Sugars are white crystalline carbohydrates that are soluble in water and generally have a sweet taste.

Monosaccharides are simple sugars

Many saccharide structures differ only in the orientation of the hydroxyl groups (-OH). This slight structural difference makes a big difference in the biochemical properties, organoleptic properties (e.g., taste), and in the physical properties such as melting point and Specific Rotation (how polarized light is distorted). A chain-form monosaccharide that has a carbonyl group (C=O) on an end carbon forming an aldehyde group (-CHO) is classified as an aldose. When the carbonyl group is on an inner atom forming a ketone, it is classified as a ketose.

Label the balls from 1 to 12 to identify them.
Weigh 1, 2, 3, 4 against 5, 6, 7, 8:

1. If they balance, 9, 10, 11, 12 contain the odd ball.
   Weigh 6, 7, 8 against 9, 10, 11.
   1. If they balance, 12 is the odd ball. Weigh 12 against any other ball to discover whether it is heavy or light.
   2. If 9, 10, 11 are heavy, they contain an odd heavy ball. Weigh 9 against 10. If they balance, 11 is the odd heavy ball, otherwise the heavier of 9 and 10 is the odd ball.
   3. If 9, 10, 11 are light, we use the same procedure to reach the same conclusion for the odd light ball.
2. If 5, 6, 7, 8 are heavy, either they contain an odd heavy ball or 1, 2, 3, 4 contain an odd light ball.
   Weigh 1, 2, 5 against 3, 6, 10.
   1. If they balance, the odd ball is 4 (light) or 7 or 8 (heavy). Weigh 7 against 8. If they balance 4 is light, otherwise the heavier of 7 and 8 is the odd heavy ball.
   2. If 3, 6, 10 are heavy, the odd ball can be 6 (heavy) or 1 or 2 (light). Weigh 1 against 2. If they balance 6 is heavy, otherwise the lighter of 1 and 2 is the odd light ball.
   3. If 3, 6, 10 are light, the odd ball is 3 and light or 5 and heavy. We thus weigh 3 against 10. If they balance, 5 is heavy, otherwise 3 is light.
3. If 5, 6, 7, 8 are light we use a similar procedure to that in II.

The container was half full at 12:59 PM. When the bacteria doubled in the next minute, the container became full. This is an example of exponential growth where the growth rate is a mathematical function that is proportional to the function’s current value.

If you ask a guard directly “Are you guarding the path to enlightenment?”, and the answer is “no”, he could be guarding the path to enlightenment and be lying about it, or he could be telling the truth and the path to enlightenment is behind the other door.

The question that you ask has to involve both guards at the same time:

“Would the other guard say that you are guarding the path to enlightenment?”When we ask a guard this question, there are 4 cases:

1. The liar is guarding the path to enlightenment. He answers “no” because the truthful guard would say “yes”.
2. The liar is not guarding the path to enlightenment: He answers “yes” because the truthful guard would say “no”.
3. The truth teller is guarding the path to enlightenment. He answers “no” because the other guard (liar) would say “no”.
4. The truth teller is not guarding the path to enlightenment. He answers “yes” because the other guard (liar) would say “yes”.

So, if a guard answers “no”, he is guarding the path to enlightenment. If he answers “yes”, the path to enlightenment is the other door. Notice that even though we have learned which is the path to enlightenment, we still don’t know which guard is the liar. To find out who is the liar we would have to ask a question like: “Would the other guard say that you always tell the truth?” A reply of “no” means you are talking to the truth teller, a reply of “yes” means you are talking to the liar.

Astronomical and geological evidence indicates that the Universe is approximately 13,820 million years old[42], and our solar system is about 4,567 million years old. Earth’s Moon formed 4,450 million years ago, just 50 million years after the Earth’s formation. Because the composition of the rocks retrieved from the Moon by the Apollo missions is very similar to rocks from the Earth, it is thought that the Moon formed as a result of a collision between the young Earth and a Mars-sized body, sometimes called Theia, which accreted at a Lagrangian point 60° ahead or behind the Earth. A cataclysmic meteorite bombardment (the Late Heavy Bombardment) of the Moon and the Earth 3,900 million years ago is thought to have been caused by impacts of planetesimals which were originally beyond the Earth, but whose orbits were destabilized by the migration of Jupiter and Saturn during the formation of the solar system. The Mars Reconnaissance Orbiter and the Mars Global Surveyor have found evidence that the Borealis basin in the northern hemisphere of Mars may have been created by a colossal impact with an object 2,000 kilometers in diameter at the time of the Late Heavy Bombardment.

The bear was white because it was a polar bear. The only place on earth where a bear can go south, west and north equal distances and end up where it started is the North Pole.

Actually, the bear could go west two or five kilometers instead of one and it would not make any difference — the bear would be making a circle around the North Pole. East and West you travel along parallels which are circles equidistant from the poles. North and South you travel along meridians which are circles that cross both the north and the south poles.

To find a pair, Mr. Brown must pick at least 2 gloves. But, if he picks 2 gloves blindly, then they may be of different colors. If he picks 3 gloves blindly, then there are only 2 possibilities: all 3 gloves are of the same color, or 2 gloves are of the same color and 1 is of a different color. Both these possibilities guarantee Mr. Brown a pair of gloves of the same color. Thus, Mr. Brown should minimally pick 3 gloves to be certain to find a pair of gloves of the same color.