Avoid The Collision of Ants

Three ants are sitting at the three corners of an equilateral triangle. Each ant starts randomly picks a direction and starts to move along the edge of the triangle. What is the probability that none of the ants collide?

'THIS' is the probability!

To Avoid The Collision of Ants

What was the problem?

Each ant can decide to go either clockwise or anti-clockwise. That is there are 2 options available for each ant to go. Hence, there will be total 2x2x2 = 8 possible combination of ants' different paths.

Now 2 ants won't collide if & only if all are either moving clockwise or anti-clockwise. In short, out of 8 possible combinations only 2 combinations are there where ants won't collide.

Hence, the probability that ant won't collide is 2/8 = 0.25.  

The Newspaper Puzzle in News

A newspaper made of 16 large sheets of paper folded in half. The newspaper has 64 pages altogether. The first sheet contains pages 1, 2, 63, 64.

If we pick up a sheet containing page number 45 what are the other pages that this sheet contains?

Here is the solution of puzzle!

The Newspaper Puzzle : Solution

What was the puzzle?

If the page starts with odd number 1 & back side followed by even 2 then at the back of page number 45, there must be number 46.

Generally, for the page p, page 65 - p shares the same sheet as p. Like page 1 and 65-1 =64, page 2 and 65-2=63 page are on the same sheet.

Therefore, page 45 must be on the same sheet as 65 - 45 = 20 page number. And the page 46 must be on the same sheet as page 65 - 46 = 19.

In short, the pages 19,20,45 and 46 must be on the same sheet. 

Sara's Desert Trek

Sara needs to trek from an oasis to a destination 10 miles away across a barren desert. 

The facts:

• Crossing one mile of desert requires using 1 gallon of water.
• Sara can only carry 6 gallons of water at a time.
• Sara can drop a water cache (of any amount of water from the supply she is carrying at that moment) at any of the nine stops along the route, and then pick up any part of the cache on a later trip.

What's the minimum number of times Sara must leave the oasis in order to cross the entire 10 mile span of desert?

This is how she optimizes her journey! 

Sara's Planning in Desert Trek

What was the challenge in journey?

1. First Sara collects 12 gallons of water at milepost 1 after having 3 trips from source. She uses 2 gallons (out of 6) for forward & backward journey from source to milepost & dropping 4 gallons in cache at milepost 1.

2.She collect 6 gallons more water at the start of 4th trip from source & drops 5 gallons at milepost 1. Now, she doesn't need to return back to source and 17 gallons of water available at milepost 1.

3.In next 2 rounds, she moves 8 gallons of water from milepost 1 to milepost 2 (1 for forward + 4 for drop + 1 for backward journey in each round). 

4.Now only 5 gallons left at milepost 2. She uses 1 gallon for journey from milepost 1 to milepost 2 and drop remaining 4 gallons at milepost 2. Now, 12 gallons of water is available at milepost 2.

3.Next, using 2 gallons (out of 6 which is maximum she can carry) she moves from milepost 2 to milepost 4 and drop 2 gallons at milepost 4 & comes back at milepost 2 using remaining 2. 

4. Again, on arriving back at milepost 2, she has left with 6 gallons of water at milepost 2 out of which she uses 2 to reach milepost 4 where 2 gallons of water still available there already collected in previous round. Now, she doesn't need to return back from
milepost 4.

5. She uses the remaining 6 gallons of water to reach at the milepost 10.

To conclude, Sara has to leave Oasis only 4 times as describe in steps 1 and 2 if she want to cross the entire 10 mile span of desert.  

Murder or Suicide? - What Do You Think?

A dead body lies at the bottom of a multistory building. It looks as though he committed suicide by jumping from one of the floors.

When the detective arrives, he goes to the first floor of the building, opens the closed window, and flips a coin towards the floor. He goes to the second floor and does the exact same thing. He continues to do this until he gets to the top floor of the building.

When he comes back down, he states that it was a murder and not a suicide. How does he know that?

(Entry to the terrace was banned due to some ongoing work).

Read the detective's logic in the case!

Detective's Logic in Murder Mystery

What is the case?

Obviously, the person can't jump with window closed or come outside of the window & close that window from inside.

The detective checks if windows is closed from inside by opening window & flipping the coin toward the floor. He flips the coin to mark the count of that particular floor where window was closed from inside & he has to open it to flip the coin.

At the end, he collects as many coin as floors of that building. So he concludes that no floor had open window from where the person might have jumped.

 
So, he concludes that it was the murder and not the suicide.

Four Glasses Puzzle

Four glasses are placed on the corners of a square table. Some of the glasses are upright (up) and some upside-down (down). A blindfolded person is seated next to the table and is required to re-arrange the glasses so that they are all up or all down, either arrangement being acceptable, which will be signaled by the ringing of a bell. 

The glasses may be re-arranged in turns subject to the following rules. 

1.Any two glasses may be inspected in one turn and after feeling their orientation the person may reverse the orientation of either, neither or both glasses.

2.After each turn the table is rotated through a random angle. 

3.The puzzle is to devise an algorithm which allows the blindfolded person to ensure that all glasses have the same orientation (either up or down) in a finite number of turns. The algorithm must be non-stochastic i.e. it must not depend on luck.

Here is that algorithm!

Solution of Blind Bartender's Problem

What was the puzzle?

Below is the algorithm which makes sure the bell will ring in at most five turns.

1.On the first turn choose a diagonally opposite pair of glasses and turn both glasses up.
At this point, the position of other 2 glasses is not known.

2.On the second turn, choose 2 adjacent glasses. One of them was turned up in the previous step, so other may or may not in up position. If the other is down then turn it up and if remaining one X is also in up position then bell will be rung.

If the bell does not ring then there are now three glasses up and one down(3U and 1D).

3.On the third turn choose a diagonally opposite pair of glasses. If one is down, turn it up and the bell will ring.

And if you find both are up, then you must have chosen other diagonally opposite pair.

If so, then turn one down so that 2 glasses are up and other 2 are down.

4.On the fourth turn choose two adjacent glasses and reverse both. If both were in the same orientation then the bell will ring. 

And in case, if you find one is up and other down like -

still reverse orientation of both as - 

Now diagonally opposite pairs are either up or down.

5.On the fifth turn choose a diagonally opposite pair of glasses and reverse both.

The bell will ring for sure.

   

Cut The Blue Cube Puzzle

A solid, four-inch cube of wood is coated with blue paint on all six sides.

Then the cube is cut into smaller one-inch cubes. These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?

Here is solution of the puzzle! 

Cut The Blue Cube Puzzle : Solution

What is the puzzle?

Apart from the 8 cubes at the center all 4 x 4 x 4 - 8 = 56 will have some paint on ones side at least. See below the 1/4 th cube is taken out.

The cubes at the 8 corners will have blue paint on three sides.

The cubes between corner cubes along 12 edges of big cube will have 2 sides painted. That is 12 x 2 = 24 cubes will painted with blue on 2 sides.

And 4 center cubes on each of 6 faces (left, right, top, bottom, front, back) will have only 1 side painted with blue. That is , there are 6 x 4 = 24 cubes having paint on one side only.

To conclude, out of 56 painted cubes,

24 cubes have paint on 1 side,

24 cubes painted with 2 sides,

8 are painted with three sides.

Zebra Puzzle or Einstein Puzzle

This puzzle has been attributed both to Lewis Carroll and to Albert Einstein:

1. There are five houses in a row. Each of the houses is painted a different color, and their occupants come from different countries, own different pets, drink different beverages, and smoke different cigarette brands.
2. The Englishman lives in the red house.
3. The Spaniard owns the dog.
4. Coffee is drunk in the green house.
5. The Ukrainian drinks tea.
6. The green house is immediately to the right (your right) of the ivory house.
7. The Old Gold smoker owns snails.
8. Kools are smoked in the yellow house.
9. Milk is drunk in the middle house.
10. The Norwegian lives in the first house.
11. The man who smokes Chesterfields lives in the house next to the man with the fox.
12. Kools are smoked in the house next to the house where the horse is kept.
13. The Lucky Strike smoker drinks orange juice.
14. The Japanese smokes Parliaments.
15. The Norwegian lives next to the blue house.

Who drinks water? Who owns the zebra?

Well, here is effort to simplify the solution 


From Wikipedia : The puzzle is often called Einstein's Puzzle or Einstein's Riddle because it is said to have been invented by Albert Einstein as a boy; it is also sometimes attributed to Lewis Carroll. However, there is no known evidence for Einstein's or Carroll's authorship and the Life International version of the puzzle mentions brands of cigarette, such as Kools, that did not exist during Carroll's lifetime or Einstein's boyhood. 
 

Simplified Solution of Zebra Puzzle or Einstein Puzzle

What was the puzzle?

Let's recollect the given clues once again.

1. There are five houses in a row. Each of the houses is painted a different color, and their occupants come from different countries, own different pets, drink different beverages, and smoke different cigarette brands.
2. The Englishman lives in the red house.
3. The Spaniard owns the dog.
4. Coffee is drunk in the green house.
5. The Ukrainian drinks tea.
6. The green house is immediately to the right (your right) of the ivory house.
7. The Old Gold smoker owns snails.
8. Kools are smoked in the yellow house.
9. Milk is drunk in the middle house.
10. The Norwegian lives in the first house.
11. The man who smokes Chesterfields lives in the house next to the man with the fox.
12. Kools are smoked in the house next to the house where the horse is kept.
13. The Lucky Strike smoker drinks orange juice.
14. The Japanese smokes Parliaments.
15. The Norwegian lives next to the blue house.

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STEP 1 :

As per clue (1), let's create a table like below, and then fill the data one by one.

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STEP 2 :

As per clue (9),(10) and (15),

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STEP 3 :

As per (2), Englishman lives in red house, that means it can't be first house. Also, (6) points that green house is next to ivory. So, ivory-green can't be at 1-2 positions. But those can be at 3-4 with red at 5 or at 4-5 with red at 3.

In short, only color left for 1 is yellow. And as per (8), Kools are smoked in the yellow house. 

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STEP 4 :

Now as per (12), Kools are smoked in the house next to the house where the horse is kept.

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 STEP 5 :

Let's assume the red house of Englishman is at no.5 then Ivory-Green are at no.3-4.
Since as per (4), green house owner drinks coffee, and only position left for Ukrainian drinking tea pointed by (5) is house no.2.

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STEP 6 :

Now the luck strike smoker drinking orange juice pointed by (13) has only position and that is house no.5. That leaves only drink available for house 1 as water.

As per (3), Spaniard who owns dog has houses 3 and 4 available but for Old Gold smoker (pointed by (7)) also available houses are 3 and 4 again.

If Old Gold smoker occupies house 3 then Spaniard who own dog would have house 4. In that case, there will be no position would be left for Japanese smoking Parliaments (Clue no.14).

Same can be proved if Old Gold smoker occupies house no.4.

Hence, our assumption made in step 5 is wrong. So, Red house must be at 3 owned by Englishman.

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STEP 7 :

So if Englishman's red house is at no.3 then at no.4 & 5 there must be ivory and green houses where coffee is drink in green house.

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STEP 8 :

Now, for Lucky Strike smoker who drinks orange juice (clue 13) and Ukrainian who drinks tea (clue 5) only available houses are (see above table) house no.2 & 4. That is, 2 and 4 are occupied by tea & orange juice drinkers (order yet unknown). With that, we can conclude that the drink in first house is Water.

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STEP 9 : 

If we assume that Lucky Striker smoker who drinks Orange juice (clue 13) live in house no.2 then Ukrainian who drinks tea (clue 5) has to be in house no.4. With that, Japanese who smokes Parliament (clue 14) need to be in the house no.5. No house is left for Spaniard who owns the dog (clue 3) as house no.2 is already occupied by pet horse. 

That's why the Lucky Strike smoker with Orange juice as drink must be in house no.4.

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STEP 10 : 

Now, if Lucky Strike smoker (clue 13) is living in house no.4 then Ukrainian who drinks tea (clue 5) must be in house no.2. With that, Japanese who smokes Parliament (clue 15) needs to be in house no.5 and Spaniard who owns dog (clue 3) needs to be in house no.4.

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STEP 11 :

In the end, for Old Gold smoker (clue 7) only house left is house no.3. And as per clue 11,Chasterfield smoker needs to be in the house followed by fox owner. Hence, they must be in the house no.1 and 2 respectively. Finally, the only pet left for house no.5 is Zebra.

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CONCLUSION : 

The final table looks like - 

Now, finding answers of given questions is easy from the table.

Who drinks water? 
- Norwegion

Who owns the zebra?  
- Japanese