Tuesday, July 12, 2011
Einstein's riddle
This is the story behind Einstein's riddle: Albert Einstein supposedly created it in the late 1800s, and it is also said that he claimed 98% of the world population couldn't find a solution. In reality, it isn't that difficult, and I am not sure of the true origin, but I have seen this one floating around the internet, and it is a good brain exercise, so here it is:
- In a street there are five houses, painted five different colors.
- In each house lives a person of different nationality.
- These five homeowners each drink a different kind of beverage, smoke different brand of cigar and keep a different pet.
Einstein's riddle is: Who owns the fish?
Necessary clues:
1. The British man lives in a red house.
2. The Swedish man keeps dogs as pets.
3. The Danish man drinks tea.
4. The Green house is next to, and on the left of the White house.
5. The owner of the Green house drinks coffee.
6. The person who smokes Pall Mall rears birds.
7. The owner of the Yellow house smokes Dunhill.
8. The man living in the center house drinks milk.
9. The Norwegian lives in the first house.
10. The man who smokes Blends lives next to the one who keeps cats.
11. The man who keeps horses lives next to the man who smokes Dunhill.
12. The man who smokes Blue Master drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next to the blue house.
15. The Blends smoker lives next to the one who drinks water.
The World's Hardest Riddle?
Some have claimed this to be the worlds hardest riddle. It isn't. It is a decent riddle, though, and a fun one for those who like riddles with systematic solutions. My own solution is below.
A chart seems like the most useful tool to help solve this riddle: Five columns for the five houses, and five rows for nationality, house color, type of drink, type of cigar, and finally, pets. Clue #8 states the man in the middle house drinks milk, so we can start by filling in that one of the 25 boxes created.
1st 2nd 3rd 4th 5th
Nation
Color
Drink Milk
Cigar
Pet
Then we deduce as much as possible from each clue as it becomes usable. The Norwegian living in the first house (# 9) could mean the first on the left or the right, since it isn't specified. I assume the left (first on chart) for now. Often, with riddles or puzzles, it is faster to make an assumption and if it doesn't work out go back and try the other way, rather than trying to hold open both possibilities while analyzing the other clues.
Clue# 14 says the Norwegian lives next to the blue house, so we can fill in the house color in the second column.
Clue # 4 says the green house is to the left of the white house, and #5 says it is occupied by a coffee drinker. The only place that works is in column four, so we can fill in color and drink there, and white for the color of the fifth.
Clue #1 says the British man is in the red house, and the third house is the only one that has neither color nor nationality specified yet, so we can fill in those two boxes. This also gives us the color of the first house, since only yellow is left. Yellow smokes Dunhill (#7), so we get that too.
Horses are next to the Dunhill smoker (#11). Put that in the second column and here we are so far:
1st 2nd 3rd 4th 5th
Nation Norwegian British
Color Yellow Blue Red Green White
Drink Milk Coffee
Cigar Dunhill
Pet Horses
I have to admit that I was stumped at this point, until I started looking for "clumps" of information. The idea is that if you can put three or more things together at this point, there is likely only one column they will fit in. In this case, I started with clue # 12: The man who smokes Blue Master drinks beer. That is two bits of information that go together.
Now we have to determine what other bit of information can be "attached" to that. From what we have on the chart, we can see that the Norwegian smokes Dunhill and the British man drinks milk, so we rule out two nationalities.# 13 says the German smokes Prince, and #3 says the Danish man drinks tea, so we are left with just the Swedish man, who we now know smokes Blue Master and drinks beer. Scanning the clues for more information about the Swedish man we see that he has dogs (# 2). The only place that these four items fit is column five, so we fill that in.
Now it gets a bit easier. The "Blends" smoker is next to a water-drinker (#15) and the cat owner (# 10), which fits only in house 2 now, so we can put "blends" in 2 and "water" in 1. That leaves only "tea" for 2. Clue # 3 says the Danish man drinks tea, so we get that as well, which leaves just one slot (house 4) for the German.
The German smokes Prince (#13), which leaves only one slot (house 3) for the Pall Malls. This is how the chart now looks:
1st 2nd 3rd 4th 5th
Nation Norwegian Danish British German Swedish
Color Yellow Blue Red Green White
Drink Water Tea Milk Coffee Beer
Cigar Dunhill Blends Pall Mall Prince Blue Master
Pet Horses Dogs
Clue # 6 says the person who smokes Pall Mall rears birds (house 3). Clue #10 says the man who smokes Blends lives next to the one who keeps cats, so "cats" can only fit in the house 1 column. That leaves but one slot open, so the coffee-drinking, Prince Cigar-smoking German in the green house owns the fish in Einstein's riddle.
This is my own way of figuring it out, and I am sure there are other ways to arrive at the answer. It is a fun puzzle, and particularly good for practicing logical reasoning. This makes it worthy of our time whether or not this is truly Einstein's riddle.
Hydrogen Peroxide found in Space
Molecules of hydrogen peroxide have been found for the first time in interstellar space. The discovery gives clues about the chemical link between two molecules critical for life: water and oxygen. On Earth, hydrogen peroxide plays a key role in the chemistry of water and ozone in our planet's atmosphere, and is familiar for its use as a disinfectant or to bleach hair blonde. Now it has been detected in space by astronomers using the European Southern Observatory-operated APEX telescope in Chile.
Sunday, May 29, 2011
The Earth, the Sun and the Moon
The Earth, Sun, and Moon
The Earth
Earth, which is our base from which we look into space, is constantly moving. Understanding this movement is one of the most useful and important things in astronomy.
The earth orbits the sun in an elliptical orbit and the moon orbits the earth with the same kind of orbit. Looking down from the north pole, the earth spins in a counterclockwise direction on an imaginary line called its axis once every day. This accounts for the fact that the sun rises in the east and sets in the west. The earth’s axis is tilted with respect to the plane of its orbit at an angle of about 23.4 degrees. If we position ourselves high above the north pole, we would see that the earth orbits the sun in a counterclockwise motion, coming to the same position among the stars every 365.26 earth days. We would also see that the moon also orbits the earth in a counterclockwise motion. This is illustrated in the following example.
Figure 1: The directions of the orbits of the earth and moon.
The average distance from the earth to the sun, the semimajor axis of its orbit, is 149,597,890 km. This distance was not known until recently and it is called the astronomical unit or AU. The distances of the other planets to the sun are usually measured in astronomical units.
Because of the tilt of the earth, not every place on earth gets light every day. Also, some places have extremely short days.
As the earth revolves around the sun, the place where light shines the brightest changes. This motion gives us the different seasons. For instance, the poles receive less light than does the equator because of the angle that the land around the poles receive the sun’s light. When the north pole is tilted toward the sun, the northern hemisphere is presented to the sun at a greater angle than the southern hemisphere and the northern hemisphere gets warmer. When this happens, the northern hemisphere gets summer while the southern hemisphere gets winter. When the south pole is tilted toward the sun, the two seasons reverse hemispheres. This is illustrated in the following image.
Figure 2: The positions of earth at the different seasons. Counterclockwise from lower left: summer, fall, winter, spring (northern hemisphere).
The earth’s orbit is called the ecliptic. The plane which contains the ecliptic is the reference plane for the positions of most solar system bodies. Viewed from earth, the ecliptic is the apparent motion of the sun among the stars.
The earth’s equator is a circle going around the earth which is on a plane that is perpendicular to the earth’s axis. The equator and the plane on which it lies are illustrated in the following image.
Figure 3: The equatorial plane.
The Equinoxes
This equatorial plane is one of the most important in astronomy because it intersects the plane of the ecliptic and gives us a reference point in space by which we can measure the positions of stars. This plane also divides the earth into halves, the northern half being the northern hemisphere, the other half being the southern hemisphere. The intersection of these planes is a line, which for convenience we will call the line of equinoxes. The real definition of equinox is the point on the celestial sphere which intersects this line, but since the celestial sphere is an imaginary sphere with any size, the equinoxes are really lines. Also, for some purposes and illustrations, it is more convenient to think of the equinoxes as a line extending into space. For other purposes, it is convinient to think of the equinoxes as directions. The two planes are illustrated below.
Figure 4: The vernal equinox from two perspectives.
One half of this line is called the vernal equinox; the other half is called the autumnal equinox. At two points in the earth’s orbit this line intersects the sun. These two places mark the start of two of the four seasons, autumn or spring. The autumnal equinox starts autumn around September 23. From earth, this marks the time when the sun looks as if it is crossing the plane of the equator on its way south. The vernal equinox starts spring around March 21. This marks the time when the sun looks as if it is crossing the plane of the equator on its way north. The earth carries the plane of the equator along with it. When the sun looks as if it is on its way north or south, the earth is actually carrying the equatorial plane along so that it crosses the sun.
Perpendicular to this line of equinoxes is a line which contains the solstices. The solstices are points on the ecliptic which start the other two seasons, summer and winter, when they cross the sun. The summer solstice is one half of this line, the winter solstice is the other half of this line. The half of this line that is north of the celestial equator is the summer solstice, the half that is south of the celestial equator is the winter solstice. Currently, the winter solstice starts winter for the northern hemisphere at about the time the earth is closest to the sun. This line is illustrated in the following example.
Figure 5: The summer and winter solstices.
Because of centrifugal force involved when an object spins, the earth is not a perfect sphere, but is somewhat flattened at the poles and bulges out at the equator. The distance from any point on the equator to the center of the earth is longer than the distance from either pole to the center of the earth. This is illustrated in the following image which is exaggerated for clarity. The form caused by this equatorial bulge is called a geoid.
Figure 6: A geoid.
The Moon
The moon is the earth’s only natural satellite. Its average distance from the earth is 384,403 km. Its revolution period around the earth is the same length and direction as its rotation period, which results in the moon always keeping one side turned toward the earth and the other side turned away from the earth. This type of motion is called synchronous rotation. The side turned away from the earth is called the moon’s dark side, even though it is lit half of the time. The moon’s sidereal period of revolution is about 27.32 days long. This means that a line drawn through the center of the earth and the moon would point to the same star every 27.32 days. Due to slight variations in the orbital velocity of the moon, over a 30 year period, 59% of the moon’s surface is made visible. This is known as libration.
The moon’s orbit is not in the plane of the ecliptic and because of the elliptical nature of the moon’s orbit, it is not always the same distance from the earth. At the two intersections of the moon’s orbit and the plane of the ecliptic are two nodes. These nodes regress along the plane of the ecliptic, making one complete rotation every 18.61 years. See Orbits.
The Effect of the Moon
The moon has a noticeable effect on the earth in the form of tides, but it also affects the motion and orbit of the earth. The moon does not orbit the center of the earth, rather, they both revolve around the center of their masses called the barycenter. This is illustrated in the following animation.
Figure 7: The earth and moon revolving around the barycenter. Notice how the earth moves slightly.
The sun acts on the earth and its moon as one entity with its center at the barycenter. Since the earth revolves around the barycenter, which in turn orbits the sun, the earth follows a wobbly path around the sun. This is illustrated in the following example. To complicate things further, the barycenter is not always in the same place due to the elliptical nature of the moon’s orbit.
Figure 8: The wobble of the earth's orbit.
*Image illustrative only; number of intersections is greater.
The sun attracts the moon in such a way that it perturbs its orbit every 31.807 days, this phenomenon is called evection. The moon also changes the position of the earth’s equinoxes. The sun and moon each attract the earth’s equatorial bulge, trying to bring it into alignment with themselves. This torque is counteracted by the rotation of the earth. The combination of these two forces is a slow rotation of the earth’s axis, which in turn results in a slow westward rotation of the equinoxes. Looking down from the north pole, the equinoxes would appear to be rotating in a clockwise motion. The equinoxes and poles complete a rotation every 25,800 years. The equinoxes move at a rate of about 50.27 arc seconds per year. This phenomenon in known as the precession of the equinoxes and is illustrated in the following image.
The Earth
Earth, which is our base from which we look into space, is constantly moving. Understanding this movement is one of the most useful and important things in astronomy.
The earth orbits the sun in an elliptical orbit and the moon orbits the earth with the same kind of orbit. Looking down from the north pole, the earth spins in a counterclockwise direction on an imaginary line called its axis once every day. This accounts for the fact that the sun rises in the east and sets in the west. The earth’s axis is tilted with respect to the plane of its orbit at an angle of about 23.4 degrees. If we position ourselves high above the north pole, we would see that the earth orbits the sun in a counterclockwise motion, coming to the same position among the stars every 365.26 earth days. We would also see that the moon also orbits the earth in a counterclockwise motion. This is illustrated in the following example.
Figure 1: The directions of the orbits of the earth and moon.
The average distance from the earth to the sun, the semimajor axis of its orbit, is 149,597,890 km. This distance was not known until recently and it is called the astronomical unit or AU. The distances of the other planets to the sun are usually measured in astronomical units.
Because of the tilt of the earth, not every place on earth gets light every day. Also, some places have extremely short days.
As the earth revolves around the sun, the place where light shines the brightest changes. This motion gives us the different seasons. For instance, the poles receive less light than does the equator because of the angle that the land around the poles receive the sun’s light. When the north pole is tilted toward the sun, the northern hemisphere is presented to the sun at a greater angle than the southern hemisphere and the northern hemisphere gets warmer. When this happens, the northern hemisphere gets summer while the southern hemisphere gets winter. When the south pole is tilted toward the sun, the two seasons reverse hemispheres. This is illustrated in the following image.
Figure 2: The positions of earth at the different seasons. Counterclockwise from lower left: summer, fall, winter, spring (northern hemisphere).
The earth’s orbit is called the ecliptic. The plane which contains the ecliptic is the reference plane for the positions of most solar system bodies. Viewed from earth, the ecliptic is the apparent motion of the sun among the stars.
The earth’s equator is a circle going around the earth which is on a plane that is perpendicular to the earth’s axis. The equator and the plane on which it lies are illustrated in the following image.
Figure 3: The equatorial plane.
The Equinoxes
This equatorial plane is one of the most important in astronomy because it intersects the plane of the ecliptic and gives us a reference point in space by which we can measure the positions of stars. This plane also divides the earth into halves, the northern half being the northern hemisphere, the other half being the southern hemisphere. The intersection of these planes is a line, which for convenience we will call the line of equinoxes. The real definition of equinox is the point on the celestial sphere which intersects this line, but since the celestial sphere is an imaginary sphere with any size, the equinoxes are really lines. Also, for some purposes and illustrations, it is more convenient to think of the equinoxes as a line extending into space. For other purposes, it is convinient to think of the equinoxes as directions. The two planes are illustrated below.
Figure 4: The vernal equinox from two perspectives.
One half of this line is called the vernal equinox; the other half is called the autumnal equinox. At two points in the earth’s orbit this line intersects the sun. These two places mark the start of two of the four seasons, autumn or spring. The autumnal equinox starts autumn around September 23. From earth, this marks the time when the sun looks as if it is crossing the plane of the equator on its way south. The vernal equinox starts spring around March 21. This marks the time when the sun looks as if it is crossing the plane of the equator on its way north. The earth carries the plane of the equator along with it. When the sun looks as if it is on its way north or south, the earth is actually carrying the equatorial plane along so that it crosses the sun.
Perpendicular to this line of equinoxes is a line which contains the solstices. The solstices are points on the ecliptic which start the other two seasons, summer and winter, when they cross the sun. The summer solstice is one half of this line, the winter solstice is the other half of this line. The half of this line that is north of the celestial equator is the summer solstice, the half that is south of the celestial equator is the winter solstice. Currently, the winter solstice starts winter for the northern hemisphere at about the time the earth is closest to the sun. This line is illustrated in the following example.
Figure 5: The summer and winter solstices.
Because of centrifugal force involved when an object spins, the earth is not a perfect sphere, but is somewhat flattened at the poles and bulges out at the equator. The distance from any point on the equator to the center of the earth is longer than the distance from either pole to the center of the earth. This is illustrated in the following image which is exaggerated for clarity. The form caused by this equatorial bulge is called a geoid.
Figure 6: A geoid.
The Moon
The moon is the earth’s only natural satellite. Its average distance from the earth is 384,403 km. Its revolution period around the earth is the same length and direction as its rotation period, which results in the moon always keeping one side turned toward the earth and the other side turned away from the earth. This type of motion is called synchronous rotation. The side turned away from the earth is called the moon’s dark side, even though it is lit half of the time. The moon’s sidereal period of revolution is about 27.32 days long. This means that a line drawn through the center of the earth and the moon would point to the same star every 27.32 days. Due to slight variations in the orbital velocity of the moon, over a 30 year period, 59% of the moon’s surface is made visible. This is known as libration.
The moon’s orbit is not in the plane of the ecliptic and because of the elliptical nature of the moon’s orbit, it is not always the same distance from the earth. At the two intersections of the moon’s orbit and the plane of the ecliptic are two nodes. These nodes regress along the plane of the ecliptic, making one complete rotation every 18.61 years. See Orbits.
The Effect of the Moon
The moon has a noticeable effect on the earth in the form of tides, but it also affects the motion and orbit of the earth. The moon does not orbit the center of the earth, rather, they both revolve around the center of their masses called the barycenter. This is illustrated in the following animation.
Figure 7: The earth and moon revolving around the barycenter. Notice how the earth moves slightly.
The sun acts on the earth and its moon as one entity with its center at the barycenter. Since the earth revolves around the barycenter, which in turn orbits the sun, the earth follows a wobbly path around the sun. This is illustrated in the following example. To complicate things further, the barycenter is not always in the same place due to the elliptical nature of the moon’s orbit.
Figure 8: The wobble of the earth's orbit.
*Image illustrative only; number of intersections is greater.
The sun attracts the moon in such a way that it perturbs its orbit every 31.807 days, this phenomenon is called evection. The moon also changes the position of the earth’s equinoxes. The sun and moon each attract the earth’s equatorial bulge, trying to bring it into alignment with themselves. This torque is counteracted by the rotation of the earth. The combination of these two forces is a slow rotation of the earth’s axis, which in turn results in a slow westward rotation of the equinoxes. Looking down from the north pole, the equinoxes would appear to be rotating in a clockwise motion. The equinoxes and poles complete a rotation every 25,800 years. The equinoxes move at a rate of about 50.27 arc seconds per year. This phenomenon in known as the precession of the equinoxes and is illustrated in the following image.
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