Jan 2009

AC < AB 2x - 4 < x + 6 x - 4 < + 6 (subtract x on both sides) x < 10 (add 4 to both sides)

AC < AE AC < AD 2x - 4 < 4x - 14 2x - 4 < 3x - 7 -2x - 4 < - 14 (subtract 4x on both sides) -x - 4 < - 7 (subtract 3x on both sides) -x < -3 (add 4 to both sides) -2x < -10 (add 4 to both sides) x > 3 (Inverting the equation by dividing the equation by -1) x > 5 (Inverting the equation by dividing by - 2)

CD < CB and CD < CE 2x – 1 < 3x - 7 2x – 1 < x + 8 -x – 1 < - 7 x – 1 < + 8 -x < - 6 x < 9 x > 6

From your previous answers x > 6 but also < 9 Now consider that DE < DB so 2x – 2 < 3x - 9 If x is now > 7, but < 9 the x = 8 -x – 2 < - 9 -x < - 7 x > 7

Total = 72 Remember x = 8 AC = 2x – 4 = 12 A → C → D → E → B → A CD = 2x – 1 = 15 12 15 14 17 14 DE = 2x – 2 = 14 EB = x + 9 = 17 Total = 72 BA = x + 6 = 14

𝐴 →𝐵 →𝐶 →𝐷 →𝐸 →𝐹 →𝐴 𝐹 →𝐷 →𝐶 →𝐴 →𝐵 →𝐸 →𝐹 Note - You can start at any letter and visit every other letter once, in any order, finally returning to your start letter. 𝐹 →𝐷 →𝐶 →𝐴 →𝐵 →𝐸 →𝐹 20 15 5 25 15 15 TOTAL = 95 It is a TOUR that exist, but MAY BE improved upon

𝐹 →𝐸 →𝐶 →𝐴 →𝐵 →𝐷 →𝐹 30 7 5 25 11 10 TOTAL = 88

Jan 2010 B → E → C → D → A → B 3.7 + 1.9 + 2.7 + 2.0 + 1.7 = 12.0 or 12 B → D → A → C → E → B 1.8 + 2.0 + 1.9 + 4.2 + 3.6 = 13.5

The best upper bound is the lowest value, so 12.0 From To The best upper bound is the lowest value, so 12.0 B → A → D → E → C → B 1.7 + 2.0 + 1.7 + 4.2 + 2.5 = 12.1

June 2010

𝑆 →𝑇 →𝑅 →𝐼 →𝑁 →𝐺 →𝑆 64 70 82 80 82 72 TOTAL = 450 𝑁 →𝐺 →𝑆 →𝑇 →𝑅 →𝐼 →𝑁

Using Kruskal or Prim draw a minimum spanning tree Delete 𝑆 Using Kruskal or Prim draw a minimum spanning tree Total minimum spanning tree = 𝟐𝟗𝟑 𝑮 𝟕𝟔 Add the two shortest deleted edges 𝑰 𝟕𝟑 𝑵 𝟕𝟒 𝑻 𝑹 𝟕𝟎 𝑰 𝟔𝟖 64 + 68 = 𝟏𝟑𝟐 𝑺 𝑻 𝟔𝟒 Lower Bound 293 + 132 = 𝟒𝟐𝟓