I have gone through some simple mental math steps:
1) we want minimum no. of wrong answered questions. so, let's do not use them to deduct marks at the start. we will only use them when their is no option. so our goal is to reach at 32 marks by deducting marks from 50.
2) their should be 32+ questions got to be correct as to be at 32 after deduction of marks and extra marks above 32 will be nullified by wrong and un-attempted questions.
3) every 6 un-attempted questions will deduct 6 marks of themselves and 6 x -1/6 = 1 marks as penalty.
so, every 6 questions will deduct in total 7 marks.
from now on we will try to deduct in multiples of 7 marks representing 6 un-attempted questions.
4) we can deduct less than 50 - 32 = 18 as to maintain 32+ marks.
7 x 2 = 14 which is in range under 18 and 7 x 3 = 21 which is out of range.
so that means we can deduct 14 marks with 12 un-attempted questions. (12 marks for the questions themselves and 2 for their penalty).
5) we have reached at 36 after deducting 14 from 50 marks with 12 un-attempted questions.
now we have to deduct rest 4 also to reach to 32.
6) if we try to deduct these 4 marks with 3 more un-attempted questions then total score would be 36 - 3 - ( 3 x 1/6 ) = 33 - 1/2 = 32.5, which is not right. so this time we have to use wrong answered questions.
7) every 3 wrong answers will deduct 3 for themselves and 3 x -1/3 = 1 for their penalty totaling to 4, which is precisely what we need now.
8) after these 3 wrong answered questions total score would be 36 - 4 = 32.
Hence, after 12 un-attempted questions and 3 wrong answered questions the net score would be 32.
Therefore, we can say that the number of questions answered wrongly by that student cannot be less than 3.
