoughts remained together,
Tugging softly at the heart.
This loneliness without you
Reminds me of special moments shared,
And says across the miles,
How much we have always cared.
The emptiness that comes
With just the memory of your touch
Stirs a feeling from within that says
I MISS YOU VERY MUCH!
Fate had only ordained that
We be part of one family,
But our mutual love and care has
Blended us in a MAJESTIC BOND
The sparks of life in me
Maybe slowly, but surely, fading
Yet the purity of my love
Still remains like it were just born.
When miles came between us
And our lives were drawn apart
Our thoughts remained together,
Tugging softly at the heart.
This loneliness without you
Reminds me of special moments shared,
And says across the miles,
How much we have always cared.
The emptiness that comes
With just the memory of your touch
Stirs a feeling from within that says
I MISS YOU VERY MUCH!
Thursday, July 26, 2007
FL29 -FL35. INTERESTING 4-LINERS
WORLD
The world we are living in,
Is mighty hard to beat.
You get a thorn with every rose,
But aren't the roses sweet?
EATING
Though destiny may shape our ends,
I think it is only fitting,
To point out that some credit goes,
To eating and to drinking.
PARTED
When we are parted,
Let me lie,
In some far corner of thy heart,
Like a sweet melody!!!!
FORGET - ME - NOT
Down in a valley,
Carved in a rock,
Three pretty words -
FORGET - ME - NOT.
WOMAN
Such dear, are your sex companions,
Part truth, part fiction,
Some thought, much whim,
And all contradiction?
SIX HONEST MEN
I keep six honest men,
They taught me all I know,
Their names are What, Where, When,
And How, Why, and Who!!!!!
HEAR THE LAUGHTER
Hear the laughter of the leaves,
And see cheerful faces around you.
Listen to the whisper of the flowers,
And let your soul soar high.
The world we are living in,
Is mighty hard to beat.
You get a thorn with every rose,
But aren't the roses sweet?
EATING
Though destiny may shape our ends,
I think it is only fitting,
To point out that some credit goes,
To eating and to drinking.
PARTED
When we are parted,
Let me lie,
In some far corner of thy heart,
Like a sweet melody!!!!
FORGET - ME - NOT
Down in a valley,
Carved in a rock,
Three pretty words -
FORGET - ME - NOT.
WOMAN
Such dear, are your sex companions,
Part truth, part fiction,
Some thought, much whim,
And all contradiction?
SIX HONEST MEN
I keep six honest men,
They taught me all I know,
Their names are What, Where, When,
And How, Why, and Who!!!!!
HEAR THE LAUGHTER
Hear the laughter of the leaves,
And see cheerful faces around you.
Listen to the whisper of the flowers,
And let your soul soar high.
Wednesday, July 25, 2007
OL7. WISDOM IN ONE LINERS
Keep your chin up, it helps to keep your mouth shut?
Knowing that you don't know much is, knowing a great deal indeed!
It is better to be approximately right than to be precisely wrong!
If the criticism is harsh, take the truth from it and forget the harshness!
If you want to leave footprints in the sands of time - don't drag your feet?
It is a healthy symptom when a man is dissatisfied without being discouraged!
A fanatic is one who refuses to alter his mind and will not change the subject!
Experience is a hard teacher, it gives the test first and the lesson afterwards!
A man gazing at the stars is proverbially at the mercy of the puddles on the road!
A man may fail many times, but he is not a failure until he begins to blame someone else?
Cease to inquire what the future has in store, and take as a gift whatever the day brings forth!
It is notorious that the memory strengthens as you lay burdens upon it and trustworthy as you trust it.
Knowing that you don't know much is, knowing a great deal indeed!
It is better to be approximately right than to be precisely wrong!
If the criticism is harsh, take the truth from it and forget the harshness!
If you want to leave footprints in the sands of time - don't drag your feet?
It is a healthy symptom when a man is dissatisfied without being discouraged!
A fanatic is one who refuses to alter his mind and will not change the subject!
Experience is a hard teacher, it gives the test first and the lesson afterwards!
A man gazing at the stars is proverbially at the mercy of the puddles on the road!
A man may fail many times, but he is not a failure until he begins to blame someone else?
Cease to inquire what the future has in store, and take as a gift whatever the day brings forth!
It is notorious that the memory strengthens as you lay burdens upon it and trustworthy as you trust it.
Monday, July 23, 2007
MS15. MAGIC SQUARES
Magic Squares - Singly even order - new method. ALSO VISIT http://pranadit.wordpress.com
While it is very easy to construct odd order magic squares (the Hindu Rule is a very easy method for an nxn square where "n" is odd) and doubly even order magic squares, singly even order squares are not that easy. Border square method is the one most commonly used. I have developed a method for construction of (n+4) x (n+4) squares from nxn square where n is 6,10,14,18......to easily give multiple solutions without much effort.
I will illustrate my method by first constructing a 14x14 square assuming that we have a 10x10 square. The 10x10 square will occupy the center place. 14x14 square has numbers from 1 to 196 and 10x10 square numbers from 1 to 100. If we take half the difference between 196 and 100 we get 48 and if we add this to all numbers in our 10x10 square we will have a 10x10 square with numbers from 49 to 148 and total of 505 + 10x48 = 985. The 14x14`square needs to have a total of 1379, this leaves us with a difference of 394. The numbers from 1 to 48 and 149 to 196 can be used to form six 4x4 squares with numbers from i) 1 to 8 & 189 to 196, ii) 9 to 16 and 181 to 188, iii) 17 to 24 and 173 to 180, iv) 25 to 32 and 165 to 172, v) 33 to 40 and 157 to 164, and vi) 41 to 48 and 149 to 156, to give a total of 394 for all the six 4x4 squares. We will label 5 from these 6 squares from 1 to 5 and, the 10x10 square by 0, the sixth square will need to be split into two 4x2 square and has been labeled from A to P to fit our requirements of 14x14 square.
1 1 2 2 3 3 A B 3 3 2 2 1 1
1 1 2 2 3 3 C D 3 3 2 2 1 1
4 4 0 0 0 0 0 0 0 0 0 0 4 4
4 4 0 0 0 0 0 0 0 0 0 0 4 4
5 5 0 0 0 0 0 0 0 0 0 0 5 5
5 5 0 0 0 0 0 0 0 0 0 0 5 5
I K 0 0 0 0 0 0 0 0 0 0 M O
J L 0 0 0 0 0 0 0 0 0 0 N P
5 5 0 0 0 0 0 0 0 0 0 0 5 5
5 5 0 0 0 0 0 0 0 0 0 0 5 5
4 4 0 0 0 0 0 0 0 0 0 0 4 4
4 4 0 0 0 0 0 0 0 0 0 0 4 4
1 1 2 2 3 3 E F 3 3 2 2 1 1
1 1 2 2 3 3 G H 3 3 2 2 1 1
We can select any one of the six squares by turns for 1 to 5, the remaining sixth 4x4 square has to be formed slightly differently as already explained. We have to ensure that A + B = C + D = E + F = G + H = I + J = K + L = M + N = O + P = 197 and A + C + E + G = B + D + F + H = I + K + M + O = J + L + N + P = 394.
Since the 10x10 square occupies the center place the and that leaves us only two top rows and two bottom rows, and similarly two L.H.S. and R.H.S. columns, the five 4x4 squares have been split into two 4x2 squares and suitably placed in the corners. The 1 square will take care of our Diagonal totals for the 14x14 square, so we don't have to worry about diagonal totals in rest of the 4x4 squares which means having selected the 4x4 square from the available 712 squares (880 less the squares which have 3 numbers<9 and 1<8, or 3 numbers>8 and 1<9) we can switch its rows and columns giving 4!x4! i.e. 576 solutions for each of the 4 squares. As regards four 2x2 squares we are again free to switch A & B, C & D, E & F, G & H, in pairs and similarly the I & J, K & L, M & N, O & P, thus giving 576 solutions for each of them. In addition we can replace X's by Y's, doubling the number of solutions, from the square as it stands. And, of course, selection of 4x4 squares i.e., A,B,....can be done in 6! ways.
(P.S. simplest way to form the 4x4 square, say with numbers from 17 to 24 and 173 to 180, would be to add in an ordinary 4x4 square 16 to all numbers from 1 to 8 and 164 to all numbers from 9 to 16.)
When we go from 14x14 to 18x18 square, the base becomes 14x14 square and to all the numbers we add 1/2(18x18 - 14x14)=64 and the 14x14 square total becomes 2275. The total required for 18x18 square is 2925, this is less by 650 to be filled by the empty cells. We have now numbers from 1 - 64 and 261 - 324 to form 8 4x4 squares with each having a total of 650. Of these one will be required for extreme corners (like 1) and one for splitting into two 4x2 squares. From rest 6 we will have 3 squares for rows and 3 squares for columns, and the number of multiple squares that can be generated from this one 18x18 squares can easily be worked out.
I will now come to the first square that can be formed, i.e. the 6x6 square. I will prepare the ground by using one of the 880 4x4 squares, but for those not fully conversant, I give the basic square using alphabets and where Aa stands for A+a.
Aa Bb Cc Dd
Dc Cd Ba Ab
Bd Ac Db Ca
Cb Da Ad bc
Here A's can be given one of the values from 1,2,3,4 and a's can be given one of the values from 0,4,8,12. But each value can be given only once, i.e. if you give 1 to A you can give B only from 2,3,4 and so on. Alternately you can give to A's from 0,4,8,12 and a's from 1,2,3,4. After constructing the 4x4 square simply add 10 to all numbers. 10 is 1/2(36-16), this will take the sum in each row, column and diagonal to 74. Numbers now available to us are 1 to 10 and 27 to 36. Next I construct a 2x2 square such that its diagonals total 37.
2 & 35 and 1 & 36.
This enables me to take care of the diagonal totals as I am going to break up the 4x4 square into four 2x2 squares and place them in the corners. For my 4x4 square I give value 1 to a, 2 to b, 3 to c, and 4 to d, 0 to A, 4 to B, 8 to C and 12 to D. So my 6x6 square now looks like this:
11 16 00 00 21 26
25 22 00 00 15 12
00 00 02 36 00 00
00 00 01 35 00 00
18 23 00 00 24 19
20 23 00 00 14 17
(Here 00 stand for values yet to be filled in.)
Next I insert numbers, from those available, in columns 3 and 4 in rows 1,2,5 and 6 such that sum of numbers in each row is 37. Next I place numbers in rows 3 and 4 in columns1, 2, 5 and 6 such that magic sum of that each column total is 37. This ensures that the magic sum of 111 is available for each row, column and 2 main diagonals. My final 6x6 square is:
11 16 34 03 21 26
25 22 33 04 15 12
09 29 02 36 05 30
28 08 01 35 32 07
18 13 10 27 24 19
20 23 31 06 14 17
( 1 has been written as 01, 2 as 02, etc..)
Even if we ignore choices for the central 2x2 square, we can from this one square alone
generate very large number of 6x6 squares. We have choice of 880 for the4x4 squares
and they can be rotated to give 4 times more squares. Next the numbers in columns 3
and 4 in the rows 1, 2, 5 and 6 can be interchanged and so can the numbers in rows 3
and 4 in columns 1, 2, 5, and 6. Also numbers in column 3 and 4 can be interchanged as
also numbers in row 3 with numbers in row 4.
To make my point clear I am replacing numbers in column 3 and 4 in rows 1,2,5 and 6 by
alphabets and also in rows 3 and 4 in columns 1, 2, 5, and 6.
00 00 A1 B1 00 00
00 00 A2 B2 00 00
C1 C2 00 00 C3 C4
D1 D2 00 00 D3 D4
00 00 A3 B3 00 00
00 00 A4 B4 00 00
(Here again 00 have been placed in the empty spaces to suit the format design.)
Here A1+B1=A2+B2=A3+B3=A4+B4=37, and C1+D1= C2+D2=C3+D3=C4+D4= 37
And so are interchangeable. So we have at least 880x4x24x24x2x2x2 squares!
Let us now go to construction of 10x10 square. First we add 32 equal to 1/2(100 -36) to
all numbers in one of the 6x6 square available to me. This will give me a total of
111+32x6=303 for the 6x6 square. Next I construct four 4x4 squares from the numbers
1 to 32 and 69 to 100 available to me by adding i) 0 to all numbers < 9 and 84 to all
numbers >8, ii) 8 to all numbers <9 and 76 to all numbers >8, iii) 16 to all numbers <9
and 68 to all numbers >8 and iv) 24 to all numbers <9 and 60 to all numbers>8, limiting
our choice to 712 4x4 squares, excluding those which may have 3 numbers < 9 and 3
numbers > 8. To go from 10x10 to 14x14, place 10x10 square in the centre increasing all
numbers by 1/2(196 100),i.e.48, to give a total of 985 for the 10x10 square, the 4x4
squares will give 394 making a total of 1,379, as required, for the 14x14 square. The 4x4
squares will number 5 and there will be 2 4x2 squares and placed just as in 10x10 square
when we proceeded from 6x6 square to 10x10square.
Similar procedures will follow from 14x14 to 18x18 etc. as clarified above. Try to form
The 10x10 square and if you have difficulty get in touch with me by email
bdtara@yahoo.com
While it is very easy to construct odd order magic squares (the Hindu Rule is a very easy method for an nxn square where "n" is odd) and doubly even order magic squares, singly even order squares are not that easy. Border square method is the one most commonly used. I have developed a method for construction of (n+4) x (n+4) squares from nxn square where n is 6,10,14,18......to easily give multiple solutions without much effort.
I will illustrate my method by first constructing a 14x14 square assuming that we have a 10x10 square. The 10x10 square will occupy the center place. 14x14 square has numbers from 1 to 196 and 10x10 square numbers from 1 to 100. If we take half the difference between 196 and 100 we get 48 and if we add this to all numbers in our 10x10 square we will have a 10x10 square with numbers from 49 to 148 and total of 505 + 10x48 = 985. The 14x14`square needs to have a total of 1379, this leaves us with a difference of 394. The numbers from 1 to 48 and 149 to 196 can be used to form six 4x4 squares with numbers from i) 1 to 8 & 189 to 196, ii) 9 to 16 and 181 to 188, iii) 17 to 24 and 173 to 180, iv) 25 to 32 and 165 to 172, v) 33 to 40 and 157 to 164, and vi) 41 to 48 and 149 to 156, to give a total of 394 for all the six 4x4 squares. We will label 5 from these 6 squares from 1 to 5 and, the 10x10 square by 0, the sixth square will need to be split into two 4x2 square and has been labeled from A to P to fit our requirements of 14x14 square.
1 1 2 2 3 3 A B 3 3 2 2 1 1
1 1 2 2 3 3 C D 3 3 2 2 1 1
4 4 0 0 0 0 0 0 0 0 0 0 4 4
4 4 0 0 0 0 0 0 0 0 0 0 4 4
5 5 0 0 0 0 0 0 0 0 0 0 5 5
5 5 0 0 0 0 0 0 0 0 0 0 5 5
I K 0 0 0 0 0 0 0 0 0 0 M O
J L 0 0 0 0 0 0 0 0 0 0 N P
5 5 0 0 0 0 0 0 0 0 0 0 5 5
5 5 0 0 0 0 0 0 0 0 0 0 5 5
4 4 0 0 0 0 0 0 0 0 0 0 4 4
4 4 0 0 0 0 0 0 0 0 0 0 4 4
1 1 2 2 3 3 E F 3 3 2 2 1 1
1 1 2 2 3 3 G H 3 3 2 2 1 1
We can select any one of the six squares by turns for 1 to 5, the remaining sixth 4x4 square has to be formed slightly differently as already explained. We have to ensure that A + B = C + D = E + F = G + H = I + J = K + L = M + N = O + P = 197 and A + C + E + G = B + D + F + H = I + K + M + O = J + L + N + P = 394.
Since the 10x10 square occupies the center place the and that leaves us only two top rows and two bottom rows, and similarly two L.H.S. and R.H.S. columns, the five 4x4 squares have been split into two 4x2 squares and suitably placed in the corners. The 1 square will take care of our Diagonal totals for the 14x14 square, so we don't have to worry about diagonal totals in rest of the 4x4 squares which means having selected the 4x4 square from the available 712 squares (880 less the squares which have 3 numbers<9 and 1<8, or 3 numbers>8 and 1<9) we can switch its rows and columns giving 4!x4! i.e. 576 solutions for each of the 4 squares. As regards four 2x2 squares we are again free to switch A & B, C & D, E & F, G & H, in pairs and similarly the I & J, K & L, M & N, O & P, thus giving 576 solutions for each of them. In addition we can replace X's by Y's, doubling the number of solutions, from the square as it stands. And, of course, selection of 4x4 squares i.e., A,B,....can be done in 6! ways.
(P.S. simplest way to form the 4x4 square, say with numbers from 17 to 24 and 173 to 180, would be to add in an ordinary 4x4 square 16 to all numbers from 1 to 8 and 164 to all numbers from 9 to 16.)
When we go from 14x14 to 18x18 square, the base becomes 14x14 square and to all the numbers we add 1/2(18x18 - 14x14)=64 and the 14x14 square total becomes 2275. The total required for 18x18 square is 2925, this is less by 650 to be filled by the empty cells. We have now numbers from 1 - 64 and 261 - 324 to form 8 4x4 squares with each having a total of 650. Of these one will be required for extreme corners (like 1) and one for splitting into two 4x2 squares. From rest 6 we will have 3 squares for rows and 3 squares for columns, and the number of multiple squares that can be generated from this one 18x18 squares can easily be worked out.
I will now come to the first square that can be formed, i.e. the 6x6 square. I will prepare the ground by using one of the 880 4x4 squares, but for those not fully conversant, I give the basic square using alphabets and where Aa stands for A+a.
Aa Bb Cc Dd
Dc Cd Ba Ab
Bd Ac Db Ca
Cb Da Ad bc
Here A's can be given one of the values from 1,2,3,4 and a's can be given one of the values from 0,4,8,12. But each value can be given only once, i.e. if you give 1 to A you can give B only from 2,3,4 and so on. Alternately you can give to A's from 0,4,8,12 and a's from 1,2,3,4. After constructing the 4x4 square simply add 10 to all numbers. 10 is 1/2(36-16), this will take the sum in each row, column and diagonal to 74. Numbers now available to us are 1 to 10 and 27 to 36. Next I construct a 2x2 square such that its diagonals total 37.
2 & 35 and 1 & 36.
This enables me to take care of the diagonal totals as I am going to break up the 4x4 square into four 2x2 squares and place them in the corners. For my 4x4 square I give value 1 to a, 2 to b, 3 to c, and 4 to d, 0 to A, 4 to B, 8 to C and 12 to D. So my 6x6 square now looks like this:
11 16 00 00 21 26
25 22 00 00 15 12
00 00 02 36 00 00
00 00 01 35 00 00
18 23 00 00 24 19
20 23 00 00 14 17
(Here 00 stand for values yet to be filled in.)
Next I insert numbers, from those available, in columns 3 and 4 in rows 1,2,5 and 6 such that sum of numbers in each row is 37. Next I place numbers in rows 3 and 4 in columns1, 2, 5 and 6 such that magic sum of that each column total is 37. This ensures that the magic sum of 111 is available for each row, column and 2 main diagonals. My final 6x6 square is:
11 16 34 03 21 26
25 22 33 04 15 12
09 29 02 36 05 30
28 08 01 35 32 07
18 13 10 27 24 19
20 23 31 06 14 17
( 1 has been written as 01, 2 as 02, etc..)
Even if we ignore choices for the central 2x2 square, we can from this one square alone
generate very large number of 6x6 squares. We have choice of 880 for the4x4 squares
and they can be rotated to give 4 times more squares. Next the numbers in columns 3
and 4 in the rows 1, 2, 5 and 6 can be interchanged and so can the numbers in rows 3
and 4 in columns 1, 2, 5, and 6. Also numbers in column 3 and 4 can be interchanged as
also numbers in row 3 with numbers in row 4.
To make my point clear I am replacing numbers in column 3 and 4 in rows 1,2,5 and 6 by
alphabets and also in rows 3 and 4 in columns 1, 2, 5, and 6.
00 00 A1 B1 00 00
00 00 A2 B2 00 00
C1 C2 00 00 C3 C4
D1 D2 00 00 D3 D4
00 00 A3 B3 00 00
00 00 A4 B4 00 00
(Here again 00 have been placed in the empty spaces to suit the format design.)
Here A1+B1=A2+B2=A3+B3=A4+B4=37, and C1+D1= C2+D2=C3+D3=C4+D4= 37
And so are interchangeable. So we have at least 880x4x24x24x2x2x2 squares!
Let us now go to construction of 10x10 square. First we add 32 equal to 1/2(100 -36) to
all numbers in one of the 6x6 square available to me. This will give me a total of
111+32x6=303 for the 6x6 square. Next I construct four 4x4 squares from the numbers
1 to 32 and 69 to 100 available to me by adding i) 0 to all numbers < 9 and 84 to all
numbers >8, ii) 8 to all numbers <9 and 76 to all numbers >8, iii) 16 to all numbers <9
and 68 to all numbers >8 and iv) 24 to all numbers <9 and 60 to all numbers>8, limiting
our choice to 712 4x4 squares, excluding those which may have 3 numbers < 9 and 3
numbers > 8. To go from 10x10 to 14x14, place 10x10 square in the centre increasing all
numbers by 1/2(196 100),i.e.48, to give a total of 985 for the 10x10 square, the 4x4
squares will give 394 making a total of 1,379, as required, for the 14x14 square. The 4x4
squares will number 5 and there will be 2 4x2 squares and placed just as in 10x10 square
when we proceeded from 6x6 square to 10x10square.
Similar procedures will follow from 14x14 to 18x18 etc. as clarified above. Try to form
The 10x10 square and if you have difficulty get in touch with me by email
bdtara@yahoo.com
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