Answers to Above Post. I cannot explain #21... It's Srinivasa Ramanujan's answer.
Some are scant. Note the formula for combinations, n!/(n -r)!r!, or nCr, is (n,r) in these answers. The answers:
1. At 6:00 (AM or PM) the clock hands are in line with one another. When will they be in line again? (7:05:27.272727, via (5 + x + (30 - (5 + 1/12(5 + x))) = 30.)
2. If you were born on Friday, January 13, in a leap year when, exactly, will that happen again? Never, of course, but when will your birthday fall on Friday, January 13 in a leap year again? (28 yrs.)
3. If you tie a string around the equator, then another string just 2π feet (6.283 feet) longer around, how high above the first string will the second be all the way around? (1 foot.)
4. How many rotations on its axis does a one-inch circumference gear make going around a stationary three-inch circumference gear? (4.)
5. Solve for x and y: √x + y = 7; √y + x = 11. (x= 9, y = 4.) Now try √x + y = 858; √y + x = 318. (x = 289, y = 841.)
6. Rolling two dice once, what is the probability of getting at least one six? (1 - (5/6 x 5/6) = 11/36, or, (1/6 x 5/6) + (1/6 x 5/6) + (1/6 x 1/6) = 11/36.)
7. I notice the alley next to my house has a ladder leaning 20 feet high against one building and another ladder leaning 10 feet high against the other building, both with feet abutting the opposite building. They intersect at a height of six feet, eight inches. I come to your house and see the same arrangement. But your alley is 3 feet wider than mine. How high is the intersection point at your house? (The same.)
8. Figure two-digit cube roots. ( Learn this second-digit pattern: 1874563290. Whatever the cube is, use its last digit to count along the pattern for the second digit of the cube root--and know that 10, 20, 30...90 cubed = 1,000, 8,000, 27,000, 64,000, 125,000, 216,000, 343,000, 512,000, and 729,000. Now have someone cube a 2-digit number, say 73. The cube is 389017. You know the second digit is a 3--from the 7th number in the pattern above, and you know the first is a seven because it must be equal to or more that 70^3 = 343,000.)
9. Prove e to the iπ ( + 1) = 0 (Given
a. e to the x = 1 + x + xsq/2! + xcubed/3!...
b. sinx = x - xcubed/3! + xfifth/5! - xseventh/7!...
c. cosx = 1 - xsq/2! + xfourth/4! - xsixth/6!...
In e to the x, substitute iu for x and do the simple algebra and remember powers of i: For i to the 1, 2, 3, 4..., the repeating pattern is +i, -1, -i, +1...)
10. The Greek's Golden Number is (√5 + 1)/2 = 1.618 = Q. Q^2 = Q + 1 and 1/Q = Q - 1. If you draw a star inside a pentagon, almost all line to line segment ratios = Q, or its inverse. A 1 to 1.618, or the same, .618 to 1 rectangle is supposed to the most pleasing to view. The new TV aspect ratio is supposed to be changed to 16 to 9 from the current 4 to 3. What height would result in a Golden number rectangle? (9.88" for a 16" length.)
11. Draw a square and make a grid of 16 little squares inside it. What is the total number of squares you can see? (30)
12. A man with an enclosed truck comes to a two-ton limit bridge. His truck weighs 3800 pounds and he's carrying 300 one-pound pigeons for a total of 4100 pounds, 100 over the limit. He decides to beat the bejesus out of the truck to get all the pigeons flying, then gets in the truck and proceeds to cross the bridge, knowing at least half the pigeons are still in flight. Does he make it? (No. It's a closed system and the truck weighs the same whether the pigeons are roosted or in flight.)
13. Consider a circle with a diameter = 1. C = π, so a semicircle has a length of π/2. Draw any number of circles on the diameter of 1 and count the length of all the new semicircles (the 'S' shape, i.e.) The length remains π/2 no matter how many little semicircles are made. Q: Do the semicircles get so small that the 'S' shape becomes a line? (No. It remains an 'S' shape with infinitely small semicircles but not a line.)
14. Twenty universities send a physicist, a chemist, a mathematician, an astronomer, and a biologist--each--to a convention. How many committees of ten, with each field represented by two scientists, can be formed? (nCr, or (20,2)=190. So there can be 2 physicists from 190 places--and with each, 2 biologists from 190 places... 190^5 = 2.47 x 10^11.)
15. If you get on your boat that goes 10 mph against a 5 mph current, and your girl drops her hat in the water as she gets on the boat, then you travel 45 minutes and she remembers, how long does it take to retrieve the hat? (45 minutes--the current is irrelevant.)
16. How is the product of any 4 consecutive integers related to the product of their perfect squares? (It's its square root.)
17. Bob's wife picks him up every day at his train's arrival. He takes an hour off one day and decides to walk, after getting off the hour-earlier train--he meets his wife on the way, hops in and their home 20 minutes early. How long did Bob walk? (He saved her 10 minutes of her travel time to the station--like she was 10 minutes early, so Bob walked 50 minutes.)
18. You have some potatoes, 99% water 1% potato. You put 100 pounds on you porch to dry. After a while enough water has evaporated so the potatoes are 98% water, How much do they weigh? (50 lbs.)
19. a. You catapult a baby straight up and it descends for 1.6 seconds. How high did it rise? (40.90 feet)
b. You set the catapult at 45 degrees. The baby lands 2.5 seconds after beginning decent. What was the height of the arc? (100 feet.)
c. You throw a twin baby from a 512-foot building at noon. Halfway down what time is it? What time does it land? (12:04:04 and 12:00:05.6.)
d. You drop the other twin when the first is 1/2 way down. What is the time between landings? (4.0 seconds.)
20. On a planet with a radius of 1080 miles and gravitational acceleration of 12m/sec/sq. what is orbit speed for the lowest possible orbit? (from a=v^2/r...1.566 mps.)
21. Evaluate √1 + 2 √1 +3 √ 1 + 4...and note that all √ bars extend all the way above all of those below. (3.)
22. Draw an isosceles triangle with an apex labeled 20 degrees. Then draw a 60 degree ray from one corner to the opposite side, and a 50 degree ray from the other corner to its opposite side. Now connect the 2 points where the rays meet the sides. Find the value of the uppermost angle of the triangle made by this connection. (30 degrees--prove it--no tools.)
23. What is the likelihood of being dealt 5 cards with no poker hand at all--'garbage?' (There are 1,302,540 such hands of 2,598,960 possible hands. So so likelihood is .501177394, or 1/1.9953.)
24. What is the likelihood of throwing 25 pennies on the floor and seeing 2 heads show up? (.5^2 x .5^23 x (25,2) = .00000894 = 1/111,894.)
25. You deal a poker hand (Ace high, always) to a friend and yourself. You have three Aces and another card and are about to draw the tenth card. What is the likelihood you'll get the last Ace, remembering he may have it? (1/43 x (47,5)/(48,5) = 1/48.)
26. What is the likelihood of getting 5 cards with one King? ([(4,1) x (48,4)]/(52,5) = 1/3.339.)
27. What is the likelihood of being dealt 5 red cards? ((26,5)/(52,5) = 1/39.51.)
28. What are the chances of choosing 6 winning numbers from a field of 42? (1/(42,6) = 1/5,245,786.)
29. What is the likelihood of rolling any pair with 5 dice? (6(1/6^2 x 5/6^3 x (5,2)) = 1/1.0368.)
30. What is the likelihood of getting any pair with 5 cards? (13 x(4,2) x (12,3) x 4^3/(52,5) = .4226.)
31. What is the likelihood of getting a hand with no aces? ((48,5)/(52,5) = 1/1.5179.)
32. What is the probability of getting a 5-card poker hand with an Ace and no other cards to improve the hand?
(You can start counting 4-sets of cards that will not improve the hand. It qiuckly becomes convoluted, confusing and quite wrong. The answer is to look at it a different way. You are looking for what is commonly called 'garbege' hands--there are 1,302,590 of them. Try counting that. See #23. The answer is .5011. 1,302,590/2,598,960) Though some players will call two 'garbage' hands by the hand with the highest card, most just call it a tie and ante-up for another round. That will always be so if the two high cards are the same rank. There is no order among suits, though some like to think Spades beat others. Not so.)
33. What if you specify, say, the Ace of Diamonds? (Again, same deal. Answer is .5011. Interesting that the deck is against you to begin with. But a little less than 1/2 of players (.4989) will be dealt a pair or better. In general, take any card from a deck, and you'll be left 1302590 4-card sets that will not better the card you chose. See my item "Frequency of Poker hands," in archives on this blog for May 2005.)
34. Three men stop at an oasis on a desert trip. They eat and fall asleep as the sun shifts. When they wake up, they've turned red with the blazing sun and begin to laugh at one another. One suddenly stops laughing. Why? (Man A reasons that if his own face was unburned, man B would wonder what man C was laughing at, and stop laughing himself.)
35. You come to a fork in the road, one way to town, the other to the dragon swamp. A truth-teller and a liar are there. What single question can you ask to find the way to town? (Ask either what the other would say and do the opposite.)
36. What is the maximum number of colors required to separate countries on a world map? (4. Try drawing 4 countries of any shape--wherever and however you draw a fifth, you can use one of the four colors.)
37. You're on a TV game show. The host shows you three doors. One has a Ferrari behind it, the other two, goats. You're asked to select one door. You choose door A, hoping the for the 1/3 chance at a Masurati. Then the host surprises you by opening door C, where there is a goat; he then asks if you'd like to change your mind and choose door B. Should you? (Positively! You double your chances to 2/3. Although this problem, published in a major newspaper syndicate, fooled the best, academic 'rolodexes' let most down. The furor was over whether probabilities change. Ordinarily the rule is "No!" But not in this case. Consider this analogy: You are an oyster shucker--your boss dumps 1000 oysters on your table, saying there's a big pearl in one. You decide to play yourself a game and put one aside on another table. Then you shuck 998 oysters finding no pearl. There are now the one you put aside at 1/1000, and the unshucked one on the table. The chances of the pearl in the unshucked oyster is 999/1000.)
38. You throw a ball straight up. It rises then falls. For how long did it stop while changing direction? (t = 0...the ball does not stop...this is a continuous function. Don't worry about it.)
39. To what power do you raise the number 'e' to equal the imaginary number 'i', or √-1? (From e^ix= cosx + i sinx, substitute π/2 for x, thus (e ^(i (π/2)) = i.
40. What is the meaning of i^i? (It is both sides of the above identitiy raised to the i power = e^-(π/2.) = i^i. You can calculate this out to .207879... but note that it is but one of a family of solutions...refer to the i and exponents thereof in question 9.)
41. Given any whole number n, what is the sum of all sets of 1, 2, 3, ... n numbers and what is the sum of those sums?. (For n = 5 that sum is 225.)
Compilation TLOICf- MMV
#'s 1, 2, 6, 7, 8, 10, 12, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, by TLOICf-©MMV
1. At 6:00 (AM or PM) the clock hands are in line with one another. When will they be in line again? (7:05:27.272727, via (5 + x + (30 - (5 + 1/12(5 + x))) = 30.)
2. If you were born on Friday, January 13, in a leap year when, exactly, will that happen again? Never, of course, but when will your birthday fall on Friday, January 13 in a leap year again? (28 yrs.)
3. If you tie a string around the equator, then another string just 2π feet (6.283 feet) longer around, how high above the first string will the second be all the way around? (1 foot.)
4. How many rotations on its axis does a one-inch circumference gear make going around a stationary three-inch circumference gear? (4.)
5. Solve for x and y: √x + y = 7; √y + x = 11. (x= 9, y = 4.) Now try √x + y = 858; √y + x = 318. (x = 289, y = 841.)
6. Rolling two dice once, what is the probability of getting at least one six? (1 - (5/6 x 5/6) = 11/36, or, (1/6 x 5/6) + (1/6 x 5/6) + (1/6 x 1/6) = 11/36.)
7. I notice the alley next to my house has a ladder leaning 20 feet high against one building and another ladder leaning 10 feet high against the other building, both with feet abutting the opposite building. They intersect at a height of six feet, eight inches. I come to your house and see the same arrangement. But your alley is 3 feet wider than mine. How high is the intersection point at your house? (The same.)
8. Figure two-digit cube roots. ( Learn this second-digit pattern: 1874563290. Whatever the cube is, use its last digit to count along the pattern for the second digit of the cube root--and know that 10, 20, 30...90 cubed = 1,000, 8,000, 27,000, 64,000, 125,000, 216,000, 343,000, 512,000, and 729,000. Now have someone cube a 2-digit number, say 73. The cube is 389017. You know the second digit is a 3--from the 7th number in the pattern above, and you know the first is a seven because it must be equal to or more that 70^3 = 343,000.)
9. Prove e to the iπ ( + 1) = 0 (Given
a. e to the x = 1 + x + xsq/2! + xcubed/3!...
b. sinx = x - xcubed/3! + xfifth/5! - xseventh/7!...
c. cosx = 1 - xsq/2! + xfourth/4! - xsixth/6!...
In e to the x, substitute iu for x and do the simple algebra and remember powers of i: For i to the 1, 2, 3, 4..., the repeating pattern is +i, -1, -i, +1...)
10. The Greek's Golden Number is (√5 + 1)/2 = 1.618 = Q. Q^2 = Q + 1 and 1/Q = Q - 1. If you draw a star inside a pentagon, almost all line to line segment ratios = Q, or its inverse. A 1 to 1.618, or the same, .618 to 1 rectangle is supposed to the most pleasing to view. The new TV aspect ratio is supposed to be changed to 16 to 9 from the current 4 to 3. What height would result in a Golden number rectangle? (9.88" for a 16" length.)
11. Draw a square and make a grid of 16 little squares inside it. What is the total number of squares you can see? (30)
12. A man with an enclosed truck comes to a two-ton limit bridge. His truck weighs 3800 pounds and he's carrying 300 one-pound pigeons for a total of 4100 pounds, 100 over the limit. He decides to beat the bejesus out of the truck to get all the pigeons flying, then gets in the truck and proceeds to cross the bridge, knowing at least half the pigeons are still in flight. Does he make it? (No. It's a closed system and the truck weighs the same whether the pigeons are roosted or in flight.)
13. Consider a circle with a diameter = 1. C = π, so a semicircle has a length of π/2. Draw any number of circles on the diameter of 1 and count the length of all the new semicircles (the 'S' shape, i.e.) The length remains π/2 no matter how many little semicircles are made. Q: Do the semicircles get so small that the 'S' shape becomes a line? (No. It remains an 'S' shape with infinitely small semicircles but not a line.)
14. Twenty universities send a physicist, a chemist, a mathematician, an astronomer, and a biologist--each--to a convention. How many committees of ten, with each field represented by two scientists, can be formed? (nCr, or (20,2)=190. So there can be 2 physicists from 190 places--and with each, 2 biologists from 190 places... 190^5 = 2.47 x 10^11.)
15. If you get on your boat that goes 10 mph against a 5 mph current, and your girl drops her hat in the water as she gets on the boat, then you travel 45 minutes and she remembers, how long does it take to retrieve the hat? (45 minutes--the current is irrelevant.)
16. How is the product of any 4 consecutive integers related to the product of their perfect squares? (It's its square root.)
17. Bob's wife picks him up every day at his train's arrival. He takes an hour off one day and decides to walk, after getting off the hour-earlier train--he meets his wife on the way, hops in and their home 20 minutes early. How long did Bob walk? (He saved her 10 minutes of her travel time to the station--like she was 10 minutes early, so Bob walked 50 minutes.)
18. You have some potatoes, 99% water 1% potato. You put 100 pounds on you porch to dry. After a while enough water has evaporated so the potatoes are 98% water, How much do they weigh? (50 lbs.)
19. a. You catapult a baby straight up and it descends for 1.6 seconds. How high did it rise? (40.90 feet)
b. You set the catapult at 45 degrees. The baby lands 2.5 seconds after beginning decent. What was the height of the arc? (100 feet.)
c. You throw a twin baby from a 512-foot building at noon. Halfway down what time is it? What time does it land? (12:04:04 and 12:00:05.6.)
d. You drop the other twin when the first is 1/2 way down. What is the time between landings? (4.0 seconds.)
20. On a planet with a radius of 1080 miles and gravitational acceleration of 12m/sec/sq. what is orbit speed for the lowest possible orbit? (from a=v^2/r...1.566 mps.)
21. Evaluate √1 + 2 √1 +3 √ 1 + 4...and note that all √ bars extend all the way above all of those below. (3.)
22. Draw an isosceles triangle with an apex labeled 20 degrees. Then draw a 60 degree ray from one corner to the opposite side, and a 50 degree ray from the other corner to its opposite side. Now connect the 2 points where the rays meet the sides. Find the value of the uppermost angle of the triangle made by this connection. (30 degrees--prove it--no tools.)
23. What is the likelihood of being dealt 5 cards with no poker hand at all--'garbage?' (There are 1,302,540 such hands of 2,598,960 possible hands. So so likelihood is .501177394, or 1/1.9953.)
24. What is the likelihood of throwing 25 pennies on the floor and seeing 2 heads show up? (.5^2 x .5^23 x (25,2) = .00000894 = 1/111,894.)
25. You deal a poker hand (Ace high, always) to a friend and yourself. You have three Aces and another card and are about to draw the tenth card. What is the likelihood you'll get the last Ace, remembering he may have it? (1/43 x (47,5)/(48,5) = 1/48.)
26. What is the likelihood of getting 5 cards with one King? ([(4,1) x (48,4)]/(52,5) = 1/3.339.)
27. What is the likelihood of being dealt 5 red cards? ((26,5)/(52,5) = 1/39.51.)
28. What are the chances of choosing 6 winning numbers from a field of 42? (1/(42,6) = 1/5,245,786.)
29. What is the likelihood of rolling any pair with 5 dice? (6(1/6^2 x 5/6^3 x (5,2)) = 1/1.0368.)
30. What is the likelihood of getting any pair with 5 cards? (13 x(4,2) x (12,3) x 4^3/(52,5) = .4226.)
31. What is the likelihood of getting a hand with no aces? ((48,5)/(52,5) = 1/1.5179.)
32. What is the probability of getting a 5-card poker hand with an Ace and no other cards to improve the hand?
(You can start counting 4-sets of cards that will not improve the hand. It qiuckly becomes convoluted, confusing and quite wrong. The answer is to look at it a different way. You are looking for what is commonly called 'garbege' hands--there are 1,302,590 of them. Try counting that. See #23. The answer is .5011. 1,302,590/2,598,960) Though some players will call two 'garbage' hands by the hand with the highest card, most just call it a tie and ante-up for another round. That will always be so if the two high cards are the same rank. There is no order among suits, though some like to think Spades beat others. Not so.)
33. What if you specify, say, the Ace of Diamonds? (Again, same deal. Answer is .5011. Interesting that the deck is against you to begin with. But a little less than 1/2 of players (.4989) will be dealt a pair or better. In general, take any card from a deck, and you'll be left 1302590 4-card sets that will not better the card you chose. See my item "Frequency of Poker hands," in archives on this blog for May 2005.)
34. Three men stop at an oasis on a desert trip. They eat and fall asleep as the sun shifts. When they wake up, they've turned red with the blazing sun and begin to laugh at one another. One suddenly stops laughing. Why? (Man A reasons that if his own face was unburned, man B would wonder what man C was laughing at, and stop laughing himself.)
35. You come to a fork in the road, one way to town, the other to the dragon swamp. A truth-teller and a liar are there. What single question can you ask to find the way to town? (Ask either what the other would say and do the opposite.)
36. What is the maximum number of colors required to separate countries on a world map? (4. Try drawing 4 countries of any shape--wherever and however you draw a fifth, you can use one of the four colors.)
37. You're on a TV game show. The host shows you three doors. One has a Ferrari behind it, the other two, goats. You're asked to select one door. You choose door A, hoping the for the 1/3 chance at a Masurati. Then the host surprises you by opening door C, where there is a goat; he then asks if you'd like to change your mind and choose door B. Should you? (Positively! You double your chances to 2/3. Although this problem, published in a major newspaper syndicate, fooled the best, academic 'rolodexes' let most down. The furor was over whether probabilities change. Ordinarily the rule is "No!" But not in this case. Consider this analogy: You are an oyster shucker--your boss dumps 1000 oysters on your table, saying there's a big pearl in one. You decide to play yourself a game and put one aside on another table. Then you shuck 998 oysters finding no pearl. There are now the one you put aside at 1/1000, and the unshucked one on the table. The chances of the pearl in the unshucked oyster is 999/1000.)
38. You throw a ball straight up. It rises then falls. For how long did it stop while changing direction? (t = 0...the ball does not stop...this is a continuous function. Don't worry about it.)
39. To what power do you raise the number 'e' to equal the imaginary number 'i', or √-1? (From e^ix= cosx + i sinx, substitute π/2 for x, thus (e ^(i (π/2)) = i.
40. What is the meaning of i^i? (It is both sides of the above identitiy raised to the i power = e^-(π/2.) = i^i. You can calculate this out to .207879... but note that it is but one of a family of solutions...refer to the i and exponents thereof in question 9.)
41. Given any whole number n, what is the sum of all sets of 1, 2, 3, ... n numbers and what is the sum of those sums?. (For n = 5 that sum is 225.)
Compilation TLOICf- MMV
#'s 1, 2, 6, 7, 8, 10, 12, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, by TLOICf-©MMV
posted by Tom at 10:42 PM
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