November Question: Daily Lottery

I participate in a simply daily lottery.  The rules are simple.  Every day I choose a number from 1-100, and at the end of the day, a random winning number from 1-100 is drawn.  If it matches the number I chose, I win.  The odds of winning a single game is 1 in 100, or 1%.

I was very lucky and won the lottery today. I decide to keep playing one number every day until I win again, and then stop playing for good. What day am I most likely to stop playing (by winning the lottery that day, of course)?

  1. Tomorrow
  2. The day after tomorrow
  3. 50 days from today
  4. 100 days from today
  5. There is no difference, every day is equally as likely

This is not a trick question so you do not need to consider unusual scenarios – you just need to understand the question.

Answer to October Question, Part 2

This is a great question because it forces one to think about what a moon phase really is, not just a standard answer regurgitated from a textbook. To answer this question, let’s take a look at the drawing I provided before, and mark out the surface visible from Earth, marked in red:

Next, let’s show the illuminated surfaces of the moon, marked in yellow, assuming sunlight coming from our left:

 

The “full moon” phase is the easiest. Only one side is illuminated by the sun, which turns out to be the visible side. You will see a bright square.

 

The next phase is more interesting. Two sides are illuminated now (remember, sunlight comes from our left), but we can only see one side, which is illuminated at an angle. We should still see a full square, but less bright.

 

At a certain point, the sun no longer illuminates the side visible to us and it goes dark. This will continue for half a lunar month cycle.

 

To summarize, for half a lunar month you will see no moon, and for the other half, you will see a full square moon going from dim to bright to dim.

 

References:

  1. If the Moon Were a Cube, What Would Its Phases Look Like? – Wired

Answer to October Question, Part 1

Bizzaro world is hard to imagine because in our daily life, we observe things on a much smaller scale. We learn about gravity, yet rarely think about what it really means and what effects it has. Here are the answers to the questions:

  1. (Where would all the water go?) Water would accumulate in the center of each cube face. Why? Well, water tends to flows to lower places. On a human scale, that would be the bottom of the cup. However on a planetary scale, that would be the closest point to the center of the Earth, i.e., the Center of Gravity. The closest point to the center of the Earth in Bizarro World would be the middle of each face, which is where the water would go and accumulate.  When seen from afar, the water would actually look like a blob sticking out from the center of each face.
  2. (What would happen when you cross an edge?) It would feel like going from an incline to a decline, like straddling a ridge. The ridge would not seem very level – the closer you are to the corners, the steeper the tilt of the ridge. This is because gravity pulls towards the center of the cube.
  3. (What would happen when you walk from the center to a corner?)  First of all, each side of the cube would be about 10,000 km, so you would have about 7,000 km to go. You’d also be at the bottom of a very, very deep ocean, but that’s just a technicality. Even though the ground is perfectly flat, it would feel steeper and steeper as you approach the corner, much like the inside of a fishbowl. The feeling of steepness comes from gravity pulling on you at a different angle. When you finally arrive at the corner, it would look like a mountaintop. You would probably be quite tired, because you’d have walked over 7,000 km, and more impressively, climbed a vertical distance of ~3,700 km. That’s a lot of climbing, considering that in real life, going from the bottom of the Mariana Trench to the tip of Mount Everest is less than 20km. You’d be in a spacesuit, standing 8,700 km from the center, well above the atmosphere. Again, just a technicality. In fact, you’d be so high up, you would weigh about half as you did on Earth, without the spacesuit. You’d see that each side of the cube is an isolated world with separately evolved flora and fauna, as the edges and corners poke out into space, preventing water and atmosphere from mixing.

It’s probably not what you were expecting. This example shows that intuition and personal experiences, however seemingly real and convincing, may not be valid at larger scales. Sometimes a deeper understanding comes only when you step back and look from afar. To use an extreme example of stepping back, when the Voyager spacecraft passed Pluto in 1989, it looked back at Earth and took the ultimate selfie – a picture of a pale blue dot. Dr. Carl Sagan, arguably the best science communicator of our times, eloquently reflected:

Look again at that dot. That’s here. That’s home. That’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every superstar, every supreme leader, every saint and sinner in the history of our species lived there-on a mote of dust suspended in a sunbeam.

The Earth is a very small stage in a vast cosmic arena. Think of the endless cruelties visited by the inhabitants of one corner of this pixel on the scarcely distinguishable inhabitants of some other corner, how frequent their misunderstandings, how eager they are to kill one another, how fervent their hatreds. Think of the rivers of blood spilled by all those generals and emperors so that, in glory and triumph, they could become the momentary masters of a fraction of a dot.

Our posturings, our imagined self-importance, the delusion that we have some privileged position in the Universe, are challenged by this point of pale light. Our planet is a lonely speck in the great enveloping cosmic dark. In our obscurity, in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves.

The Earth is the only world known so far to harbor life. There is nowhere else, at least in the near future, to which our species could migrate. Visit, yes. Settle, not yet. Like it or not, for the moment the Earth is where we make our stand.

It has been said that astronomy is a humbling and character-building experience. There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we’ve ever known.”

 

Powerful words indeed.

I will let you think about the cubical moon question a bit more before I reveal the solution.

References:

  1. What if Earth Were a Cube? – Discovery.com
  2. What Would It Be Like Walking around on a Cube Planet? – The Straight Dope
  3. Pale Blue Dot: A Vision of the Human Future in Space (1997), Carl Sagan

October Question: Bizarro World

When I was a kid, I used to read a lot of comics, which back then only came in dead tree format. In the classic Superman comics, there was a character that came from Bizzaro World. Bizzaro World was just like Earth, but instead of being spherical, it was a giant cube. So here is the thought experiment.

Imagine you were on a cubical planet, with the same mass as Earth, with the same atmosphere and oceans. Think about the following questions (hint – think about what “down” means):

  1. Where would all the water go? Remember, water accumulates in lower places, yet all six sides are flat.
  2. What would happen when you cross an edge? Would you simply flip 90 degrees?
  3. Imagine you are standing in the center of one of the sides. You decide to walk to the corner. What would happen along the way?

Bonus question. Imagine the following scenario: Earth is nice and spherical, however the moon is now a cube. Because the moon is tidally locked to the Earth, the same side always faces the earth. Let’s assume the side facing Earth is one of the square faces. The question is, what would the phases of the moon look like from Earth? Here is a drawing to help you visualize.

 

Feel free to post any questions you may have, I will try to answer them.

 

Images provided by Yahoo! Images and Classroom Clipart. Images not to scale. Also, the Sun does not really smile.