Out of Box Thinking !

OBT(Out of Box Thinking) can be the best complement to a puzzle solver. It's a very widely used word to show the extraordinary brilliance of some guy. But this parameter has no scale, you can't say this solution is more OBT than that one. It's just a feeling that comes from inside of the observer or listener. You can never measure or define the OBT clearly but you will identify immediately as soon as it happens. I never get impressed by ordinary things, or just above average things, but this OBT impresses me the most. I love seeing people who has radical thinking and they come up with OBT very often. Talking to them is always a pleasure.


I will love to share some of the Out of thinking i came across in my life and i want you people to share your own experiences.


Puzzle:

There's a tennis tournament with one hundred twenty seven players, Shockley began, in measured tones. You've got one hundred twenty-six people paired off in sixty-three matches, plus one unpaired player as a bye. In the next round, there are sixty-four players and thirty-two matches. How many matches, total, does it take to determine a winner?


OBT :

126
It takes one match to eliminate one player. One hundred twenty-six players have to be eliminated to leave one winner. Therefore, there have to be 126 matches.


Ref : HowWould You Move Mount Fuji?


Don't forget to share your experience !

Logic Challenge !

There was a good job going in the office and the boss could not decide which of the three candidates should have it, each of them being worthy of it and all of them very bright indeed. So he set them a problem and the one who solved it would get the job. He showed them five discs, three black and two white and said: I am going to put a disc on the forehead of each of you. You will be able to see the others discs but not your own. There will be no talking. By pure deduction you will have to work out what colour disc you have, and the one who does so gets the job.? He withheld the two white discs and put a black one on each of them. After a time one of the men stepped forward and successfully claimed the job. How did he figure out that he had a black disc on?


Come up with your Logic then We will Discuss !

Some Tricky Puzzles

1.Boston United's latest player lives on the 13th floor of a tower block. Every morning he takes the lift down to the ground floor and leaves the building. When he returns home in the evening, if there is someone else in the lift or it's raining he goes straight back to his floor directly. However, if there is nobody else in the lift or it hasn't rained he goes up to the 10th floor and walks up the remaining three flights of stairs. He hates walking up stairs so why does he do it?

2.There are 2 identical strings. If you light one of the strings at its end, it will take exactly one hour for it to finish burning completely. The string will not burn evenly - it is thicker in some places, thinner in others. For example, the string may not be half consumed exactly 30 minutes from lighting it at one end. You have no other means of telling time, and you want to know when exactly 45 minutes have passed. All that you have is a lighter and these 2 identical strings. What is the most accurate method you can use, given these conditions?

3.A water lily growing in a circular pond doubles in size every day. It takes thirty days to cover the whole pond. How long does it take to cover half the pond? 


Triumph 1: The person is dwarf and he can reach up to the 10th floor button in the lift and in rainy days he has an umbrella to use to press the 13th floor button. and while coming down he just has to press ground floor button, which he can reach easily.


Triumph 2: You have to realize the fact that how uneven the rope can be but if you light the rope from both sides it will take half an hour to complete burn. If you are agree with me then you got the answer.
Light one rope from both sides and the other from only one side, when the first rope burnt completely it's 30mins, and the 2nd rope would have taken 30 more mins to burn completely so light the other end of the second rope, so it will get finished in next 15mins.


Triumph 3: 29 days (You can impress your girl friend with this one, but don't forget to say sorry after you finished laughing)

Puzzles for School Children

Q1: Each child in a family has at least 5 brothers and 4 sisters. What is the smallest number of children the family might have?

Q2: Louise runs the first half of a race at 5 miles per hour. Then she picks up her pace and runs the last half of the race at 10 miles per hour. What is her average speed on the course?

Q3:what is ?
8
2 5 2
1 2 4 2 1
1 2 1 3 1 2 1
1 2 1 1 ? 1 1 2 1

Q4: How Many Days?Froggie fell down a 10-foot well. He cannot hop out. He has to climb out. He climbs three feet a day, but during the night, while resting, he slips back two feet. At this rate, how many days will it take Froggie to climb out of the well?

Q5: How Many Marbles?
Marta distributed 100 marbles among five bags.
Bag #1 and Bag #2 together contain 52 marbles.Bag #2 and Bag #3 together contain 43 marbles.Bag #3 and Bag #4 together contain 34 marbles.Bag #4 and Bag #5 together contain 30 marbles. How many marbles are there in each bag?

Q6: How Many Students?
A new school has opened with fewer than 500 students. One-third of the students is a whole number. So are one-fourth, one-fifth, and one-seventh of the students. How many students go to this school?

Q7: Pick a Pair
Ben has socks in five different colors: two pairs of blue socks, two pairs of black, three pairs of brown, four pairs of green, and four pairs of white. Ben, who is not very neat, doesn't bother to pair up his socks when he puts them away. He just throws them in the drawer. Now Ben is packing to go away for the weekend, but there's been a power failure and he can't see the socks in his drawer.
How many socks does he have to take out of his drawer to be sure he has at least two that will make a pair?

Q8: Sale!
An online shopping site reduced the price of one computer model by 25 percent for a sale. By what percentage of the sales price must it be increased to put the computer model back at its original price?

Q9: Which Way?
Once a boy was walking down the road, and came to a place where the road divided in two, each separate road forking off in a different direction.
A girl was standing at the fork in the road. The boy knew that one road led to Lieville, a town where everyone always lied, and the other led to Trueville, a town where everyone always told the truth. He also knew that the girl came from one of those towns, but he didn't know which one.
Can you think of a question the boy could ask the girl to find out the way to Trueville?


Post Your Solutions. I will Get back to you.

10 Common Interview Puzzles

Q1:A person who was making a list of population of NOIDA came to RAM’s house and that man wants to record the age of all people staying with RAM. That man was RAM’s childhood friend meeting after a longtime.
RAM’S FRIEND: "How have you been?"
RAM: "Great! I got married and I have three daughters now.
"RAM’S FRIEND: "Really? How old are they?"
RAM: "Well, the product of their ages is 72, and the sum of their ages is the same as the number on that building over there..."
RAM’S FRIEND: "Right, ok... Oh wait... Hmm, I still don't know."
RAM: "Oh sorry, the oldest one just started to play the piano."
RAM’S FRIEND: "Wonderful! My oldest is the same age!"

Q2:Five pirates discover a chest full of 100 gold coins. The pirates are ranked by their years of service, Pirate 5 having five years of service, Pirate 4 four years, and so on down to Pirate 1 with only one year of deck scrubbing under his belt. To divide up the loot, they agree on the following:
The most senior pirate will propose a distribution of the booty. All pirates will then vote, including the most senior pirate, and if at least 50% of the pirates on board accept the proposal, the gold is divided as proposed. If not, the most senior pirate is forced to walk the plank and sink to Davy Jones’ locker. Then the process starts over with the next most senior pirate until a plan is approved.
Remember that these pirates are not ordinary people they are extremely intelligent and greedy, they are also perfectly rational and know exactly how the others will vote in every situation. Emotions play no part in their decisions.
The most senior pirate thinks for a moment and then proposes a plan that maximizes his gold, and which he knows the others will accept. How does he divide up the coins?

Q3:
The warden meets with 23 new prisoners when they arrive. He tells them, "You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no communication with one another.
"In the prison is a switch room, which contains two light switches labeled A and B, each of which can be in either the 'on' or the 'off' position. I am not telling you their present positions. The switches are not connected to anything.
"After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell
"No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back.
"But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch room.' and be 100% sure.
"If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators."
What is the strategy they come up with so that they can be free?

Q4:
you are presented with three doors (door 1, door 2, door 3). one door has a million dollars behind it. the other two have goats behind them. you do not know ahead of time what is behind any of the doors.
monty asks you to choose a door. you pick one of the doors and announce it. monty then counters by showing you one of the doors with a goat behind it and asks you if you would like to keep the door you chose, or switch to the other unknown door.
should you switch? if so, why? what is the probability if you don't switch? what is the probability if you do.

Q5:There is a pile of N (can be Even or Odd) coins placed on a table, in which K coins head upward. Can you make two piles of coin out of this pile having equal number of heads upward? But you can’t see which coin is heading upward, you can just count coins. No restriction on K 

Q6:
"a line of 100 airline passengers is waiting to board a plane. they each hold a ticket to one of the 100 seats on that flight. (for convenience, let's say that the nth passenger in line has a ticket for the seat number n.)
unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. all of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. if it is occupied, they will then find a free seat to sit in, at random.
what is the probability that the last (100th) person to board the plane will sit in their proper seat (Seat No 100)?"

Q7:"at one point, a remote island's population of chameleons was divided as follows:
13 red chameleons
15 green chameleons
17 blue chameleons
each time two different colored chameleons would meet, they would change their color to the third one. (i.e.. If green meets red, they both change their color to blue.) is it ever possible for all chameleons to become the same color? why or why not?"

Q8:You have 12 coins. One of them is counterfeit. All the good coins weigh the same, while the counterfeit one weights either more or less than a good coin. Your task is to find the counterfeit coin using a balance-scale in 3 weighs. Moreover, you want to say whether the coin weighs more or less. (have patience b’coz its solvable)

Q9:There are 10 ball producing machines out of which 9 machines produces 10gm balls and the remaining one produces 11gm balls, you are given with a weighing balance with all kinds of weights so that u can measure any weight you like, you have to identify which machine produces the 11gm balls but you can weigh only once. You have sufficient number of balls from each machine.

Q10:
you have $10,000 dollars to place a double-or-nothing bet on India in the Pepsi cup (max 7 games, series is over once a team wins 4 games). Unfortunately, you can only bet on each individual game, not the series as a whole. How much should you bet on each game, so that, if the yanks win the whole series, you expect to get 20k, and if they lose, you expect 0? Basically, you know that there may be between 4 and 7 games, and you need to decide on a strategy so that whenever the series is over, your final outcome is the same as an overall double-or-nothing bet on the series.


Please post your Solutions. Also you can ask for solutions

10 Common Interview Puzzles

Q1:A person who was making a list of population of NOIDA came to RAM’s house and that man wants to record the age of all people staying with RAM. That man was RAM’s childhood friend meeting after a longtime.
RAM’S FRIEND: "How have you been?"
RAM: "Great! I got married and I have three daughters now.
"RAM’S FRIEND: "Really? How old are they?"
RAM: "Well, the product of their ages is 72, and the sum of their ages is the same as the number on that building over there..."
RAM’S FRIEND: "Right, ok... Oh wait... Hmm, I still don't know."
RAM: "Oh sorry, the oldest one just started to play the piano."
RAM’S FRIEND: "Wonderful! My oldest is the same age!"
Q2:Five pirates discover a chest full of 100 gold coins. The pirates are ranked by their years of service, Pirate 5 having five years of service, Pirate 4 four years, and so on down to Pirate 1 with only one year of deck scrubbing under his belt. To divide up the loot, they agree on the following:
The most senior pirate will propose a distribution of the booty. All pirates will then vote, including the most senior pirate, and if at least 50% of the pirates on board accept the proposal, the gold is divided as proposed. If not, the most senior pirate is forced to walk the plank and sink to Davy Jones’ locker. Then the process starts over with the next most senior pirate until a plan is approved.
Remember that these pirates are not ordinary people they are extremely intelligent and greedy, they are also perfectly rational and know exactly how the others will vote in every situation. Emotions play no part in their decisions.
The most senior pirate thinks for a moment and then proposes a plan that maximizes his gold, and which he knows the others will accept. How does he divide up the coins?
Q3:
The warden meets with 23 new prisoners when they arrive. He tells them, "You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no communication with one another.
"In the prison is a switch room, which contains two light switches labeled A and B, each of which can be in either the 'on' or the 'off' position. I am not telling you their present positions. The switches are not connected to anything.
"After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell
"No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back.
"But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch room.' and be 100% sure.
"If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators."
What is the strategy they come up with so that they can be free?
Q4:
you are presented with three doors (door 1, door 2, door 3). one door has a million dollars behind it. the other two have goats behind them. you do not know ahead of time what is behind any of the doors.
monty asks you to choose a door. you pick one of the doors and announce it. monty then counters by showing you one of the doors with a goat behind it and asks you if you would like to keep the door you chose, or switch to the other unknown door.
should you switch? if so, why? what is the probability if you don't switch? what is the probability if you do.

Q5:There is a pile of N (can be Even or Odd) coins placed on a table, in which K coins head upward. Can you make two piles of coin out of this pile having equal number of heads upward? But you can’t see which coin is heading upward, you can just count coins. No restriction on K 

Q6:
"a line of 100 airline passengers is waiting to board a plane. they each hold a ticket to one of the 100 seats on that flight. (for convenience, let's say that the nth passenger in line has a ticket for the seat number n.)
unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. all of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. if it is occupied, they will then find a free seat to sit in, at random.
what is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)?"
Q7:"at one point, a remote island's population of chameleons was divided as follows:
13 red chameleons
15 green chameleons
17 blue chameleons
each time two different colored chameleons would meet, they would change their color to the third one. (i.e.. If green meets red, they both change their color to blue.) is it ever possible for all chameleons to become the same color? why or why not?"
Q8:You have 12 coins. One of them is counterfeit. All the good coins weigh the same, while the counterfeit one weights either more or less than a good coin. Your task is to find the counterfeit coin using a balance-scale in 3 weighs. Moreover, you want to say whether the coin weighs more or less. (have patience b’coz its solvable)
Q9:There are 10 ball producing machines out of which 9 machines produces 10gm balls and the remaining one produces 11gm balls, you are given with a weighing balance with all kinds of weights so that u can measure any weight you like, you have to identify which machine produces the 11gm balls but you can weigh only once. You have sufficient number of balls from each machine.

Q10:
you have $10,000 dollars to place a double-or-nothing bet on India in the Pepsi cup (max 7 games, series is over once a team wins 4 games). Unfortunately, you can only bet on each individual game, not the series as a whole. How much should you bet on each game, so that, if the yanks win the whole series, you expect to get 20k, and if they lose, you expect 0? Basically, you know that there may be between 4 and 7 games, and you need to decide on a strategy so that whenever the series is over, your final outcome is the same as an overall double-or-nothing bet on the series.

For Sudoku Lovers !

I designed this puzzle in Aug 2009 for "Contest for Best Logic Innovative Puzzle" organised by Serbian Puzzle Association. So just take a print out and try to solve it.

KNIGHTS LOOP SUDOKU

The objective is to fill the 9×9 grid so that each column, each row, each of the nine 3×3 boxes, and both the main diagonals contains the digits from 1 to 9 only one time each. Also it has 4 knight loops of 4 different digits each of length 4.

Knight Loop of digit K of length N : It’s a loop formed by moving a knight of chess on this 9x9 Sudoku Grid to N different squares and back to the initial position in N steps and replacing all these places by the digit K.

For Detailed Solution mail me @ rakhel.parida@gmail.com

For more about the Contest

Result

On Rohan Rao's Blog

Two Beautiful Puzzles

Let’s solve two beautiful puzzles today both of different kind and very interesting. The similarity between the two puzzles is that when you know the answer, they will seem very obvious to you but I can bet it will take some time to get to the answers. If you are able to solve them by yourself, a sense of triumph is guaranteed. So enjoy solving the puzzles.

Puzzle 1: The two Eggs

There is a building of 100 floors (1-100) and you have two eggs with you. You want to find out the minimum floor (we will call it “the floor”) from top of which if you drop the egg it will break. So you have to come up with a strategy by which you can definitively find out the floor with minimum number of drops. You can use the same egg again if it’s not broken. So you have to tell the maximum number of drops required to find out the floor by your strategy and it should be minimum.

Hint: If you have one egg then you need 100 drops to find the floor, because the strategy will be to start with floor 1 and then go to floor 2 and so on. You can’t just go directly to any of the higher floor because if the egg breaks then you can’t find out the floor. But with two eggs you can do some jumping on the first place and use the second egg for deterministically find out the floor.

Puzzle 2: 5 pirates and 100 coins

Once a group of 5 pirates got 100 gold coins from a dead ship. They wanted to distribute these coins among them. So they set some rules before start dividing. The pirates were A, B, C, D and E from eldest to youngest respectively. i.e. A is the eldest and E is the youngest. All the pirates are intelligent and very greedy.

The Rules are:

First A, the eldest will propose how to  distribute the coins, then if half or more than half including A gets agree to the proposal then the coins will be distributed according to A’s proposal but if it’s less than half then A will be killed. Now B will propose his proposal, and similarly if half or more than half including B agree to the proposal then the coins will be distributed according to B or else he will be killed and then it’s turn of C and so on.

What is the maximum number of coins A can take with his clever proposal?

Hint: very tricky hit the root and don’t ignore any information in this puzzle except the dead ship.

Triumph 1:

I know you must be having some number by your strategy, so let me share the answer with you before I discuss how. Answer is 14

Almost everyone whom I have asked this puzzle, takes some time to understand what I am asking for, is it the floor or the strategy or the minimum number of drops. It’s actually very hard to phrase this puzzle simpler. I tried but I think many would still be having the same problem.

When asked this puzzle most of us start with the first drop from and arbitrary number mostly a number between 10 to30 sometimes its 50. Then we think if it does not break then where and we move ahead, and then when we approach 100 we start thinking if it breaks then what. After 5 iteration almost everyone reaches to an answer of 19 or 20, with a strategy of dropping the first egg at 10,20,30 and so on and if it breaks at 100 then start from 91 and reach to an answer of 19. Then people ask for the answer and if i say the answer some people really fight hard to reach to the answer and then realise it’s the answer.

Let’s say the answer is N.

1^st Drop: So the maximum floor from which I can drop for the first time is N, so that if it breaks I can use the remaining one to cover from 1 to N-1.

2^nd Drop: As I lost one drop at N, now I have to find out the floor in N-1 drops, so I can drop at [N+(N-1)]^th floor, so that if it breaks I can use the remaining egg to cover the floors N+1 to [N+(N-1)]-1

3^rd Drop : N+(N-1)+(N-2) and so on

So at the end we have to drop the egg from N+(N-1)+(N-2)+…+2+1 which is ∑N

So if we have to cover 100 floors, ∑N≥100 and N to be minimum makes N=14

So we will drop the first egg from 14^th floor if it breaks we will start from 1^st floor and within 14 drops we can find the floor and if does not break at the 14^th floor then we will drop at 27^th (14+13) floor, and if it breaks we will start from the 15^th and so on.

I consider this puzzle as a classic puzzle which teaches multiple things about puzzle solving. The concept of N is a number, in puzzles like this you can assume the answer to be N and start solving like you solve an equation for the value of N. Also this is a classic example where we start with hit and trial and then we realise the constraints of this puzzle, and I feel there is nothing wrong with hit and trial as long as you are getting something out from every heat and trial and moving towards the goal. We use many kinds of search algorithms; binary search is the most famous and most applied search algorithm. This puzzle also tells something about a new search algorithm, I named it Broken-Egg search; we can increase the number of eggs to reach to the result faster. I also used this search algorithm in one of my project where it was difficult to use binary search. If at any point of time you find difficulty in implementing binary search you can try this search algorithm.

PS: Try solving the same puzzle if you have 3 eggs

Triumph 2:

This is a very common puzzle but for those who never heard of this puzzle, it’s the Unthinkable, because this puzzle has no start point. Also the information given in this puzzle seems not so useful. Even it’s almost impossible for 95 percent of the people to get to the answer without any prior experience on these kinds of puzzles. Let’s start with the solution

A proposes: need 2 more people among B, C, D and E to agree with A’s proposal

As we can’t justify with certainly who will support A with how much coins, so it’s of no use putting forward our stupid figures of 33,34,20,25 etc.

So let’s find out where is the bottom, if everyone else is killed and it’s a matter between D and E, then D will take all the coins without giving E anything. So E knows that when the thing will come down to this he will not get anything. Lets see what happens one step up with C, D and E as everyone is so much intelligent and greedy so C will just give one coin to E and E will support as he finds one coin or else will get nothing if C get killed. So C can maximum take away 99 coins. Moving to B, C, D and E B will give one coin to D and take 99. D will support B with one coin as D will get nothing if B gets killed. So when all are there A will give C and E one coin each and take 98 with him because B can play his trick where C and E gets nothing.

Answer to the puzzle is 98

Some obvious arguments : people many a time ask me why E will say yes when he is getting the same from C, but E is so much intelligent that he can see B’s proposal getting succeed, in that case it will not come down to C, so E is playing safe.

This puzzle really puzzles me a lot, because I still wonder will this happen in a real life, Because 99% guys who heard this puzzle for the first time didn’t able to answer it. Does the solution is a perception or a paradox Or is there a catch in the solution which we are not able to discover. Logically till day I am convinced with this solution. I think in real life we are much more greedier than the pirates are in the puzzles, and the greediness makes us blind from analysing the real consequences, so the puzzle with a constraint that all are intelligent and greedy is quite not true in real life. Also in Real life we think of bending decisions by bribing, like B can bribe C and E more than one coin and get them against A but in puzzles until and unless told you can think of things like that. Because its strictly mentioned A will propose first then B, so B can’t play around before A. I think this is a big factor in our inability to answer the puzzle. We think of everybody at the same time we think of A, B, C.. every pirate purposing at same time and get confused. There is a lot more psychology involved behind this puzzle. your inputs are very much necessary.

PS: Try to solve the puzzle with 5 coins and 100 pirates, you have to tell who will definitely die?

Participate actively in this blog I promise an entertaining time ahead.

Thank You !

Reversing a Puzzle

It frequently happens that a puzzle consists of obtaining a given final position from a given initial position by making the smallest possible number of moves. One can then starts from the final position instead, and try to obtain the initial position by backward moves. If this has succeeded, all one has to do to get the solution of the original puzzle is to perform all moves in the reverse order and in the reverse direction. It is virtually certain that the new puzzle (the reversed one) will be simpler; otherwise the puzzle would surely have been presented in the reversed form. Hence, in most cases the reversal of the puzzle will produce a simplification.

Example:  Golf and Tennis

Three golf balls (the small circles) and two tennis balls (the larger circles) are placed in a row, with golf balls and tennis balls alternating (Fig-initial). Now with every move a golf ball and an adjacent tennis ball are moved as one whole, so that they remain adjacent. You are not allowed to reverse the balls; that is, you should not interchange the golf ball and the tennis ball. Moving the golf ball and the tennis ball is considered as one move. The problem is to get the golf balls and the tennis balls in a row in such a way that the golf balls are adjacent, and the tennis balls, too (Fig-final), in as few moves as possible.

 As rotation is not allowed, the new row is parallel to the original row. However, it is not necessary that the golf balls get to the left of the tennis balls, as in the Fig-final; as a matter of fact, if you have managed to get the golf balls on the left-hand side, all you have to do to get the golf balls to the right of the tennis balls is to perform the mirror image of each move. If it were allowable to interchange a golf ball and an adjacent tennis ball, the final position could be reached without trouble in three moves; the relative order of golf balls would not have changed, any more than that of the tennis balls. Since interchanging a golf ball and a tennis ball is forbidden, it is slightly more complicated to reach the final position.

Solution :

In the initial position, in which these are four cases where a golf ball and a tennis ball are adjacent, a large number of moves can be made. However, from the final position only two moves are possible. Therefore we reverse the puzzle and consider the final position as the initial one; in the new initial position we designate the golf balls and tennis balls as in Fig-num

In determining the moves, one should observe that there is no point in performing two consecutive moves with the same pair (because then one can combine both moves into one). Furthermore, one should take care that after every move there will be another pair (consisting of a golf ball and an adjacent tennis ball) with which a next move can be made. This leads to the following puzzle tree

­The dots indicate free positions. From the position 3. . 2A 1 B infinitely many moves are possible. You can place the pair 2 A (or I B) wherever you want, because in each case you can make another move with the pair IB (or 2A). However, we have only included those third moves (following 3 . . 2A 1 B) after which it is possible to make a fourth move which yields the desired final position (the original initial position). Hence, this can be done in four ways; in the first two moves these agree completely and for the rest they come to practically the same thing: the last two moves take the pairs 2 A and 1 B to the left of3, the order being unimportant, while it also makes no difference which of these pairs gets next to 3. The four ways are:

then the solution in question is