CAT 2024 Slot 2 · DILR · Question 9

Solving the set

Step 1 – Lengths of all paths. The diagram is a grid of 4 columns of points and 4 rows of points. Using the given distances (OP = 150, ON = MN = 300 for the widths; ML = 400, EL = 200, DE = 400 for the heights):

Line	Segments and lengths (m)
Columns (widths)	A–B, H–G, I–J, P–O = 150; B–C, G–F, J–K, O–N = 300; C–D, F–E, K–L, N–M = 300
Rows (heights)	A–H, B–G, C–F, D–E = 400; H–I, G–J, F–K, E–L = 200; I–P, J–O, K–N, L–M = 400
Diagonal I–G	√(150² + 200²) = 250
Diagonal O–K	√(300² + 400²) = 500

Step 2 – Special points. Entry/exit is only at C. Residences lie on paths A–H and M–L (and at A, H, M, L). The post office is at P and the bank at B.

Step 3 – Lake-side paths. The three shaded lakes are C-D-E-F, G-F-K-J and K-L-M-N. The paths along their sides are listed below.

Lake	Paths next to it	Length (m)
Lake 1 (C, D, E, F)	C–D 300, D–E 400, E–F 300, F–C 400	1400
Lake 2 (G, F, K, J)	G–F 300, F–K 200, K–J 300, J–G 200	1000
Lake 3 (K, L, M, N)	K–L 300, L–M 400, M–N 300, N–K 400	1400
Total	12 paths	3800

Step 4 – Useful shortest routes (used in the first question).

Leg	Shortest route	Distance (m)
L → P (post office)	L–K–O–P = 300 + 500 + 150	950
P → B (bank)	P–I–G–B = 400 + 250 + 400	1050
B → L (home)	B–C–F–K–L = 300 + 400 + 200 + 300 (or B–G–F–E–L)	1200
Total		3200

This question

The visitor starts and ends at C and may not use A, H, M, L or the paths A–H and M–L. The usable points are B, C, D, E, F, G, I, J, K, N, O, P (12 points).

D can only be reached from C or E, and E (without L) only from D or F, so the loop must contain C–D–E–F. The longest loop through all 12 points is:

Step	Path	Length (m)	Running total
1	C–B	300	300
2	B–G	400	700
3	G–J	200	900
4	J–I	150	1050
5	I–P	400	1450
6	P–O	150	1600
7	O–N	300	1900
8	N–K	400	2300
9	K–F	200	2500
10	F–E	300	2800
11	E–D	400	3200
12	D–C	300	3500

Hence, the answer is 3500.