GATE CS 2022 Set 1 — Question 10

Put the poles on a grid with coordinates $0$ to $6$ in each direction. The land is the $6 \times 6$ square with two opposite corners cut off: the corner at $(6,6)$ is cut by the segment $(5,6)$–$(6,5)$, and the corner at $(0,0)$ by $(0,1)$–$(1,0)$. Rope $R1$ runs from $(0,2)$ to $(2,2)$; rope $R2$ runs from $(4,6)$ down to $(4,4)$.Every plot must be similar to the others, so a natural choice is the shape of the land's corner pieces: a square with one corner cut off at half its side. The total area is $36 - 2\times\tfrac12 = 35$; with two small plots (a $2\times 2$ square with a $1\times1$ corner cut, area $3.5$) and two large plots (a $4\times4$ square with a $2\times2$ corner cut, area $14$) we get $3.5+3.5+14+14 = 35$.The existing ropes already form one side of each small plot, so add:The diagonal splits the remaining region into two $4\times4$ squares, each with one corner cut off by a right triangle of legs 2: $[0,4]\times[2,6]$ missing the corner at $(4,2)$, and $[2,6]\times[0,4]$ missing the corner at $(2,4)$. All four plots are similar (each is a square with a corner cut at half its side).Each small plot needs one rope to close it and the remaining region needs one rope to split it into two plots, so three additional ropes are needed, and three suffice.Answer: (D) 3