Riddle #1073

cleanlogicmath

Train's length

A 400 yard long train, travelling at 30 mph, enters a 4.5 mile long tunnel. How long will elapse between the moment the front of the train enters the tunnel and the moment the end of the train clears the tunnel?
9 minutes and 27.2727 seconds.
88.79 %
37 votes

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Flipping Lockers

There are 1 million closed school lockers in a row, labeled 1 through 1,000,000. You first go through and flip every locker open. Then you go through and flip every other locker (locker 2, 4, 6, etc...). When you're done, all the even-numbered lockers are closed. You then go through and flip every third locker (3, 6, 9, etc...). "Flipping" mean you open it if it's closed, and close it if it's open. For example, as you go through this time, you close locker 3 (because it was still open after the previous run through), but you open locker 6, since you had closed it in the previous run through. Then you go through and flip every fourth locker (4, 8, 12, etc...), then every fifth locker (5, 10, 15, etc...), then every sixth locker (6, 12, 18, etc...) and so on. At the end, you're going through and flipping every 999,998th locker (which is just locker 999,998), then every 999,999th locker (which is just locker 999,999), and finally, every 1,000,000th locker (which is just locker 1,000,000). At the end of this, is locker 1,000,000 open or closed?
Locker 1,000,000 will be open. If you think about it, the number of times that each locker is flipped is equal to the number of factors it has. For example, locker 12 has factors 1, 2, 3, 4, 6, and 12, and will thus be flipped 6 times (it will end be flipped when you flip every one, every 2nd, every 3rd, every 4th, every 6th, and every 12th locker). It will end up closed, since flipping an even number of times will return it to its starting position. You can see that if a locker number has an even number of factors, it will end up closed. If it has an odd number of factors, it will end up open. As it turns out, the only types of numbers that have an odd number of factors are squares. This is because factors come in pairs, and for squares, one of those pairs is the square root, which is duplicated and thus doesn't count twice as a factor. For example, 12's factors are 1 x 12, 2 x 6, and 3 x 4 (6 total factors). On the other hand, 16's factors are 1 x 16, 2 x 8, and 4 x 4 (5 total factors). So lockers 1, 4, 9, 16, 25, etc... will all be open. Since 1,000,000 is a square number (1000 x 1000), it will be open as well.
91.55 %
50 votes

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Appleland to Bananaville

You have been given the task of transporting 3,000 apples 1,000 miles from Appleland to Bananaville. Your truck can carry 1,000 apples at a time. Every time you travel a mile towards Bananaville you must pay a tax of 1 apple but you pay nothing when going in the other direction (towards Appleland). What is highest number of apples you can get to Bananaville?
833 apples. Step one: First you want to make 3 trips of 1,000 apples 333 miles. You will be left with 2,001 apples and 667 miles to go. Step two: Next you want to take 2 trips of 1,000 apples 500 miles. You will be left with 1,000 apples and 167 miles to go (you have to leave an apple behind). Step three: Finally, you travel the last 167 miles with one load of 1,000 apples and are left with 833 apples in Bananaville.
91.22 %
48 votes

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A fly in between trains

Two trains are traveling toward each other on the same track, each at 60 miles per hour. When they are exactly 120 miles apart, a fly takes off from the front of one of the trains, flying toward the other train at a constant rate of 100 miles per hour. When the fly reaches the other train, it instantly changes directions and starts flying toward the other train, still at 100 miles per hour. It keeps doing this back and forth until the trains finally collide. If you add up all the distances back and forth that the fly has travelled, how much total distance has the fly travelled when the trains finally collide?
The fly has travelled exactly 100 miles. We can figure this out using some simple math. Becuase the trains are 120 miles apart when the fly takes off, and are travelling at 60 mph each, they will collide in exactly 1 hour. This gives the fly exactly 1 hour of flying time, going at a speed of 100 miles per hour. Thus, the fly will travel 100 miles in this hour.
91.22 %
48 votes

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New York Hair

You are visiting NYC when a man approaches you. "Not counting bald people, I bet a hundred bucks that there are two people living in New York City with the same number of hairs on their heads," he tells you. "I'll take that bet!" you say. You talk to the man for a minute, after which you realize you have lost the bet. What did the man say to prove his case?
This is a classic example of the pigeonhole principle. The argument goes as follows: assume that every non-bald person in New York City has a different number of hairs on their head. Since there are about 9 million people living in NYC, let's say 8 million of them aren't bald. So 8 million people need to have different numbers of hairs on their head. But on average, people only have about 100,000 hairs. So even if there was someone with 1 hair, someone with 2 hairs, someone with 3 hairs, and so on, all the way up to someone with 100,000 hairs, there are still 7,900,000 other people who all need different numbers of hairs on their heads, and furthermore, who all need MORE than 100,000 hairs on their head. You can see that additionally, at least one person would need to have at least 8,000,000 hairs on their head, because there's no way to have 8,000,000 people all have different numbers of hairs between 1 and 7,999,999. But someone having 8,000,000 is an essential impossibility (as is even having 1,000,000 hairs), So there's no way this situation could be the case, where everyone has a different number of hairs. Which means that at least two people have the same number of hairs.
91.22 %
48 votes

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Land of Brainopia

In the land of Brainopia, there are three races of people: Mikkos, who tell the truth all the time, Kikkos, who always tell lies, and Zikkos, who tell alternate false and true statements, in which the order is not known (i.e. true, false, true or false, true, false). When interviewing three Brainopians, a foreigner received the following statements: Person 1: I am a Mikko. Person 2: I am a Kikko. Person 3: a. They are both lying. b. I am a Zikko. Can you help the very confused foreigner determine who is who, assuming each person represents a different race?
Person 1 is a Miko. Person 2 is a Ziko. Person 3 is a Kikko.
91.22 %
48 votes

logicmath

How long was he walking

Every day, Jack arrives at the train station from work at 5 pm. His wife leaves home in her car to meet him there at exactly 5 pm, and drives him home. One day, Jack gets to the station an hour early, and starts walking home, until his wife meets him on the road. They get home 30 minutes earlier than usual. How long was he walking? Distances are unspecified. Speeds are unspecified, but constant. Give a number which represents the answer in minutes.
The best way to think about this problem is to consider it from the perspective of the wife. Her round trip was decreased by 30 minutes, which means each leg of her trip was decreased by 15 minutes. Jack must have been walking for 45 minutes.
90.86 %
46 votes

logicmath

In a bank

A women walks into a bank to cash out her check. By mistake the bank teller gives her rupee amount in change, and her paise amount in rupees. On the way home she spends 5 paise, and then suddenly she notices that she has twice the amount of her check. How much was her check amount ?
The check was for Rupees 31.63. The bank teller gave her Rupees 63.31 She spent .05, and then she had Rupees 63.26, which is twice the check. Let x be the rupees of the check, and y be the paise. The check was for 100x + y paise He was given 100y + x paise Also 100y + x - 5 = 2(100x + y) Expanding this out and rearranging, we find: 98y = 199x + 5 or 199x ≡ -5 (mod 98) or 98*2*x + 3x ≡ -5 (mod 98) 3x ≡ -5 ≡ 93 (mod 98) this quickly leads to x = 31
90.47 %
44 votes

logicmath

Equation riddle

If, Fernando + Alonso + McLaren = 6 Fernando x Alonso = 2 Alonso x McLaren = 6 Then, McLaren x Fernando =?
3 Explanation: Fernando + Alonso + McLaren = 6 Fernando x Alonso = 2 Alonso x McLaren = 6 Given. Fernando x Alonso = 2 Alonso x McLaren = 6 Rewriting Both equations in terms of Alonso, Fernando = 2/Alonso McLaren = 6/Alonso Replacing above values in equation "Fernando + Alonso + McLaren = 6" 2/Alonso + Alonso + 6/Alonso =6 (2 + Alonso^2 + 6)/Alonso = 6 8 + Alonso^2 = 6Alonso Alonso^2 - 6Alonso + 8 = 0 (Alonso - 4) (Alonso - 2) = 0 Therefore; Alonso = 4 or 2 Let's take value of Alonso as 2 Fernando = 2/2 = 1 McLaren = 6/2 = 3 Therefore; McLaren x Fernando = 3 x 1 = 3
90.47 %
44 votes

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A grandfather's clock

A grandfather's clock chimes the appropriate number of times to indicate the hour, as well as chiming once at each quarter hour. If you were in another room and hear the clock chime just once, what would be the longest period of time you would have to wait in order to be certain of the correct time?
You would have to wait 90 minutes between 12:15 and 1:45. Once you had heard seven single chimes, you would know that the next chime would be two chimes for 2 o'clock.
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cleanlogic

Smartest kid in the world

Jake and his friend Paco had very famous challenge sessions at their school. One would suggest something they could do, and the other would prove it wrong somehow. One day, Jake surprised Paco by stating: "I can answer any question in the world." Sure that he would win the challenge, Paco accepted the task of proving it wrong. He wrote up a test full of impossible questions. After a while, Jake returned the test. Paco unbelievably lost the challenge and told Jake he could indeed answer any question. How did Jake win?
For all the impossible questions, Jake simply wrote "I don't know".
90.47 %
44 votes