Explanation:

Let Vishal has V toffees initially and he buys A toffees each day
Let Shekhar has S toffees initially and he buys B toffees each day
After 4 days, they have equal number of toffees.
hence V+4A = S+4B
=> V-S=4(B-A) —- (i) After 12 days Vishal has 8 (16/23)% =200/13% more toffees than Shekhar
V + 12A = (S+ 12 B)* (1+2/23)
=> V+12A =25/23 * (S+12B) —– (ii) After 13 days, Sunidhi who has no toffee initially, took 6 toffees from Vishal and 3 toffees from Shekhar, as a result now Vishal has 6 toffees more than Shekhar.
Toffees with Vishal after 13 days = V+13B ; when 6 toffees given to Sunidhi now he has V+13A-6
Toffees with Shekhar after 13 days = S +13B; when 3 toffees given to Sunidhi now he has S+13B -3
also ;
V+13A-6 = (S+13B -3) + 6
=>V+13A = S+13B+9
=>V-S=13(B-A)+9 ———– (iii)
After 15 days, they all clubbed together their toffees and divided equally among the three. Now each of them has 71 toffees.
All the toffees that they have after 15 days is due to Vishal and Shekhar, as Sundihi had 0 toffees initially
So V+15A + S+15B=3*71=213 —- (iv)
Equate eq (i) and (iii) for V-S = V-S in each equation , so we have now
4(B-A)= 13(B-A)+9
We get A-B= 1
A=1+B—-(v)
put this in eq (i)
S-V=4
S=V+4— (vi)
Put A=1+B and S=V+4
in eq (ii) and (iv) to get two equations with two variable
we will get
V+12B=88
and
V+15B=97
solve and get
B=3 and V=52
hence
A=4 and S=56
so now we have all the required value; keep them handy
V=52; A=4
S=56; B=3
So Vishal has 52 toffees initially

Hence, opiton 4 is the correct answer.