A Famous Prize Problem

Here is a famous prize problem that Sam Loyd issued in 1882, offering $1000 as a prize for the best answer showing how to arrange the seven figures and the eight 'dots' .4.5.6.7.8.9.0. which would add up to 82.

 

Answer:



Answer

 

Length of Bridge

Charles walks over a railway-bridge. At the moment that he is just ten meters away from the middle of the bridge, he hears a train coming from behind. At that moment, the train, which travels at a speed of 90 km/h, is exactly as far away from the bridge as the bridge measures in length. Without hesitation, Charles rushes straight towards the train to get off the bridge. In this way, he misses the train by just four meters! If Charles would, however, have rushed exactly as fast in the other direction, the train would have hit him eight meters before the end of the bridge.

What is the length of the railway-bridge?

 

Answer:

44 meters

Let the length of the bridge be x meters.
Running towards the train, Charles covers 0.5x-10 meters in the time that the train travels x-4 meters. Running away from the train, Charles covers 0.5x+2 meters in the time that the train travels 2x-8 meters.
Because their speeds are constant, the following holds:
(0.5x-10) / (x-4) = (0.5x+2) / (2x-8)
which can be rewritten to
0.5x2 - 24x + 88 = 0
Using the abc formula we find that x=44, so the railway-bridge has a length of 44 meters.

7 + 8 = ?

If:

2 + 3 =8,
3 + 7 =27,
4 + 5 =32,
5 + 8 =60,
6 + 7 =72,

then:

7 + 8 = ?





Answer:

98
2+3=2*[3+(2-1)]=8
3+7=3*[7+(3-1)]=27
4+5=4*[5+(4-1)]=32
5+8=5*[8+(5-1)]=60
6+7=6*[7+(6-1)]=72

therefore:

7+8=7*[8+(7-1)]=98
x+y=x[y+(x-1)]=x^2+xy-x"

Replace 'X'

Replace the 'X's by any mathematical symbols to make the expression equal to 111.

18 X 12 X 2 X 3 = 111

 

Answer:

18 * 12 ÷ 2 + 3 = 111

Find a Number

A number with an interesting property:

When I divide it by 2, the remainder is 1.
When I divide it by 3, the remainder is 2.
When I divide it by 4, the remainder is 3.
When I divide it by 5, the remainder is 4.
When I divide it by 6, the remainder is 5.
When I divide it by 7, the remainder is 6.
When I divide it by 8, the remainder is 7.
When I divide it by 9, the remainder is 8.
When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.
When I looked for a smaller number with this property I couldn't find one.

Can you find it?
 

Answer:

Find the LCM of 2,3,4,5,6,7,8,9,10. It is 2520.

Number is: 2520-1 = 2519

 

Other approach

The number has to end in 9.
Look brute force for small numbers.
39 + multiples of 40,
69 + multiples of 70 and
89 + multiples of 90
Smallest one is 2519.

 

 

 

What should Henry choose?

Henry has been caught stealing cattle, and is brought into town for justice. The judge is his ex-wife Gretchen, who wants to show him some sympathy, but the law clearly calls for two shots to be taken at Henry from close range. To make things a little better for Henry, Gretchen tells him she will place two bullets into a six-chambered revolver in successive order. She will spin the chamber, close it, and take one shot. If Henry is still alive, she will then either take another shot, or spin the chamber again before shooting.

Henry is a bit incredulous that his own ex-wife would carry out the punishment, and a bit sad that she was always such a rule follower. He steels himself as Gretchen loads the chambers, spins the revolver, and pulls the trigger. Whew! It was blank. Then Gretchen asks, 'Do you want me to pull the trigger again, or should I spin the chamber a second time before pulling the trigger?'

What should Henry choose?

Answer:

Henry should have Gretchen pull the trigger again without spinning.

We know that the first chamber Gretchen fired was one of the four empty chambers. Since the bullets were placed in consecutive order, one of the empty chambers is followed by a bullet, and the other three empty chambers are followed by another empty chamber. So if Henry has Gretchen pull the trigger again, the probability that a bullet will be fired is 1/4.

If Gretchen spins the chamber again, the probability that she shoots Henry would be 2/6, or 1/3, since there are two possible bullets that would be in firing position out of the six possible chambers that would be in position.

Cross Bridge

Four people need to cross a rickety bridge at night. Unfortunately, they have only one torch and the bridge is too dangerous to cross without one. The bridge is only strong enough to support two people at a time. Not all people take the same time to cross the bridge. Times for each person: 1 min, 2 mins, 7 mins and 10 mins. What is the shortest time needed for all four of them to cross the bridge?

Answer:

17 mins

The initial solution most people will think of is to use the fastest person as an usher to guide everyone across. How long would that take? 10 + 1 + 7 + 1 + 2 = 21 mins. Is that it? No. That would make this question too simple even as a warm up question.

Let's brainstorm a little further. To reduce the amount of time, we should find a way for 10 and 7 to go together. If they cross together, then we need one of them to come back to get the others. That would not be ideal. How do we get around that? Maybe we can have 1 waiting on the other side to bring the torch back. Ahaa, we are getting closer. The fastest way to get 1 across and be back is to use 2 to usher 1 across. So let's put all this together.

1 and 2 go cross
2 comes back
7 and 10 go across
1 comes back
1 and 2 go across (done)

Total time = 2 + 2 + 10 + 1 + 2 = 17 mins

G T Y O R J O T E O U I A B G T

What does this message say?

G T Y O R J O T E O U I A B G T


Hint:
Count the letters and try splitting the letters up into groups.

Answer:

'Great Job You Got It'
This type of code is known as a Caesar Box (Julius Caesar was the first to write codes this way.) To decipher the message, simply divide the code into four groups of four (you can also divide them into groups such as 5 groups of 5 or 6 groups of 6 depending on the number of letters in the phrase), and rearrange them vertically like this ...

G T Y O
R J O T
E O U I
A B G T

Then you read vertically column by column.

Two Eggs

> You are given 2 eggs.

> You have access to a 100-storey building.

> Eggs can be very hard or very fragile means it may break if dropped from the first floor or may not even break if dropped from 100 th floor.Both eggs are identical.

> You need to figure out the highest floor of a 100-storey building an egg can be dropped without breaking.

> Now the question is how many drops you need to make. You are allowed to break 2 eggs in the process.

 
 

Answer:

Answer is: 14
Let x be the answer we want, the number of drops required.

So if the first egg breaks maximum we can have x-1 drops and so we must always put the first egg from height x. So we have determined that for a given x we must drop the first ball from x height. And now if the first drop of the first egg doesn’t breaks we can have x-2 drops for the second egg if the first egg breaks in the second drop.

Taking an example, lets say 16 is my answer. That I need 16 drops to find out the answer. Lets see whether we can find out the height in 16 drops. First we drop from height 16,and if it breaks we try all floors from 1 to 15.If the egg don’t break then we have left 15 drops, so we will drop it from 16+15+1 =32nd floor. The reason being if it breaks at 32nd floor we can try all the floors from 17 to 31 in 14 drops (total of 16 drops). Now if it did not break then we have left 13 drops. and we can figure out whether we can find out whether we can figure out the floor in 16 drops.

Lets take the case with 16 as the answer

1 + 15 16 if breaks at 16 checks from 1 to 15 in 15 drops
1 + 14 31 if breaks at 31 checks from 17 to 30 in 14 drops
1 + 13 45 .....
1 + 12 58
1 + 11 70
1 + 10 81
1 + 9 91
1 + 8 100 We can easily do in the end as we have enough drops to accomplish the task


Now finding out the optimal one we can see that we could have done it in either 15 or 14 drops only but how can we find the optimal one. From the above table we can see that the optimal one will be needing 0 linear trials in the last step.

So we could write it as

(1+p) + (1+(p-1))+ (1+(p-2)) + .........+ (1+0) >= 100.

Let 1+p=q which is the answer we are looking for

q (q+1)/2 >=100

Solving for 100 you get q=14.
So the answer is: 14
Drop first orb from floors 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, 100... (i.e. move up 14 then 13, then 12 floors, etc) until it breaks (or doesn't at 100)

Aeroplane Quiz

The puzzle question is : On Bagshot Island, there is an airport. The airport is the homebase of an unlimited number of identical airplanes. Each airplane has a fuel capacity to allow it to fly exactly 1/2 way around the world, along a great circle. The planes have the ability to refuel in flight without loss of speed or spillage of fuel. Though the fuel is unlimited, the island is the only source of fuel.
What is the fewest number of aircraft necessary to get one plane all the way around the world assuming that all of the aircraft must return safely to the airport? How did you get to your answer?
Notes:
(a) Each airplane must depart and return to the same airport, and that is the only airport they can land and refuel on ground.
(b) Each airplane must have enough fuel to return to airport.
(c) The time and fuel consumption of refueling can be ignored. (so we can also assume that one airplane can refuel more than one airplanes in air at the same time.)
(d) The amount of fuel airplanes carrying can be zero as long as the other airplane is refueling these airplanes. What is the fewest number of airplanes and number of tanks of fuel needed to accomplish this work? (we only need airplane to go around the world)

 

Answer:

As per the puzzle given above The fewest number of aircraft is 3!

Imagine 3 aircraft (A, B and C). A is going to fly round the world. All three aircraft start at the same time in the same direction. After 1/6 of the circumference, B passes 1/3 of its fuel to C and returns home, where it is refueled and starts immediately again to follow A and C.

C continues to fly alongside A until they are 1/4 of the distance around the world. At this point C completely fills the tank of A which is now able to fly to a point 3/4 of the way around the world. C has now only 1/3 of its full fuel capacity left, not enough to get back to the home base. But the first 'auxiliary' aircraft reaches it in time in order to refuel it, and both 'auxiliary' aircraft are the able to return safely to the home base.

Now in the same manner as before both B and C fully refuelled fly towards A. Again B refuels C and returns home to be refuelled. C reaches A at the point where it has flown 3/4 around the world. All 3 aircraft can safely return to the home base, if the refuelling process is applied analogously as for the first phase of the flight.

Medieval Empire

You are the ruler of a medieval empire and you are about to have a celebration tomorrow. The celebration is the most important party you have ever hosted. You've got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned. The poison exhibits no symptoms until death. Death occurs within ten to twenty hours after consuming even the minutest amount of poison. You have over a thousand slaves at your disposal and just under 24 hours to determine which single bottle is poisoned. You have a handful of prisoners about to be executed, and it would mar your celebration to have anyone else killed. What is the smallest number of prisoners you must have to drink from the bottles to be absolutely sure to find the poisoned bottle within 24 hours?


Answer:

10 prisoners must sample the wine. Bonus points if you worked out a way to ensure than no more than 8 prisoners die.

Number all bottles using binary digits. Assign each prisoner to one of the binary flags. Prisoners must take a sip from each bottle where their binary flag is set.

Here is how you would find one poisoned bottle out of eight total bottles of wine.

Bottle 1 Bottle 2 Bottle 3 Bottle 4 Bottle 5 Bottle 6 Bottle 7 Bottle 8
Prisoner A X X X X
Prisoner B X X X X
Prisoner C X X X X
In the above example, if all prisoners die, bottle 8 is bad. If none die, bottle 1 is bad. If A & B dies, bottle 4 is bad.

With ten people there are 1024 unique combinations so you could test up to 1024 bottles of wine.

Each of the ten prisoners will take a small sip from about 500 bottles. Each sip should take no longer than 30 seconds and should be a very small amount. Small sips not only leave more wine for guests. Small sips also avoid death by alcohol poisoning. As long as each prisoner is administered about a millilitre from each bottle, they will only consume the equivalent of about one bottle of wine each.

Each prisoner will have at least a fifty percent chance of living. There is only one binary combination where all prisoners must sip from the wine. If there are ten prisoners then there are ten more combinations where all but one prisoner must sip from the wine. By avoiding these two types of combinations you can ensure no more than 8 prisoners die.

Image Strips Puzzle [1]

In this post you try a simple Image Strips Puzzle (by G. R. Roosta).

Study the following image for no more than 20 second:

Image Strips Puzzle

Scroll down!

Scroll down!

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Scroll down!

Scroll down!

Scroll down!

Scroll down!

Scroll down!

Scroll down!


Now, from memory, identify which of the following pictures can be a complete version of the above image!

(a)

 

 

(b)

 

(c)

 

 



 

Answer:

Picture (b).
 

Ferrying Soldiers

A detachment of 25 soldiers must cross a wide and deep river with no bridge in sight. They notice two 12-year-old boys playing in a rowboat by the shore. The boat is so tiny, however, that it can only hold two boys or one soldier. How can the soldiers get across the river and leave the boys in joint possession of the boat? How many times does the boat pass from shore to shore in your algorithm?

 

Answer:

First, the two boys take the boat to the other side, after which one of them returns with the boat. Then a soldier takes the boat to the other side and stays there while the other boy returns the boat. These four trips reduce the problem size—measured by the number of soldiers to be ferried—by 1. Thus, if this fourtrip procedure is repeated the total of 25 times, the problem will be solved after the total of 100 trips. (Of course, for the general instance of n soldiers, 4n trips will need to be made.)

Artistic, Historic and Scientific Interest

The institution houses collections of objects of artistic, historic and scientific interest, and displayed for the edification and enjoyment of the public. One word has been removed from the passage above. Select that word from the choice below and reinstate it into its correct place in the passage.

a. huge b. permanent c. produced d. conserved e. priceless f. accumulated

 

Answer:

d, conserved: The institution houses collections of objects of artistic, historic and scientific interest, conserved and displayed for the edification and enjoyment of the public.

Glove Selection

There are 20 gloves in a drawer: 5 pairs of black gloves, 3 pairs of brown, and 2 pairs of gray. You select the gloves in the dark and can check them only after a selection has been made. What is the smallest number of gloves you need to select to guarantee getting the following?

(a) At least one matching pair
(b) At least one matching pair of each color

 

Answer:

The answers are 11 and 19 gloves for parts (a) and (b), respectively. a. In the worst case, before you get at least one matching pair, you will select 5 black gloves, 3 brown gloves, and 2 gray gloves—all for the same hand. The next glove will have to yield a matching pair. Thus, the answer is 11 gloves. b. In the worst case, before you get one matching pair of each color, you will select all 10 black, all 6 brown, and 2 gray gloves for the same hand. The next gray glove will have to yield a matching pair. Thus, the answer is 19 gloves.
 

Which two numbers should replace the question marks?

Which two numbers should replace the question marks?


19, 20, 21, ?, ?, 26, 28, 32, 33, 40


 

Answer:

22 and 24: there are two interwoven sequences. Starting at 19, alternate numbers progress +2, +3, +4, +5. Starting at 20, alternate numbers progress +2, +4, +6, +8.

Identify the missing piece!

Look at the following picture, where a piece is missing. Can you identify the missing piece?

 

Answer:

D

Incorrect View

Here is a cube presented from five different perspectives. One of the views is incorrect. Can you tell which one?

Answer:

View C is not correct.

Balance Thermometers

Your challenge is to balance the thermometers in this puzzle in such a way that they all read an identical number. For each unit rise in any thermometer, one of the other thermometers must fall one unit, and vice versa.

 

Answer:

 

Find Pair

Find the pair to complete the following sequence.

Choose from:

Answer:

(b) to complete every possible pairing of the four symbols.

Statistics

Suppose that 3% of births give rise to twins. What percentage of the population is a twin: 3%, less than 3%, or more than 3%?



Answer:

More than 3% of the population are twins. Out of 100 births, 97 are single and 3 are twins. That's 103 babies in total, six of which are twins, which represents 5.8% of the population.