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Donald Cartmill

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  1. I want to share with you the solution to any odd numbered grid,where by the columns , rows ,and diagonals each has the same total. we have all seen the very simple 3 x 3 grid filled with # 1 thru 9 ; Much more difficult would be a 5 x 5 grid with numbers 1 thru 25 ; A 7 x 7 grid would be virtually impossible ,but with the few simple rules below any odd numbered grid can be filled as fast as you can write and all cols ,rows and diagonals total the same 1a) Always start in the square to right of center square !) you are always traveling diagonally down to the right; when you e
  2. A monk started out at 7:00 am at the bottom of the mountain and climbed to the top arriving at 7:00 pm He prayed all thru the night and at 7:00 am started down the mountain arriving at the bottom at 7:00 pm. Is there a point on the mountain trail where the monk was at the same time going up and coming down Proof please
  3. Fold rope in 1/2 then again in 1/4 allow 1/4 to burn =15 minutes
  4. 60-40= 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 mp
  5. 60-40= 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 60-40
  6.     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 ; 90-65  =     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

  7. Machine A worked 40 hours. 45x = 40(x + 5 ); 45x = 40x +200.; 5x = 200 ;x = 40
  8. The area would be that of the original triangle,however that area would vary as determined by the length of the 3rd side
  9. operates on the 1st ; Turns the gloves inside out for # 2; operates bare handed on the 3rd
  10. 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
  11. 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) t
  12. 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) t
  13. 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
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