Logic contest

Problem 3:

Construct a chess position that is illegal (cannot be achieved in any number of moves from the normal position), but is legal if any one unit (other than the kings) is removed.

Problem 5

Alphonse and his $n-1$ friends are sitting in a circular fashion when Edgar meets them (sitting of n people). Each of them owns coin. First person passes 1 coin to the person on his left side. The second person in turn passes 2 coins to the person siting on the left. Third person passes 1 coin to the left, 4th passes 2 coins to his left and so on, so each person receiving coin from the right has to pass 2 coins to the left. Similarly, a person receiving 2 coins from the right has to pass 1 coin to the left.

If at any point of time, a person runs out of coin he is thrown out of the game. The game will terminate if at the end only 1 person is with coins in his possession. There might be values of n where the game may not terminate (e. g. 3 persons left with 4 4 4 coins respectively) For how many different values of n from 4 to 100 does the game terminate?

Problem 9

In a palace, there are some finite number of rooms (but you do not know how many) arranged in a circular fashion. Each room is connected to two other rooms and the last room is connected to the first room.

In each of the room there is a lightbulb, either switched on or off. Some of the rooms are lit, some of them are not and there is no pattern that the illumination sequence follows.

You start off at an arbitrary room and are required to turn of the bulbs in all of the rooms.

What is your strategy and how do you ensure that you've actually switched off all the bulbs?

Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?

Problem 11

You have two equal sized buckets. One contains water and the other contains the same amount of wine. You transfer a cup of wine to the water bucket and mix it in. Next you transfer a cup of the mixture back to the wine bucket. Is there more wine in the water, or water in the wine?

Problem 13.

You have a switch that can be turned on and off. In 1 minute you turn it on, in 30 seconds you turn it off, in 15 seconds you turn it on, and so on. After 2 minutes is it on or off?

Problem 1:

Two men find an old gold coin and want to have a coin toss with it to decide who gets it. The only problem is the coin is heavier on one side so it comes up heads more than tails. What is a fair way for the men to toss the coin and decide who gets the coin?

Problem 2 :

You want to make a choice between two options using a coin.

However, due to the nature of the options, you want your choice to be biased, that is you want the probability of choosing the first option to be $p$ such that $p \in (0,1)$ and $2^n \times p \notin \mathbb{N} \quad \forall n \in \mathbb{N}$

How do you do this?

Problem 2b (originally Problem 3, but now just a variant of Problem 2 because too much math):

Same as Problem 2, but the coin is not necessarily fair; it gives heads with probability $q$ where $q \in (0, 1)$. (Note that $q \neq 0, 1$.) And now the probability $p$ can be any number in $[0, 1]$, too.

Problem 4

In the year 1995, A murder happened. Mr. Shalls was found dead on his bed. Famous detective Conan was called. He narrowed the suspects to Charlie, Gotham, Sharky, Hitler and Mrs. Shalls. A letter came to the government of state in which the murder happened. It was written 222---666---66---1---66; 444---7777; 8---44---33; 55---444---555---555---33---777; Can you find out who is the killer, given that The letter is 100% true?

Problem 6

Imagine you have a piece of string long enough to stretch around the earth (40,074 km or 4,007,400,000 cm). Then you take an extra meter of string and add it to the string around the Earth. Now you spread this extra string around the Earth, supporting it somehow, so that the string forms a circle off the ground. How high off the ground would the string be?

Problem 7

Chopsticks is a hand game for two players, in which players extend a number of fingers from each hand and transfer those scores by taking turns to tap one hand against another.

Each player uses both hands to play the game, the number of digits extended on a hand showing the number of points that the hand has. Both players start with each hand having one point — one finger extended on each hand. The goal of the game is for a player to force their opponent to extend all of their fingers and thumbs on both hands. A hand with all fingers and its thumb extended is called a "dead hand". Players take turns to tap one of their hands against another hand that is not dead (either their own other hand, or one of their opponent's). The number of points on the tapping hand is added to the number on the tapped hand, and the player with the tapped hand extends their digits to show the new score. The tapping hand remains unchanged.

A player may tap their own hand to transfer points from one hand to the other. For example, if a player had three points on his or her right hand and one on his or her left, the player could rearrange them to have two on each hand. A "dead hand" is treated as having no points, for this purpose, which allows a player to bring a dead hand back into play by transferring points to it. For instance

If you have 5/2 (Dead hand and other hand has 2 points) you can split 1/1. (one hand has 1 point the other has 1 point) If you have 5/3 you can split 2/1 or 1/2. If you have 5/4 you can split 2/2 or 3/1 or 1/3. If you have 2/3 you can switch it and make it 3/2

You cannot prolong the game by not taking your turn.

It is already known that the first player (the player who starts first) has the winning strategy. As the second player, what is the strategy to prolong the game as long as possible? (Assume the first player and second player are both perfect logicians, and that the first player wants to win as fast as possible.)

References

Problem 8

The weird and inaccurate soothsayer Nostradamus once announced…..

“On Wednesday 2nd February 2000, an event will take place for the first time in over 1000 years. In fact, the last time this event happened was 28 August 888″. For once, he was right. What was the event?

Problem 10

One change: My solution to problem 5 is wrong. I have creditted marks to AA. He can post its solution

Problem 12:

On Friday 15,2016 , Aman Rajput went to Cannaught Place. He started talking to strangers and asked some questions. Here is the conversation:

Me: Hello Ma'am my name is aman. I want to ask you a question. I want you to answer in yes or no . Okay ? Stranger : Okay fine! Ask.

Me: Say yes, if you have 2 children including the fact that one should be a girl born on Friday ? Stranger: No ,Sorry,

Proceeding in similar manner. Finally,after a long time I find a stranger who says yes.

What is the probability that this stranger has two girls ?Assume equal probability of giving birth to either sex and equal probability to give brith on any day.

problem 14

Imagine that you are trapped in a prison cell. The only way out is through the door, but the door is locked. There is a combination lock on the door, and the following sequence of numbers appears above the lock:

1

11

21

1211

111221

312211

…to open the door you have to enter the next number in the sequence into the combination lock. What number should you enter?

Problem 15

If you were to put a coin into an empty bottle and then insert a cork into the neck, how could you remove the coin without taking out the cork or breaking the bottle?

Problem 16

A safe can be opened by inserting strings of an arbitrary length made from the digits from 1 to 9 which don't need to be included just once.

It is known that apart from the combinations which are correct and open the safe there are 2 other types of combinations , some which when entered will have no affect on the safe which are called neuter combinations and some which when inserted will destroy the safe which will be named bad combinations.

It is known that some strings are related to other strings of other by a relation which will be named R by some rules which are about to be presented.

For clarity define the following , for 2 strings of arbitrary length x and y

a) xy is the combination made from x followed by y

b) i(x) is the inverse of the combination x (if x is 21 for example then i(X)=12) anyway

The relation R has the following characteristics :

A) For any combination x , 2x2 is related to x.

B) If x R y then 1x R 2y

C) If x R y then 5x R i(y)

D) If x R y then 9x R yy

E) If x R Y and x is a neuter combination then y is a bad combination

F) If x R y and x is a bad combination then y is a neuter combination.

Find a configuration which will certainly open the safe.

Consider that from a correct string you can't obtain a neuter one or a wrong string.

Also the relation between the string is not reflexive. That is if x R y it doesn't mean that y R x.

Problem 17

A knight arrives at a castle carrying a lance that is five feet long. The guard tells him that no-one is allowed in the castle with an object that is over four feet long. The knight is a tad upset, but then has an idea. He goes into town, finds the local carpenter and asks him to make something. The knight then returns to the castle and the guard lets him in.

Due to his clever thinking, the knight finds himself inside the castle with his lance. The lance is fully functioning, and has not been cut in any way.

How is this possible?

So , I should post a solution for that question since I said I will anyway and since Prince awarded me points for it anyway.

The problem is actually pretty beautiful as it implies a synthetic and simple reasoning but quite elegant and cute anyway and more importantly it so as can be solved just by involving any quite of technicality or , if some technicality would be involved , it would be so at the point at which it is immediately conceived in the form of a direct concept.

To understand better the problem , let's define a "profit" as being the gain/lose of coins a player has after he both receives and gives a coin , the act of receiving a giving being a turn. Also define a round to be a continuous sequence of moves in which every player receives such a profit once but observe that this definition of round is incomplete since it is not reduced to the set of turns since if a round would be the set of moves when each player has made a turn then the first player will always gain more profit at the end of each round.

Now , having those stuff defined let's see how the game would go in the conceiving of this so defined terms or of the "language"/interpretation provided to understand it and used anyway. What will be done will anyway be to analyze how the profit of every player varies over the passing of rounds which is simply exactly the only stuff to which the problem itself reduces.

Observe that for each number of players , n , in the first round the first 2 players will anyway be eliminated (the 1st gives his only coin , the 2nd gives the 2 coins) and then after this depending on their positions , each player will alternate receiving 2 coins and giving 1 coin having a profit of +1 coin after a turn and receiving 1 coin and giving 2 coins having a profit of -1 coin each this alternating sequence starting with the 3nd player which receives 2 and gives 1 coin therefore with the 3nd having a +1 profit. Then it can be concluded that after the first round you have that each player in the odd position except the first two , which are in an odd position have a +1 and each player which is in a even position has a -1 profit meaning that at the profit-round representation you have +1 -1 +1 -1 +1-1 +1-1 etc anyway.

That means that after the first round all odd and one even position (player 1) are off the game so the remaining number of players will be n-1/2 which would mean in a properly said interpretation of stuff that you lose a pair of 2 already and from each other pair of 2 players remaining you lose one player anyway. The distribution of this profit can be quickly checked for the parity cases as for even number of players cases at some round after the first the distribution for a configuration after the 1st round will be stable meaning that at each round the same players will have a -1 profit while for the odd the players will alternate losing.

Let's analyze this intuitive and almost immediately seen thinking checking the cases of odd numbers and even number of players after some round after the first.

To check this suppose that at some round after the 1st the number of persons remaining in the game is even. Then at each round the first player has a +1 gain since he receives 2 from the last because the last is on an even place and gives 2 coins and therefore all those on even positions will get +1 , while the odd ones will get -1 until the end of the round where the same thing keeps continuing since at the next round the first also receives 2 coins from the last since the last is on an odd position which runs indefinetely with playerson odd positions loosing that is having -1 profit until they are off the game.

So the idea that a sequence of odd players will have players getting out of the game is right.

Now , check the case of an even number of players. For an even number of players the first player at the second round also gains +1 and also the same alternating lose/win profit stuff happens but because the position of the last person in the sequence is odd after the end of round anyway the person on the first place will not gain but lose coins this same course of alternating happening so at every 2 round you will have that the odds have +1 and the even have -1 at the first , then the odds have -1 and the even +1 at the second. Because every palyer therefore alternates receiving and losing coins their profit remains always the same.

So it's checked that for an odd number of players at a round after the first the game runs in a continuous pattern and never ends as it's unstable every alternating round anyway.

Because of this it means that you need to have just odd sequences at every step of the game and therefore that because at every sequence which has players getting out of the you eventually lose half of the players the number of players which shall remain after the first round should be of the for 2^m anyway.

Then you have that all n which after the first round gives you powers of 2 are ok and all the others , because contain in prime factorization an odd number are not since they will lead eventually to an odd sequence of players which as seen will get to run indefinitely never ever ending anyway so to say!

Expressed mathematically anyway for it to work you need to satisfy , for any number n the condition 2^m= (n-1)/2 with n = 2^m+1 or n = 2^m+2 for the odd and even numbers.

For the numbers between 4 and 100 therefore you have 10 or 11 cases depending whether or not you have to include 4 meaning the powers 1 , 2 , 4 , 3 , 5 , 6 anyway.

Of course there are some details to cover in the proof to make it rigorous such that at every round after the first the players have more than 1 coin for example but such things are pretty much easily understood by anyone and is sufficient to let it so anyway.

Btw , sorry for the rushed proof. Hopefully it's still clear enough to be understood.