Brainden-Logic-1 Flashcards
A Ping-Pong Ball in a Hole:
Your last good ping-pong ball fell down into a narrow metal pipe imbedded in concrete one foot deep.
How can you get it out undamaged, if all the tools you have are your tennis paddle, your shoe-laces, and your plastic water bottle, which does not fit into the pipe?
Think outside of the box. None of these equipments can be useful, use the water from the bottle to make the ball float if there is no water, pee!
A Man in Elevator:
A man who lives on the tenth floor takes the elevator down to the first floor every morning and goes to work. In the evening, when he comes back; on a rainy day, or if there are other people in the elevator, he goes to his floor directly. Otherwise, he goes to the seventh floor and walks up three flights of stairs to his apartment.
Can you explain why?
Think outside of the box.
He must be short! If it was raining, he had an umbrella to press #10, and if there were others around, then he asked them to press 10.
You are in a room with no metal objects except for two iron rods. Only one of them is a magnet.
How can you identify which one is a magnet?
Hang them on the ceiling and the one that turns towards north is a magnet.
Virile Microbes: A Petri dish hosts a healthy colony of bacteria. Once a minute every bacterium divides into two. The colony was founded by a single cell at noon. At exactly 12:43 (43 minutes later) the Petri dish was half full.
At what time will the dish be full?
At each minute, the microbes double, if it takes them 43 minutes to fill the half of the Pie shaped container, it will take them ONLY ONE minute to fill the other half.
An Arab sheikh tells his two sons to race their camels to a distant city to see who will inherit his fortune. The one whose camel is slower wins. After wandering aimlessly for days, the brothers ask a wise man for guidance. Upon receiving the advice, they jump on the camels and race to the city as fast as they can.
What did the wise man say to them?
If the goal is to make your camel run very very slow, then if you switch camels, then your goal is to race the camels and win.
One absentminded ancient philosopher forgot to wind up his only clock in the house. He had no radio, TV, telephone, internet, or any other means for telling time. So he traveled on foot to his friend’s place few miles down the straight desert road. He stayed at his friend’s house for the night and when he came back home, he knew how to set his clock.
How did he know?
Let’s say if he leaves at 6PM from his house, and reaches his friend’s house at 6:30 PM. (At this point, he does not know how long it takes him to reach to his friend’s house).
Next morning, he leaves his exactly at 6:30 AM, he knows that his home clock would be 6AM. He walks home and (for ex) he notices his home clock at 6:20AM, this means that it takes him 20 mins to walk to his friends’ home. Hence his clock’s time must be 6:50am when he arrives. Because, he left his friend’s house at 6:30 and it takes him 20 mins to get there.
I will blindfold you and paint either red, or blue dot on each man’s forehead. When I take your blindfolds off, if you see at least one red dot, raise your hand. The one, who guesses the color of the dot on his forehead first, wins.” And so it was said, and so it was done. The Grand Master blindfolded the three contestants and painted red dots on every one. When he took their blindfolds off, all three men raised their hands as the rules required, and sat in silence pondering. Finally, one of them said: “I have a red dot on my forehead.”
How did he guess?
Wisest one must have thought like this:
(1) I see all hands up and 2 red dots, so I can have either a blue or a red dot.
(2) If I had a blue one, the other 2 guys would see all hands up and one red and one blue dot. So they would have to think that if the second one of them (the other with red dot) sees the same blue dot, then he must see a red dot on the first one with red dot.
(3) However, they were both silent (and they are wise), so I have a red dot on my forehead.
He showed the three men 5 hats - two white and three black. Then he turned off the lights in the room and put a hat on each Puzzle Master’s head. After that the old sage hid the remaining two hats, but before he could turn the lights on, one of the Masters, as chance would have it, the winner of the previous contest, announced the color of his hat. And he was right once again.
What color was his hat? What could have been his reasoning?
There are 2W and 3B hats. Let’s agree that WWB is not a fair distribution. Hence there are these possible scenarios for the first guy to be confused:
(1) W, B, B
(2) B, B, B
(3) W, B, W
(4) W, W, B
(5) B, B, W
(6) B, W, B
For the 2nd guy, if he sees: W, B, W (#3, then he would know he is wearing Black), Similarly for #5, if he sees B, B, W, he knows that he cannot wear white or #1 would have knows, hence, there are below distributions left after the 2nd guy goes.
(1) W, B, B
(2) B, B, B
(4) W, W, B
(6) B, W, B
Answer is obvious for # 3. It has to be black.
The Grand Master takes a set of 8 stamps, 4 red and 4 green, known to the logicians, and loosely affixes two to the forehead of each logician so that each logician can see all the other stamps except those 2 in the Grand Master's pocket and the two on her own forehead. He asks them in turn if they know the colors of their own stamps: A: "No." B: "No." C: "No." A: "No." B: "Yes." What color stamps does B have?
A…B…C.
Let’s play with these scenarios, there is 0RG, 1RG, 2RG, or 3RG. Makes the life easier.
if two have RR…(no RG)
RR RR GG == C instantly has answer (sees all 4 R)
RR GG RR == B instantly has answer (sees all 4 R)
GG RR RR == A instantly has answer (sees all 4 R)
similarly if two have GG…
if two/three have RG…
RG RG RG == nobody can have any answer
RG RG RR == nobody can have any answer
RG RG GG == nobody can have any answer
RR RG GG (one RG)
1st turn, nobody has the answer.
2nd turn, the person who sees one RR and one GG thinks:
If I had RR, the person having GG instantly must have had the answer.
If I had GG, the person having RR instantly must have had the answer.
But they did not…
So I have neither RR nor GG.
Therefore I have RG!
The chieftain showed 5 headbands - 2 red and 3 white. The 3 men were blindfolded and positioned one after another, face to back. The chief put a headband on each of their heads, hid two remaining headbands, and removed their blindfolds. So the third man could see the headbands on the two men in front of him, the second man could see the headband on the first, and the first could not see any headbands at all.
According to the rules any one of the three men could speak first and try to guess his headband color. And if he guessed correctly - they passed the test and could go free, if not - they failed. It so happened that all 3 Palefaces were prominent logicians from a nearby academy. So after a few moments of silence, the first man in the line said: “My headband is …”.
What color was his head band? Why?
My head-band is white. 2R - 3W
(1) W | W | R
(2) W | W | W
(3) W | R | R
(4) W | R | W
(5) R | R | W
Four angels sat on the Christmas tree amidst other ornaments. Two had blue halos and two - yellow. However, none of them could see above his head. Angel A sat on the top branch and could see the angels B and C, who sat below him. Angel B, could see angel C who sat on the lower branch. And angel D stood at the base of the tree obscured from view by a thicket of branches, so no one could see him and he could not see anyone either.
Which one of them could be the first to guess the color of his halo and speak it out loud for all other angels to hear?
If A sees 2Y or 2B, then he would know the color of his halos. If A doesn’t know, B would know the color by looking at C.
Mr. Smith works on the 13th floor of a 15 floor building. The only elevator moves continuously through floors 1,2,…,15,14,…,2,1,2,…, except that it stops on a floor on which the button has been pressed. Assume that time spent loading and unloading passengers is very small compared to the travelling time.
Mr. Smith complains that at 5pm, when he wants to go home, the elevator almost always goes up when it stops on his floor. What is the explanation?
Now assume that the building has n elevators, which move independently. Compute the proportion of time the first elevator on Mr. Smith’s floor moves up.
Simplify the problem:
(1) There are 28 possible scenarios -
(2) Elevator is on the 1st floor going up, 2nd floor going up/down, …15th floor going down.
(3) Mr. Smith wants to go home (hence he wants to go down from the 13th floor).
(4) Mr. Smith will get the elevator going down when
- –> 13th floor going down, 14th floor going up/down, and 15 floor going down.
- -> The probability of Mr. Smith catching the elevator going down is 4/28 or 1/7 which is equal to .1429
NCAA basketball pool. There are 64 teams who play single elimination tournament, hence 6 rounds, and you have to predict all the winners in all 63 games. Your score is then computed as follows: 32 points for correctly predicting the final winner, 16 points for each correct finalist, and so on, down to 1 point for every correctly predicted winner for the first round. (The maximum number of points you can get is thus 192.) Knowing nothing about any team, you flip fair coins to decide every one of your 63 bets. Compute the expected number of points.
- -> There will be total of [(2^k)-1] games: That is total of 63 games.
- -> Intuitively speaking - there are 64 teams and since there is only one winner, there are 63 losers
- -> For the first round I collect 1 point for being right, 2nd round I collect 2^1 = 2 points, 3 round = 2^2 = 4 points all the way upto 32 points for picking the finalist.
- -> The expected number of point is 16 in each round, with total of 6 rounds. That is the total of 96 points
Three players enter a room and a red or blue hat is placed on each person’s head. The color of each hat is determined by a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players’ hats but not his own. No communication of any sort is allowed, except for an initial strategy session before the game begins. Once they have had a chance to look at the other hats, the players must simultaneously guess the color of their own hats or pass.
–> Strategy is improves the odds to 3/4
—> If a player sees two different hat colors, he passes
—> If he sees two the same color - he chooses the opposite color
Scenarios:
(1) BBB - Loose
(2) RRR - Loose
(3) BRB - Pass, Red, Pass - Win
(4) RBR - Pass, Blue, Pass - Win
(5) BBR - Pass, Pass, Red - Win
(6) RRB - Pass, Pass, Blue - Win
(7) BRR - Blue, Pass, Pass - Win
(8) RBB - Red, Blue Blue - Win
Hence there are six outcomes that wins from eight of them. Hence 3/4 probability of winning.
Somebody chooses two nonnegative integers X and Y and secretly writes them on two sheets of paper. The distrubution of (X, Y ) is unknown to you, but you do know that X and Y are different with probability 1. You choose one of the sheets at random, and observe the number on it. Call this random number W and the other number, still unknown to you, Z. Your task is to guess whether W is bigger than Z or not. You have access to a random number generator, i.e., you can generate independent uniform (on [0,1]) random variables at will, so your strategy could be random.
Exhibit a stategy for which the probability of being correct is 1/2 + ε, for some ε > 0. This ε may depend on the distribution of (X, Y ), but your strategy of course can not.
P(w) = Probability of Winning
Assume A
