CAT 2024 Slot 2 — DILR Question 3
Answer the following questions based on the information given below.
The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes (3 of them, each in the shape of rectangles – shaded in the diagram) of a gated area. Different points on the walkway are indicated by letters (A through P) with distances being OP = 150 m, ON = MN = 300 m, ML = 400 m, EL = 200 m, DE = 400 m.
The following additional information about the facilities in the area is known.
1. The only entry/exit point is at C.
2. There are many residences within the gated area; all of them are located on the path AH and ML with four of them being at A, H, M, and L.
3. The post office is located at P and the bank is located at B.
| Segment | Length (m) | Segment | Length (m) |
|---|---|---|---|
| AB, HG, IJ, PO | 150 | HI, GJ, FK, EL | 200 |
| BC, CD | 300 | GF, FE | 300 |
| JK, KL | 300 | ON, NM | 300 |
| AH, IP, BG, JO | 400 | CF, KN, DE, LM | 400 |
| GI (diagonal) | 250 | OK (diagonal) | 500 |
Coordinates (m), origin at P, taking the grid columns at x = 0, 150, 450, 750 and rows at y = 0, 400, 600, 1000: A(0,1000) B(150,1000) C(450,1000) D(750,1000); H(0,600) G(150,600) F(450,600) E(750,600); I(0,400) J(150,400) K(450,400) L(750,400); P(0,0) O(150,0) N(450,0) M(750,0). Lakes: C-D-E-F, G-F-K-J, K-L-M-N.
One resident takes a walk within the gated area starting from A and returning to A without going through any point (other than A) more than once. What is the maximum distance (in m) she can walk in this way?
Answer & solution
Answer: 5100
Medium
This is the longest simple cycle through A. A's only neighbours are B and H, so both AB and AH are forced. The two diagonals (GI = 250, OK = 500) are shorter than the grid detours they replace, so a maximum walk avoids them. Build the longest cycle using the long 400 and 300 segments.
Segments (m): AB,HG,IJ,PO = 150; HI,GJ,FK,EL = 200; BC,CD,GF,FE,JK,KL,ON,NM = 300; AH,IP,BG,JO,CF,KN,DE,LM = 400. The four corners A, D, M, P each have only two walkways, so their pair of edges is forced once they are used.
Forced edges. Each corner that the loop passes through uses both of its only two segments.
Complete the loop with long links. Choosing BG, CF, KN, JO (all 400) plus GF, JK (300) and EL, HI (200) closes a single cycle visiting all 16 points.
Total length. Add the forced edges (step 1) and the links (step 2).