Interesting Facts

This week may be termed as the week of optical illusions. And here I am coming with yet another optical illusion.

How many colors do you see?



There are only 3 colors: White, green, and pink.
There seem to be two different shades of pink and green, but there is only one pink and one green.

Here is another illusion. I have seen it presented as a puzzle or math problem but commonsense tells us that this must be an illusion. The only thing we have to do is try and explain the illusion.





Solution:

The area of a right triangle is computed by multiplying the base times the height and dividing by two. The pieces of the puzzle have an area of 32 square units. Although they can be assembled to form what appears to be a 13×5 right triangle, they actually form quadrilaterals that are slightly smaller or slightly bigger than a 13×5 right triangle.

The top figure has an area of 32 square units. The bottom figure, including the empty square, has an area of 33 square units. A real 13×5 right triangle would have an area of 32.5 square units. The distortion is difficult to see because one square of the picture is approximately 3% of the area.

The distortion can be seen more clearly when the empty square constitutes a larger percentage of the area, as in the figure below where 1 square represents 13% of the area.

This picture is an ambigram, an image which can be viewed in more than one way depending on how you perceive it. The thing about this sort of image, in particular, is that it manages to convince you visually that you're looking at two completely contradictory views at the exact same time. What does this tell you about perception, and the way our brain processes conflicting stimuli? Can you see it as both images simultaneously, or merely as one, then the other, alternating based on how you squint or tip your head?

Today is the birthday of Mahathma Gandhiji, on this day I would like to recall the partition of India and Pakistan.



Millions left for their promised new homeland with smiles on their faces as trains left both India and Pakistan.



The massive exchange of population that took place in the summer of 1947 was unprecedented.



This is a train to Pakistan being given a warm send-off.
In 1947, the border between India and its new neighbour Pakistan became a river of blood, as the exodus erupted into rioting.



These pictures are by Margaret Bourke-White from Khushwant Singh's book Train to Pakistan, Roli Books.Over 10 million people were uprooted from their homeland and travelled on foot, bullock carts and trains to their promised new home.



An aged and abandoned Muslim couple and their grand children sitting by the the roadside on this arduous journey. "The old man is dying of exhaustion. The caravan has gone on," wrote Bourke-White.



It left behind a trail of death and destruction. The Indian map was slashed to make way for a new country - Pakistan.



In a couple of months in the summer of 1947, a million people were slaughtered on both sides in the religious rioting.



The street was short and narrow. Lying like the garbage across the street and in its open gutters were bodies of the dead. Here, bodies of the victims of rioting are picked up from a city street.



With the tragic legacy of an uncertain future, a young refugee sits on the walls of Purana Qila, transformed into a vast refugee camp in Delhi.



Men, women and children who died in the rioting were cremated on a mass scale.
Villagers even used oil and kerosene when wood was scarce.



The migration

This is an expression that is normally used to caution someone. When you say that one swallow does not make a summer, what you mean is that just because something good has happened, it doesn't mean good things will continue to happen. Chances are things may go bad, instead of improving. You are requesting the individual to err on the side of caution, and not to carried away.

Just because you've won the first round doesn't mean you are going to win the championship. Remember one swallow does not make a summer.



The expression comes from the world of Aesop's Fairy tales. In the story, a young man sees a swallow on a warm winter day. As you know, a swallow is a bird which usually appears in the spring. Thinking that the winter season is over, the young man sells off his woollen coat, and with the money he has made, he goes to the bar and drinks. unfortunately, in the days that follow, the temperature drops. The young man, shivering in the cold realises that one swallow does not make a summer.

Article idea: The Hindu dated on September 30, 2008

Larry and Sergey decide that the BackRub search engine needs a new name. After some brainstorming, they go with Google – a play on the word “googol,” a mathematical term for the number represented by the numeral 1 followed by 100 zeros. The use of the term reflects their mission to organize a seemingly infinite amount of information on the web.



Today is the tenth anniversary of google.
And hence the post is special!

Zeller's rule is used to calculate the day on which any date falls for any year. With this technique you will have the calendar for any given year available to you.

The rule is as follows

f = k + [(13*m - 1)/5] + d + [d/4] + [c/4] - 2*c

where,

k = day of month
m = month number, taking Mar=1, ..., Dec=10, Jan=11, Feb=12
d = last two digits of year, using the previous year for Jan and Feb
c = first two digits of year

Rules:

1.In Zeller's rule the year begins in March and ends in February. Hence, the month number from March is 1, April is 2, May is 3 and so on up to January, which is 11, and February is 12.

2.January and February are counted as the 11th and 12th months of the previous year. Hence, if you are calculating the day of any date on January 2026, the notation will be (month=11 and year= 2025) instead of (month=1 and year=2026).

3.While calculating, we drop off every number after the decimal point.

4.Once we have found the answer we divide it by 7 and take the remainder. Remainder 0 corresponds to Sunday; Remainder 1 corresponds to Monday ; Remainder 2 corresponds to Tuesday and so on....

Example:

Find the day on 26th June 1983

f = k + [(13*m - 1)/5] + d + [d/4] + [c/4] - 2*c

Here k=26, m=4, d=83, c =19

f= 26+(13*4-1/5)+83+83/4+19/4-2*19
= 105

105 divided by 7 leaves a remainder 0. Hence the day is a Sunday!


Derivation of the formula:

Here we're defining

k = day of month
m = month number, taking Mar=1, ..., Dec=10, Jan=11, Feb=12
d = last two digits of year, using the previous year for Jan and Feb
c = first two digits of year

The formula is then

f = k + [(13*m - 1)/5] + d + [d/4] + [c/4] - 2*c

and we use the remainder after dividing f by 7 to find the day of the
week.

Where does this come from? Let's first note the reason for the odd
handling of months: we want leap day not to affect the formula, so we
move it to the end of the 'year', and act as if the year began on
March 1.

Now note that in defining f, all we care about is the remainder after
dividing, so it will be enough to make sure that f increases by 1
whenever the day of the week advances by one day; we don't care about
the actual value of f.

Now let's build the formula piece by piece.

How does the year affect the day? Well, since 365 = 7*52+1, each
normal year advances the day by 1, so our formula can start with the
year number:

f = d

Whenever the year advances by 1, so does the day of the week.

But we have to adjust this to account for leap years. Every four years
we have an extra day, so we'll want to add 1 to f. This is done by
adding [d/4], since this increases by 1 only when d becomes a multiple
of 4, which is a leap year. So now we have

f = d + [d/4]

Now how do centuries affect the day? A century contains 100 years, 24
of which normally are leap years (since century days, like 1900, are
NOT leap years). So each century the day advances by 124 days, which
is 7*18-2, and therefore the day of the week goes BACK 2 days. So we
have

f = d + [d/4] - 2*c

But every fourth century year IS a leap year (as 2000 was), so we
have to adjust just as we did for leap years:

f = d + [d/4] - 2*c + [c/4]

Now we come to the months, and this is the cute part. Consider, for
each month, how many days it has BEYOND 28, and then add that up to
see the effect the months have on the day of the week:

1 2 3 4 5 6 7 8 9 10 11 12
Month Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb
Days 31 30 31 30 31 31 30 31 30 31 31 (28)
Excess 3 2 3 2 3 3 2 3 2 3 3 0
Accum 0 3 5 8 10 13 16 18 21 23 26 29
\_________________/\__________________/\_______

The number of accumulated days is counted at the start of the month,
so if we divide it by 7, the remainder shows how many weekdays the
start of the month is from the starting day for the 'year'.

Notice the pattern in the excess: 3,2,3,2,3 repeats every five months,
and the accumulation reaches 13 in that time. So every 5 months, we
want to add 13 days. That suggests that we want to add a term like
[13m/5]. That doesn't quite give us what we want:

1 2 3 4 5 6 7 8 9 10 11 12
Month Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb
Days 31 30 31 30 31 31 30 31 30 31 31 (28)
Excess 3 2 3 2 3 3 2 3 2 3 3 0
Accum 0 3 5 8 10 13 16 18 21 23 26 29
13m 13 26 39 52 65 78 ...
[13m/5] 2 5 7 10 13 15 ...

If we subtract 2 from this, it isn't quite right; we have to shift it
a bit. So after playing with it a bit, we find

13m-1 12 25 38 51 64 77 ...
[(13m-1)/5] 2 5 7 10 12 15 ...
[(13m-1)/5]-2 0 3 5 8 10 13 ...

That's just what we want. So we'll use

f = d + [d/4] - 2*c + [c/4] + [(13m-1)/5] - 2

Finally, we have to add the day, since each day obviously adds one to
the day of the week; and adjust to get the right day of the week for,
say, Mar 1, 2000, since nothing we've done so far actually determined
WHICH day we start the whole pattern on. It turns out that we can just
remove the -2, and we get

f = d + [d/4] - 2*c + [c/4] + [(13m-1)/5] + k

And there's the formula!

It comes from Central America and is found from Mexico to Panama. It is quite common in its zone, but it not easy to find because of its transparent wings, which is a natural camouflage mechanism.

A butterfly with transparent wings is rare and beautiful. As delicate as finely blown glass, the presence of this rare tropical gem is used by rain forest ecologists as an indication of high habitat quality and its demise alerts them of ecological change. Rivaling the refined beauty of a stained glass window, the translucent wings of the Glasswing butterfly shimmer in the sunlight like polished panes of turquoise, orange, green, and red.