Mathematical Questions and Solutions in Continuation of the Mathematical Columns of "The Educational Times" Volume 64
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Mathematical Questions and Solutions in Continuation of the Mathematical Columns of "The Educational Times" Volume 64

Mathematical Questions and Solutions in Continuation of the Mathematical Columns of "The Educational Times" Volume 64


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 Excerpt: ...figure, and (2) the respective probabilities of 0, 1, 2, 3 intersections, if three diagonals are drawn at random. Solution by H. Fortey. (1) In what follows, diagonals are not produced, and intersection means intersection within the polygon. (2) Let the polygon have n sides. Then the number of pairs of intersecting diagonals is C4, for each set of 4 angular points gives one pair. But the whole number of diagonals is n (---3), and the whole number of pairs of diagonals is in.(--s)Oj; therefore the chance of a random pair intersecting = C4-f i 0-3,0., = fe=llfczjl. n--on--& (3) In dealing with the question of 3 diagonals, let us first include the sides of the polygon amongst the random lines that may be drawn so that 3 random lines are not all (or even any of them) necessarily diagonals. (4) Now the 3 random lines may have their extremities at any 6, 6, 4, or 3 angular points of the polygon, and the types for each number of points when there are 3, 2, or 1 intersections, with the number of cases included in each type, are as follows: --(5) Therefore, if I, be the number of triplets of lines which make p intersections, we see by inspection that IS = C6. I2= 3. C6 + 6. C5, I, = 6. C6+10.MC6 + 4. C4. (6) But, remembering that both diagonals and sides are included in this calculation, let us see what deduction must be made from the values of I3, I2, It, when diagonals only are considered. In the diagrams of paragraph 4, any two lines which cross one another are obviously diagonals, but an uncrossed line may be a side. Therefore I3, I2 remain unaltered, but from I, we must deduct n. -2C4 for the 6-point type, In. -2C3 for the two 5-point types, n. -2C2 for the 4-point type; 1279...


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Product Details
  • ISBN-13: 9781153633536
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 28
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 1153633531
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 68 gr


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