Powers of Pascal’s Triangle


Pascal’s Triangle

One day a man’s boss needed help gambling.  The boss asked his employee how he could win in gambling.  This employee made Pascal’s Triangle.

Blaise Pascal (1623-1662) studied mathematics, philosophy, and physics.  His father collected taxes in Rouen, France.  Pascal always loved creating calculators.  After 3 years, Pascal made 20 of the first calculators.  He called these calculators Pascalines.

Nevertheless, of all of Pascal’s discoveries and inventions, humanity most commonly remembers him for making Pascal’s Triangle.

Pascal’s triangle consists of a one on the first line, then two ones on the second line, then a one, a two, and a one on the third line.  Each line is formed of numbers which are the sum of the two numbers above it, as shown below.


As you can see, each row is formed of numbers which are the sum of the two numbers diagonally on top of the number.  Pascal’s triangle continues forever, always containing increasingly large numbers.  The border of Pascal’s triangle is always one.

Powers of 11

1 = 1                                                                  110
11 = 11                                                              111
11 * 11 = 121                                                    112
11 * 11 * 11 = 1331                                          113
11 * 11 * 11 * 11 = 14641                               114
11 * 11 * 11 * 11 * 11 = 161051                     115
11 * 11 * 11 * 11 * 11 * 11 = 1771561           116

The zeroth row of Pascal’s triangle is one.  11is one.
The first row of Pascal’s triangle is one one, or 11.  11is one.
The second row of Pascal’s triangle is one two one, or 121.  112 is 121.

This pattern progresses until we get to the fifth row of Pascal’s Triangle.  The fifth row of Pascal’s triangle is one five ten ten five one, or 15,101,051.  115 is 161,051.  So does that mean that Pascal’s triangle only tells us the powers of 11 up to 114?  Of course not! One just has to know how to read Pascal’s Triangle.

1.jpgWhenever you have a two digit number produced by adding together the two numbers above it in Pascal’s triangle, you keep the unit digit and take all the digits after it and add it to the next number’s one’s digit.  For example, in the fifth row of Pascal’s triangle, you have:

1   5   10   10   5   1
Take the one in the first ten and carry it over to the next ten, you are left with:
1 5 11 0 5 1
then you take the tens digit one in 11 and carry it over to the leftmost 5. You get
1 6 1 0 5 1
11is 161,051.

This pattern works for every power of 11 greater than 4.  Lets try 116. The sixth row of Pascal’s Triangle is 1 6 15 20 15 6 1.

1 6 15 20 15 6 1
Take the one from the leftmost 15 and carry it over to the twenty.  You get:
1 6 15 21 5 6 1
Then take the 2 from the 21 and bring it over to the 15. You should now have:
1 6 17 1 5 6 1
Finally, take the 1 from the 17 and move it over to the 6.  You end up with:
1 7 7 1 5 6 1
11is 1771561

Now you know how to read the powers of 11 from Pascal’s Triangle!

Prime Numbers

In pascal’s triangle, the uppermost row is called the 0th row.  The second highest row is called the first row.  The leftmost number in each row is called the 0th number.  The second number to the left is called the first number.

Now examine the first number of each row.  If it is a prime number, then try dividing that number into each of the other numbers in that row except for the last number and the 0th number.  You should get a whole number for each answer.

Whenever the first number in a row is prime, all the preceding numbers except for the last one are multiples of that first number.  pascals2.jpg

Powers of 2

1 = 1                                                       20
2 = 2                                                      21
2 * 2 = 4                                               22
2 * 2 * 2 = 8                                        23
2 * 2 * 2 * 2 = 16                               24
2 * 2 * 2 * 2 * 2 = 32                        25
2 * 2 * 2 * 2 * 2 * 2 = 64                 26

The zeroth row of Pascal’s triangle consists of the number one.   2is 1.
The first row of Pascal’s triangle consists of 2 ones.  1 + 1 = 2.  2is 2
The second row of Pascal’s triangle consists of 2 ones and a two.  1 + 2 + 1 = 4. 2is 4.
The third row of Pascal’s triangle consists of 2 ones and 2 threes.  1 + 3 + 3 + 1 = 8. 2is 8.

This pattern continues in Pascal’s triangle forever.

Hockey Sticks

Pick a number, any number.  That is, a number on Pascal’s triangle.  Then choose a diagonal direction going downwards.  Continue however many rows down you want.  On the last number, change the direction of the diagonal.  That number will be the sum of all the numbers in the first diagonal.  2.jpg


Triangular numbers are the number of units in an equilateral triangle.  In an equilateral triangle with a side length of 3, there are 6 units in the triangle.triang.dots.gif

The third diagonal going down on Pascal’s triangle contains all triangular numbers.

Counting numbers are positive integers such as 1, 2, 3, 4, &c.  The second row of Pascal’s triangle has every counting number.

Tetrahedral numbers are “numbers that can be represented that can be represented by a regular geometric arrangement of equally spaced points. Tetrahedral numbers correspond to placing discrete points in the configuration of a tetrahedron.”





Roberto Devereux

Eugene Toth, Society, Uncategorized

by Eugene Toth

For the New York Metropolitan Opera’s March 24, 2016 gala opening of “Roberto Devereux,” the eyes of opera enthusiasts sparkled.  Not only was the performance a new production.   For the first time ever, on the 413th anniversary of Queen Elizabeth I’s death, the Met staged Gaetano Donizetti’s “Roberto Devereux”

Donizetti set several operas in Britain. After Anna Bolena and Maria Stuarda Roberto Devereux was Donizetti’s third opera about British Queens.  “Roberto Devereux,” depicts the golden age of Elizabeth’s reign and the Tudor era, when poetry, music and theater flowered.

In 1599, Roberto Devereux, a former lover of Elizabeth I, returned from an unsuccessful war in Ireland, to England.   The opera tells why Elizabeth I executed him for treason.

With historical detail, exquisite costumes invoked the splendor of the Elizabethan era.  In

his first appearance, Devereux wore a black overcoat with silver lining over black plate armor, based upon a 1590 Portrait by William Segar of  Roberto Devereux, Earl of Essex.

In the Metropolitan Opera’s new production, the stage wore no curtains.  Between two balconies, a wall approached and receded from the audience.  In different scenes the wall represented a palace of Elizabeth I, the Palace of the Duke of Nottingham, and the Tower of London.  At the sides, in two galleries, chorus members acted as an audience and witnesses focusing attention upon the four soloists – soprano Elizabeth, the mezzo soprano Sara, the tenor Robert Devereux, and baritone Duke of Nottingham.

Elizabeth I’s passion for her lover  Robert, Earl of Essex, drives the plot.  In scene 1, Elizabeth displayed the character of an imperious, fearsome, and proud monarch.  She held more power than anyone else in England.  Parliament sought to execute Essex as a rebel for treason.  The elderly Queen loved a younger man.  The Queen confided to Sara, a beautiful lady in waiting, that the Queen would pardon Devereux of the treason charges, if he still loved the Queen.

Devereux loved Sara.  Elizabeth had forced Sara to marry Devereux’s best friend and supporter, the Duke of Nottingham.   Trapped in a marriage she never wanted, Sara still loved Devereux.   At their secret meeting, a duet between Sara and  Devereux supplies one of the opera’s high points.   Delightful flute mirrored the intense love they shared.  Telling him to flee, that they must never meet again, Sara gave Devereux, as a token of her love, her blue shawl.

By order of the Queen, Sir Walter Raleigh arrested Devereux.  Raleigh discovered Sara’s blue scarf.  The scarf proved Devereux loved a woman.  Blind with rage and jealousy, Elizabeth signed Devereux’s death warrant.

Recognizing his wife’s scarf, the Duke of Nottingham, drunk in his palace, assaulted Sara with a knife and threw her about.  Devereux’s unwise passion for Sara turned against him his best supporters – the Queen and his former friend the Duke of Nottingham.

Still in love with Devereux, too late, the Queen canceled his execution.   Moments before the executioner chopped off Devereux’s head, she pardoned Devereux.  A cannon shot signaled his death.   The Queen saw visions of Devereux’s headless ghost and a bloody crown.

Elizabeth could not order Devereux to love her.  Even the greatest power meets limits. In the background of the stage statues symbolized Time and Death. Renouncing her throne, she exclaimed “Let James be King!”  A blast of the orchestra’s brass marked her death.

Setting a fast and thrilling pace, the Queen’s transforming feelings supply the opera’s dramatic tension.  Her love transformed into fury, regret, sorrow, remorse, despair, and finally madness.  Donizetti called this work, “the opera of emotions.”


Note: For pdf version, click here: Roberto Devereux PDF


Art Showcase, Authors, Blog, Omar Abdelhamid, Philosophy, Poetry, Society, Uncategorized


A Poem by Omar Abdelhamid

It is Monday.  

The sky is freshly rolled out

the air is soft and silent

A single crimson leaf

dances above my head


And the world is in front of my eyes

but my arms cannot reach out and grab it


I am a speck in the ground below

Some hear my whispers

Stoop down to listen.

Pick up the words I drop

And place them into my humble basket


  1. Am.  Sinking.

My arms and neck  are stuck in the cement

I cannot reach out, nor do I have a soul to lift me

I am bound to something, something more than me

and it rushes through my bones

through my mind bouncing back and forth and breaking through every thought every memory

And I can feel the frozen bodies of those who fell behind me,  screaming, hoping,

 leaving their legacies

to be plucked by the vulture time

who eats his meal with no haste

they leave it for me

for me  to drop

and for you to stoop down,

pick up, and place into my humble basket


They will find me here

if they care to look


a moment in time

an echo, a memory


And every echo is smaller

when there is no one there to hear

please hear


Send help

Give my words a way

To break me out

out of this cage


Authors, Blog, Omar Abdelhamid, Philosophy, Society, Uncategorized


A Short Story by Omar Abdelhamid

      Exhausted from the walk and relieved to have found a seat, I braced myself for the roar of the train as it took off. Suddenly we were in the dark, and I could spot blotches of graffiti on the inside of the tunnel as we whizzed by. I could not think, because my thoughts,unable to keep up with the speed of the iron beast, were left to float in the emptiness of the tunnels, to be ignored, if even noticed, by later commuters. The screech of the friction of the rails as we shot through exhausted my mind, until I could see nothing, I could hear nothing, and I could know nothing. And West shot the beast towards home.

The Achievements of the Han Dynasty

Authors, Blog, Drew Morris, Miscellaneous, Science and Technology, Society, Uncategorized

The Han dynasty lasted from 206 BC-220 AD and had many amazing achievements that changed the world for all and benefits our society in many ways. The Han dynasty had many achievements in science, and one of them was the seismograph. The Seismograph was an impressive instrument because it detected earthquakes from hundreds of miles away. Another scientific achievement was that they learned how to predict when the sun was going to have an eclipse. This discovery helped people because they were always ready when an eclipse came.  In medicine, doctors found new kinds of medications. This helped doctors treat more diseases and patients who were ill. Han craft workers also made an amazing invention when they learned how to create paper. They created paper by pounding the bark of Mulberry trees.  The invention of paper had a huge effect on the way people lived. Paper made it easier to record what was happening. With paper, students invented the first Chinese dictionary. Another idea that came up under the rule of emperor Wudi was Grand School. Grand School were schools that were created to help students get jobs in the local government. Grand School was the empire’s best school. They were set up in every province or state in the empire. Without Grand School and the Confucian emphasis on education, their society might not have had innovators to create these inventions. Overall, thanks to the Han dynasty’s hard work and achievements, society obtained tools and scientific advancements that benefit us to this day.

Annie and Communism

Authors, Blog, Omar Abdelhamid, Society, Uncategorized

The cold war was a period of tension between the two major world powers, the U.S. and The Soviet Union, and the two major world political structures, Communism and Capitalism during the 20th century. It was, of course, much more complex and interesting than quickly outlined here. But this simple description gives us a bit of context for what we are about to discuss.

The Cold War was all about getting people on either side of the tension. Both sides used all kinds of propaganda to get their own citizens more strongly associated with the beliefs of their government.

This propaganda was perpetuated through various different mediums, including cartoons, posters, and (as made possible by the popularity of advancing technology) movies.

Anthropology, Placebos, and Magic Voodoo Doctors

Authors, Blog, Omar Abdelhamid, Science and Technology, Society, Uncategorized

When the anthropology students of Horace Miner harshly judged and mocked the cultures of the people they studied and read about, Miner showed his students the humanity of these cultures and put them in a better light in a very clever way. He wrote an article  about the Nacirema, a Tribe with very strange customs and traditions, such as a mouth-rite ritual done by sticking horse hairs in the mouth. How strange indeed. A culture with medicine men and women and a charm-box in the washing room.

whispering night

Art Showcase, Authors, Omar Abdelhamid, Poetry, Uncategorized

Whispering night

A whispering night

Silences itself to life 

Until normality 

Having long been the rhythm of the sun

Sets and is but a whisper

And the cold trickles up the veins 

When the buttress of the fragile soul 

Breaks and drops it’s  child

And not the dark

But the dimming light is seen

With squinting

” let it live ” is cried

And the light does push so gallantly for life 

Till the final desperate flicker and the  end

Puts it to rest at last 

And sight excludes it 

And when the mind is pressed and the night is dark and the music is soft but deadly 

A child quickly falls and falls and falls


-Omar Abdelhamid 

On the Elements of Love

Authors, Omar Abdelhamid, Philosophy, Society, Uncategorized

by Omar Abdelhamid


To encompass the benefit and the reason for the survival of love in humans, one can describe all forms of “Love” as the basis of all motivation. The reason for doing anything that has ever been done is love.

We can further break down love into 3 different forms.

One kind of love is loving something for what it has done before. Another is loving for what it is doing. And the last is loving for what it can do.

All these forms are similar, despite being listed in the dictionary as three entirely different definitions of the word love. Because all three types of love is giving very passionately because there is something that can, is, or has been given to you before. You love because you were given or will be given and you are grateful. So love is in a sense gratitude as well.

These categories seem vague at first glance, so it would help to provide examples for each kind.