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Everything posted by Donald Cartmill

6040= 20 car lengths; therefor man walked only 10 car lengths. ; mans rate is 3 mph; t = d / speed ; We have to look at the 1st car in each direction .as a unit of length Now assume 10 cl to be 1 mile 1m / 3 mph =.333 hrs; traveling in same direction train traveled = 40 + 10 = 50 lts =5 miles ; s = d / t 5 m/ .33 hrs = 15 mph train traveling opposite direction =60  10 = 50 lts = 5m; 5m / .33 hrs = 15 mph So let us assume a different value for 10cl = 0.5 miles ; 0.5 / 3mph = 016666667 hrs ; 50 cl = 2.5 m; 2.5 m / 0.16666666667 = 15 mph Regardless of what value in miles is assumed for 10 car lengths the answer is 15 mph If you cannot assume both are traveling at the same speed,then it is unsolvable. or actually a solutions for every increment of time he walks i.e. he walks 15 seconds and in that time 40 passes one way and 60 passes the other way...Obviously they would really be flying and one going approximately 50% faster

6040= 20 car lengths; therefor man walked only 10 car lengths. ; mans rate is 3 mph; t = d / speed ; Now assume 10 cl to be 1 mile . / 3 mph =.333 hrs; traveling in same direction train traveled = 40 + 10 = 50 lts =5 miles ; s = d / t 5 m/ .33 hrs = 15 mph train traveling opposite direction =60  10 = 50 lts = 5m; 5m / .33 hrs = 15 mph So let us assume 10cl = 0.5 miles ; 0.5 / 3mph = 016666667 hrs ; 50 cl = 2.5 m; 2.5 m / 0.16666666667 = 15 mph Regardless of what value in miles is assumed for 10 car lengths the answer is 15 mph 6040= 20 car lengths; therefor man walked only 10 car lengths. ; mans rate is 3 mph; t = d / speed ; Now assume 10 cl to be 1 mile . / 3 mph =.333 hrs; traveling in same direction train traveled = 40 + 10 = 50 lts =5 miles ; s = d / t 5 m/ .33 hrs = 15 mph train traveling opposite direction =60  10 = 50 lts = 5m; 5m / .33 hrs = 15 mph So let us assume 10cl = 0.5 miles ; 0.5 / 3mph = 016666667 hrs ; 50 cl = 2.5 m; 2.5 m / 0.16666666667 = 15 mph Regardless of what value in miles is assumed for 10 car lengths the answer is 15 m time = distance / speed; "t " will be in hours. speed = distance / time Regardless of what value in miles is assumed for 10 car lengths the answer is 15 mph

90 70= 20 deg using side of square as "a" , we have tan20 = o/a and then O= atan 20 or 0.3639a; The right leg of the triangle and the top portion of the right side 20 +45 =65 ; 9065 = 25 degrees the other angle at top left and therefore the 3rd angle is 65; tan 25 = o/a or o = a tan25; o = 0.4663a. since a is the length of a side then the bottom leg of the lower right triangle = 1  0.4663 = 0.5336a ; the vertical leg of the lower triangle is 1  0.3639a = 0.6360a. we will designate the left angle of the bottom triangle as Y , So we can say Tan Y = o/a or 0.6360a / 0.5336a; tan Y = 1.1917; Y = 50 Adding the bottom angles we have 65 + 50 = 115 : 180  115 =65 degrees X = 65 degrees

Machine A worked 40 hours. 45x = 40(x + 5 ); 45x = 40x +200.; 5x = 200 ;x = 40

Area of an infinitely shaded region
Donald Cartmill replied to BMAD's question in New Logic/Math Puzzles
The area would be that of the original triangle,however that area would vary as determined by the length of the 3rd side 
Operating on three people with two surgical gloves.
Donald Cartmill replied to BMAD's question in New Logic/Math Puzzles
operates on the 1st ; Turns the gloves inside out for # 2; operates bare handed on the 3rd 
Help! Honeymoon destination riddle!
Donald Cartmill replied to Nickynoo's question in New Logic/Math Puzzles
scotland 
If the question is : What is the ratio of the arc length to the perpendicular length from the center of one side of the square to the arc ??? Then assuming diameter of 1" ; the diagonal of the largest square would also be 1 ; this divides the square into 2 isosceles triangles ,whose angles are 45,45,90 In a 45 degree triangle the sides are = Hsin angle or 1 sin 45 = 0.707; therefore i/2 of the lt of a side is 0.353 . The radius is 0.5. 0.5  0.353 = the rise 0.147 the distance from arc to square Now the circumference is pie D or 3.14 D, 3.14 (1) = 3.14; acr length 3.14 divided by 4 =0.7853 ratio of arc length to rise ....0.7853/0.147 = 5.342 :1 Previously I assumed the wrong question ,However assuming a diameter of 1" R = 0.5" . I calculated the rise ( r ),which is the distance from the one side of the square at the mid point of that side to the arc or circumference . That value r =0.147 therefore the ratio of r/R = 0.147/0.5 0.147 /0.5 = 1/X; 0.147X = 1x 0.5; X = 1 /0.147; Ratio r / R is 1 /6.802 Quote

total of $ 0.75 ; Arrange 4 pennies in a circle around a penny in the center (A ) such that none of the outside pennies are on the same diameter passing thru (A) Let's assume that as we look at our screen (B) is directly below (A) at a distance of the radius ; Moving CW up to the left more than 90 deg is (C) ; continuing CW approx 70 degrees which stops us short of being diametrically opposite (B) is location of (D) : continue CW about 90 degrees for (E). You have 4 triangles which do NOT over lap ABC; ACD ; ADE ; AEB . You then have 6 triangles created by lines from(B) to (D) and from (C) to (E) These triangles being BCD ; BDE; BDA; CDE ; CEA ; CEB. So we have 4 @ 0.10 = $040 and 6 @ .05 = 0.30 for a total of $ 0.70 plus the 5 pennies = 75 cents

Total of $ 0.70 ; Arrange 4 pennies in a circle around a penny in the center (A ) such that none of the outside pennies are on the same diameter passing thru (A) Let's assume that as we look at our screen (B) is directly below (A) at a distance of the radius ; Moving CW up to the left more than 90 deg is (C) ; continuing CW approx 70 degrees which stops us short of being diametrically opposite (B) is location of (D) : continue CW about 90 degrees for (E). You have 4 triangles which do NOT over lap ABC; ACD ; ADE ; AEB . You then have 6 triangles created by lines from(B) to (D) and from (C) to (E) These triangles being BCD ; BDE; BDA; CDE ; CEA ; CEB. So we have 4 @ 0.10 = $040 and 6 @ .05 = 0.30 for a total of $ 0.70

If the question is : What is the ratio of the arc length to the perpendicular length from the center of one side of the square to the arc ??? Then assuming diameter of 1" ; the diagonal of the largest square would also be 1 ; this divides the square into 2 isosceles triangles ,whose angles are 45,45,90 In a 45 degree triangle the sides are = Hsin angle or 1 sin 45 = 0.707; therefore i/2 of the lt of a side is 0.353 . The radius is 0.5. 0.5  0.353 = the rise 0.147 the distance from arc to square Now the circumference is pie D or 3.14 D, 3.14 (1) = 3.14; acr length 3.14 divided by 4 =0.7853 ratio of arc length to rise ....0.7853/0.147 = 5.342 :1

Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle. This is clear Several clarifications are required: What's the ratio of the edge of the circle .....(1) edge of the circle a) the circumference or b) one of the arcs clarification required: to the middle of the edge of the square ? (assume minimum length) to the radius of the circle. . Is the question : What is the ratio of the arc length to the perpendicular length from the center of one side of the square to the arc ??? Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle. What's the ratio of the edge of the circle to the middle of the edge of the square (assume minimum length) to the radius of the circle.

20 RIDDLES TO TEST YOUR MATH, IQ AND THINKING SKILLS.
Donald Cartmill replied to ash9898's question in New Logic/Math Puzzles
OK I disagree with our most sage and top contributor rocdocmac on 5, 8 , 12 is ok if you add 30 seconds ,17 ,20 . In addition I will be making corrections to #1 . What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 CORRECTION 2 = 999 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 mis interpreted the Q 1,2,3 is correct 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium I assumed in the absence of additional info there were 3 positions . i.e low , med , high. any even number of flips will put the sw back on med 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 This is NOT a math calculation I envisioned 1/2 of 5 = 2 1/2 + 1/2 = 3; therefore 1/3 of 10 = 3.33 + 1/3 =3.367 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 Of the 55 ,20 choose both leaving 35 ; of the 44 ,20 choose both leaving 24; 35 music only ;24 sports only ; 20 M&S total=79 100  79 = 21 who do neither 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 120 5 seats taken 4 at a time 5 ! All four are let in free. Could you have given the correct answer? 120 5 seats taken 4 at a time 5 ! 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes NOTE: the question is not how long it chimes at 12 :00 ,but how long from the 30 seconds of chiming at 6:00 3. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. my bad 20 How many times he has to paint 8? 20 15. You have two books. One of them is upsidedown and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 Up side down ,laying on its side ,propped at an angle the first page is 1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers A pound of feathers weighs 16 ounces avoirdupois; Gold is weighed on the troy system whereby a troy pound is 12 oz ; 1 troy oz = 1.0971 oz avoirdupois therefore 1 pound of gold weighs 13.1652 oz avoirdupois 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has $300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you $600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your $300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women. Quote 
20 RIDDLES TO TEST YOUR MATH, IQ AND THINKING SKILLS.
Donald Cartmill replied to ash9898's question in New Logic/Math Puzzles
. What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 CORRECTION x =900 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 mis interpreted the Q 1,2,3 is correct 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium I assumed in the absence of additional info there were 3 positions . i.e low , med , high. any even number of flips will put the sw back on med 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 I envisioned 1/2 of 5 = 2 1/2 + 1/2 = 3; therefore 1/3 of 10 = 3.33 + 1/3 =3.367 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 Of the 55 ,20 choose both leaving 35 ; of the 44 ,20 choose both leaving 24; 35 music only ;24 sports only ; 20 M&S total=79 100  79 = 21 who do neither 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 120 5 seats taken 4 at a time 5 ! All four are let in free. Could you have given the correct answer? 120 5 seats taken 4 at a time 5 ! 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes NOTE: the question is not how long it chimes at 12 :00 ,but how long from the 30 seconds of chiming at 6:00 3. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. my bad 20 How many times he has to paint 8? 20 15. You have two books. One of them is upsidedown and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 Up side down ,laying on its side ,propped at an angle the first page is 1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers A pound of feathers weighs 16 ounces avoirdupois; Gold is weighed on the troy system whereby a troy pound is 12 oz ; 1 troy oz = 1.0971 oz avoirdupois therefore 1 pound of gold weighs 13.1652 oz avoirdupois 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has $300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you $600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your $300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women. 
20 RIDDLES TO TEST YOUR MATH, IQ AND THINKING SKILLS.
Donald Cartmill replied to ash9898's question in New Logic/Math Puzzles
. What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 mis interpreted the Q 1,2,3 is correct 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium I assumed in the absence of additional info there were 3 positions . i.e low , med , high. any even number of flips will put the sw back on med 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 I envisioned 1/2 of 5 = 2 1/2 + 1/2 = 3; therefore 1/3 of 10 = 3.33 + 1/3 =3.367 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 120 5 seats taken 4 at a time 5 ! All four are let in free. Could you have given the correct answer? 120 5 seats taken 4 at a time 5 ! 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes 13. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. my bad 20 How many times he has to paint 8? 20 15. You have two books. One of them is upsidedown and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has $300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you $600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your $300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women. 
20 RIDDLES TO TEST YOUR MATH, IQ AND THINKING SKILLS.
Donald Cartmill replied to ash9898's question in New Logic/Math Puzzles
. What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 240 All four are let in free. Could you have given the correct answer? 240 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes 13. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. 11 How many times he has to paint 8? 11 15. You have two books. One of them is upsidedown and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has $300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you $600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your $300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women. 
You're super genius if you solve this Puzzle
Donald Cartmill replied to rubaidi's question in New Logic/Math Puzzles
90 students however there are 95 involvements i.e. ( 30 fb; 40 vb; 25 bb ). That there are 23 who do not participate is NOT crucial to the solution. A ) 15 fb + vb ; B) 10 fb + bb ; C) 8 bb + vb ;; 15 fb+ 10 fb =25 fb combined plus 5 ; 15 vb + 8 vb = 23 vb plus 17 ; 10 bb + 8 bb = 18 bb plus 7 The plus 5 ; plus 17; plus 7 represents respectively those who play only 1 or all 3 sports . = 29 Now 25 + 23 + 18 = 66 involvements; plus 29 involvements = 95 involvements Therefore the plus 5 ; plus 17; plus 7 represents respectively those who play only 1 sport and there were none who played all 3 sports I yield to Rubaidi's Venn diagram above great solution 
While the combined speed ratios provide the correct answer . I still maintain the speed ratio of A to B is not crucial to the solution. i.e. Let us suppose that A is merely to step out at the finish line to congratulate B and C the competitors in a two man race . A steps out when B has run 80 meters; Congratulates B when B completes the race and notes at the time C has completed 80 meters. Where was C when A first made his appearance ? The answer is 4/5 x 80 = 64 meters While the combined speed ratios provide the correct answer . I still maintain the speed ratio of A to B is not crucial to the solution. i.e. Let us suppose that A is merely to step out at the finish line to congratulate B and C the competitors in a two man race . A steps out when B has run 80 meters; Congratulates B when B completes the race and notes at the time C has completed 80 meters. Where was C when A first made his appearance ? The answer is 4/5 x 80 = 64 meters To beat a dead horse let A step out when B is at 85 meters; He then congratulates B as he finishes the race and notes C is at 80 meters. He wonders where was C , when I first stepped out ??? The answer is of course 4/5 x 85 = 68 meters

Yes ,I agree Capt Eds' answer is not quite as straight forward as could be. After A reaches 100 m ...20 m ahead of B , the speed of A is not significant to the solution. The key is that when B is at 100 ,C is at 80. Therefore C is 8/10 the speed of B and it follows that when B was at 80 m...C was at 8/10 x 80 = 64 m. A is at 100 m when C is at 64 therefore A beats C by 36 m

A crosses finish 20 meters a head of B ; Now B crosses 20 meters ahead of C ; i.e. B at 100 meters ...C at 80 meters; therefore C was traveling at 8/10 the speed of B. Therefore when B was at 80 meters ...C was 8/10 x 80 meters or 64 meters. Thus A beat C by 36 meters

The answer given is 5! , which closely resembles 51 or 5 ! exclamation mark. It is however 5 factorial or 5 x 4 x 3 x 2 x 1= 120 25  55 + ( 85 + 65 ) = 120 =5 ! 25  55 + (150 ) ; 55 + 175 ; 175  55 == 120

the odds of eating a poison apple for the 1st bowl = 3/5 + 3/4 = 12/20 + 15/20 = 27/20 = !.35 chances of eating the P/A 2nd bowl =2/5 + 2/4= 2/3 = 24 /60 +30/60 + 40/60 = 84/60 = 1.4 CHANCES Therefore you would eat from the 1st bowl with a 0.05 better chance of avoiding the P?A

I'm not sure I recognize your solution 6 from its description of how it's constructed, but your final description of it seems right. Nice solve. Let me attempt to clarify 6) 4 points A,B,C,D, lay on two intersecting circles having the same radius. point A is the center of one circle with C and D laying on its circumference , D is the center of the 2nd circle with A and B laying on its circumference. The 4 points create a trapezoid in which The diagonals AC , & BD plus the base AD are equal all being a radius. Points B and C lay on their respective circumferences at points where the chords AB = CD = equals the top BC I don't know if that is any more clear

4 would be the smallest number. assign a unit weight of 1 ;2 ;3 ;4; to the 4 weights place units 1 and 4 on left pan and 2 and 3 on the right pan each weigh 5 and balance sorry I misunderstood the question in my earlier answer