Mathematics Quiz II - How Many Squares Are In This Picture

Calculate the total number of squares in the following image.

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Answer : 40

Explanation :

If there is an n x n square (number of squares are in horizontal row should be equal to vertical row)

there is a way to calculate total number of squares in it.

Suppose there is a 2 x 2 square

How many squares are in it. If you say 4 then you are not absolutely right. Correct answer is 5. There is also a big square ACIG

Now how do you calculate this answer in mathematical terms.
Total number of squares in an n x n square =  n² + (n-1)² + (n-2)² + ......+ 1²
For 2 x 2 square = 2² + 1² = 4 + 1 =  5

Now suppose there is 3 x 3 square

How many squares are in it. Using above formula, we can easily calculate.

Total number of squares in an n x n square =  n² + (n-1)² + (n-2)² + ......+ 1²
For 3 x 3 square = 3² + 2²  + 1² = 9 + 4 + 1 =  14
So where are these 14 squares.
ABFE, BCGF, CDHG, EFJI, FGJK, GHLK, IJMN, JKON, KLPO, ACKI, BDLJ, EGOM, FHPN and ADPM

Now similar way we can calculate squares in Original question. There is one 4 x 4 square and two 2 x 2 squares in the image.
Total number of squares in a 4 x 4 square = 4² + 3² + 2²  + 1² = 16 + 9 + 4 + 1 =  30
Total number of squares in a 2 x 2 square = 2² + 1² = 4 + 1 =  5
There is another 2 x 2 square in image = 2² + 1² = 4 + 1 =  5
So Total number of squares = 30 + 5 + 5 = 40

Exercise for you : How many squares are in a Chess Board? :)

Interstellar (2014) - Calculations and Analysis

                                             Interstellar (2014)
                                              Welcome to the Fountain
                                           Quench Increase your Thirst

Fact : 1
Miller's Planet to outside observers orbits Gargantua every 1.7 hours. On Miller's Planet, that means the planet orbits ten times a second around Gargantua , which is normally faster than the speed of light. But since the spin from Gargantua caused space to whirl around it similar to wind, Miller's Planet does not travel faster than light relative to its space as the laws of physics say you cannot travel faster than light relative to space, but space itself is not bound by the speed limit. As such, faster than light travel is possible by bending and twisting space. However, Gargantua would have to fill half the sky in order for it to be so close.

Fact : 2
The time dilation on Miller due to the gravitational forces of Gargantua would be tantamount to the planet moving through empty space at roughly 99.99999998% the speed of light. 

Fact : 3
Gargantua’s mass must be at least 100 million times bigger than the Sun’s mass. If Gargantua were less massive than that, it would tear Miller’s planet apart. The circumference of a black hole’s event horizon is proportional to the hole’s mass. For Gargantua’s 100 million solar masses, the horizon circumference works out to be approximately the same as the Earth’s orbit around the Sun: about 1 billion kilometers.

Fact : 4
Miller’s planet is about as near Gargantua as it can get without falling in and if Gargantua is spinning fast enough, then one-hour-in-seven-years time slowing is possible. But Gargantua has to spin awfully fast. There is a maximum spin rate that any black hole can have. If it spins faster than that maximum, its horizon disappears, leaving the singularity inside it wide open for all the universe to see; that is, making it naked—which is probably forbidden by the laws of physics

Fact : 5
Einstein’s laws dictate that, as seen from afar, for example, from Mann’s planet, Miller’s planet travels around Gargantua’s billion-kilometer-circumference orbit once each 1.7 hours. This is roughly half the speed of light! Because of time’s slowing, the Ranger’s crew measure an orbital period sixty thousand times smaller than this: a tenth of a second. Ten trips around Gargantua per second. That’s really fast! Isn’t it far faster than light? No, because of the space whirl induced by Gargantua’s fast spin. Relative to the whirling space at the planet’s location, and using time as measured there, the planet is moving slower than light, and that’s what counts. That’s the sense in which the speed limit is enforced.


QUERIES 
1. How old is Miller’s planet? If, as an extreme hypothesis, it was born in its present orbit when its galaxy was very young (about 12 billion years ago), and Gargantua has had its same ultrafast spin ever since, then the planet’s age is about 12 billion years divided by 60,000 (the slowing of time on the planet): 200,000 years. This is awfully young compared to most geological processes on Earth. Could Miller’s planet be that young and look like it looks? Could the planet develop its oceans and oxygen-rich atmosphere that quickly? If not, how could the planet have formed elsewhere and gotten moved to this orbit, so close to Gargantua?

2. What is the gravitational time dilation equation for Miller's planet? As there is 60000 ratio between time on earth and time on Miller's planet , to balance the equation what should be the distance of planet from Gargantua, angular momentum as it is revolving around very fast spinning object and Mass of Gargantua ?

3. We all know Gravitational Time Dilation does not affect Mann's planet as it is far from Gargantua’s vicinity. But we also know, almost immediately after the Endurance’s explosive accident in orbit around Mann’s planet, the crew find the Endurance being pulled toward Gargantua’s horizon. From this it appears when crew leaves Mann’s planet, the planet must be near Gargantua. Following diagram is the orbit of Mann's Planet.

According to this, what should be the orbital period and orbital velocity of Mann's planet for a person on Earth and a person on Mann's planet.

4. How long Dr. Mann spent time on Mann's planet according to him and according to an observer on Earth(Keep its orbital path in mind)? How long did he spend in hibernation for both observers?

5. Why was Endurance able to receive signal from Earth but Earth was not able to receive signal sent by Endurance.

6. When Cooper left Earth, he was 35 Years old and when he returned, He was 124 years old for Murph. 35 + 2 years for Saturn Journey + 23 years for Miller's Planet Journey + 51 years for Black Hole Journey = 111, Where are 13 Years Missing? How Long was Cooper out for himself?

8. What is the Orbital velocity of Miller's planet for a person on Earth and a person on Miller's planet?

9. What is the age of Brand when Cooper arrives at Edmund's Planet?

10. How long Dr. Laura Miller spent time on Miller's planet according to her and according to an observer on Earth? Lets keep in mind that she died minutes ago before Cooper and Brand reached there.

11. Miller’s planet travels around Gargantua’s billion-kilometer-circumference orbit once each 1.7 hours. Could Rom see it moving very fast from Mothership?

12. If Coop and team would try to communicate with Rom from Miller's planet, how would their communication appear? According to Coop how fast would they get response from Rom and similarly how long would Rom get response from Coop & team?