## مسائل رياضية ومنطقية صعبة جدا وحلها

مسائل رياضية ومنطقية صعبة جدا وحلها

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بالتوفيق

• 101. Logic required. Of three men one man always tells the truth, one always tells lies, and one answers yes or no randomly. Each man knows which man is who. You may ask three yes/no question to determine who is who. If you ask the same question to more than one person you must count it as question used for each person whom you ask. What three questions should you ask? (Answer).
• 102. Calculus required. At dark is thrown at a dart board of radius 1. The dart can hit anywhere on the board with equal probability. What is the mean distance between where the dart hits and the center? (Answer), (Solution).
• 103. Simple math required.. Create the number 24 using only these numbers once each: 3, 3, 7, 7. You may use only the following functions: +, -, *, /. This is not a trick question, for example the answer does not involve a number system other than base 10 and does not allow for decimal points. (Answer).
• 104. Probability required, computer helpful. In the Price is Right three players compete for one place in the showcase showdown. Each player takes their turn spinning a wheel which has an equal probability of stopping on every amount evenly divisible by .05 from .05 to 1.00 . If the player does not like their first spin they may spin again, adding the second spin to their first, however if they go over 1.00 they are immediately disqualified. In the event of a tie the winner is determined randomly with each player left having an equal probability of winning. What is perfect strategy for all players in this game, assuming that all other players also play by perfect strategy? (Answer).
• 105. Calculus required. There is a street of length 4. The street is initially empty. Cars then come along to fill the street until there is no space left that is large enough to park a car in. Every car is length 1. Drivers will choose a location to park at random among all possible locations left. No consideration, whether good or bad, is given to other cars. What is the expected number of cars that will be able to park? (Answer), (Solution).
• 106.
Geometry required. The three circles in the diagram above all are tangent to the line in the picture. The radius of circle A is a, of circle B is b, and of circle C is c. All three circles are tangent to each other. What is c as a function of a and b? (Answer), (Solution).
• 107. Simple math required. Create the number 24 using only a 1, 3, 4, and 6. You may only use +, -, /, and *. Parenthesis are allowed. For example if I asked for 23 an answer would be ((6-1)*4)+3. This is not a trick question, for example the answer does not involve a number system other than base 10 and does not allow for decimal points. (Answer).
• 108. Simple math required. On a deserted island live five people and a monkey. One day everybody gathers coconuts and puts them together in a community pile, to be divided the next day. During the night one person decides to take his share himself. He divides the coconuts into five equal piles, with one coconut left over. He gives the extra coconut to the monkey, hides his pile, and puts the other four piles back into a single pile. The other four islanders then do the same thing, one at a time, each giving one coconut to the monkey to make the piles divide equally. What is the smallest possible number of coconuts in the original pile? (Answer), (Solution).
• 109. Probability required. You have two bags of m&m candy. One bag has 99 red candies and 1 blue candy. The other bag has 99 blue candies and 1 red candy. You choose a bag at random, open it, and draw just one candy from the bag. If the candy drawn is red what is the probability what is the probability that it was drawn from the bag with 99 reds and 1 blue? (Answer), (Solution).
• 110. Geometry required. There is a straight cable buried under a unit square field. You must dig one or more ditches to locate the buried cable. Where should you dig to guarantee finding the cable and to minimize digging? For example you could dig an X shape for total ditch length of 2*sqr(2) but there is a better answer. (Answer).
• 111.
Geometry required. The three colored circles in the diagram above have radii of 1, 2, and 3, and each are tangent to the other two. A fourth interior circle is tangent to all three colored circles. What is the radius of the interior circle? For extra credit what is the radius of the exterior circle (not pictured) that is tangent to the three colored circles? (Answer), (Solution).
• 112. Statistics required. In a drawer are two red socks and three blue socks. A sock is drawn at random from the drawer, with replacement, one million times. What is the range, with the expected outcome as the midpoint of the range, such that the probability is 95% that the number of red socks drawn falls within this range? (Answer), (Solution).
• 113. Statistics required. The following is a distribution of the first 10,000 digits of e:
0 974
1 989
2 1004
3 1008
4 982
5 992
6 1079
7 1008
8 996
9 968
It is speculated that the number 6 appears a disproportionately high number of times and thus the digits are not distributed randomly. Test the hypothesis that these digits form a random sample such that the outcome of 10,000 truly random digits would pass the test 95% of the time. (Answer), (Solution).
• 114. Statistics required. A gold mining company is testing locations for its next mine. From location A eight samples were taken of units of gold per ton of ore. The results were 1.23, 1.42, 1.41, 1.62, 1.55, 1.51, 1.60, and 1.76 . From location B six samples were taken with the following results 1.76, 1.41, 1.87, 1.49, 1.67, and 1.81 . It is assumed that the amount of gold in a sample in either location have a standard normal distribution with a fixed, yet unknown, mean and variance, and that the variance in both locations is equal. Test the hypothesis that the mean gold content of both locations is equal. Use a 10% level of significance, in other words assume that if the two means were the same the test would pass 90% of the time. (Answer), (Solution).
• 115. The police commissioner hired a mathematician to help at a crime scene. At the scene were between 100 and 200 glasses of wine. Exactly one glass was poisoned. The police lab could test any sampling for poison. A group of glasses could be tested simultaneously by mixing a sample from each glass. The police commissioner desires only to minimize the maximum possible tests required to determine which exact glass was poisoned. The mathematician started by asking a detective to select a single glass at random for testing. "Wouldn't that waste a test?", the detective asked. "No, besides I'm in a gambling mood.", the mathematician replied. How many glasses were there? (Answer), (Solution).
• 116. Probability required. A player sits down at a roulette table with \$20. He bets \$1 at a time on either red or black. Either bet pays even money and has a probability of 9/19 of winning. What is the probability that the player wins \$10 before losing all of his \$20? (Answer), (Solution).
• 117. The keeper of the web page The premature death of rockstars argues that rockstars rock stars do not live as long as the general population. He states that the average age at death of rock stars is 36.9 and 75.8 for the general population. What is wrong with this use of these statistics? This is an illustrated example of lying with statistics. (Answer).
• 118. Thirteen pirates put their treasure in a safe. They decide that the safe should be able to be opened if any majority of pirates agree but not be able to be opened if any minority agree. The pirates don't trust each other so they consult a locksmith. The locksmith puts a specific number of locks on the safe such that every lock must be opened to open the safe. Then he distributes keys to the pirates such that every pirate has some but not all of the keys. Any given lock can have multiple keys but any given key can only open one lock. What is the least number of locks required? (Answer), (Solution).
• 119. A round table sits flush in a corner of a square room. One point on the edge of the table is 5" from one wall and 10" from the other. What is the radius of the table? (Answer), (Solution).
• 120. What is the expected number of turns needed to form a Yahtzee. (Answer), (Solution).