Jain housing complex on OMR has a democratically elected governing council…
2024
Jain housing complex on OMR has a democratically elected governing council comprising of president, secretary and the treasurer. During their annual meeting they take up 3 different initiatives for discussion and voting, namely, painting of exteriors, 24 hour security and additional water tank. They vote as below: • Each member of the council votes for at least one of the initiatives and against at least one of the initiatives. • Exactly two members of the council vote for the painting initiative. • Exactly one member of the council votes for the security initiative. • Exactly one member of the council votes for the water tank initiative. • The president votes for the painting initiative and against the security initiative. • Secretary votes against the painting initiative. • Treasurer votes against the water tank initiative. Which one of the following statement could be true?
- A.
President and Secretary vote the same way on the water tank initiative.
- B.
Secretary and Treasurer vote the same way on the painting initiative.
- C.
Secretary and Treasurer vote the same way on the security initiative.
- D.
President votes for one of the initiatives and secretary votes for two of the initiatives.
Attempted by 3 students.
Show answer & explanation
Correct answer: D
In a constraint-satisfaction voting puzzle, when a fixed number of 'for' votes must be distributed among a fixed set of voters and some voters' choices are already known, the remaining voters' votes become forced by the count. When exactly one of two voters must supply a category's single remaining 'for' vote, those two voters are locked into an always-opposite pattern on that category, not a free choice.
Painting requires exactly 2 'for' votes. President is for, Secretary is against, so Treasurer must vote for painting to reach the total of two.
Security requires exactly 1 'for' vote. President is against security (given), so the single 'for' vote must come from either Secretary or Treasurer, but not both — they always vote oppositely on security.
Water tank requires exactly 1 'for' vote. Treasurer is against water tank (given), so the single 'for' vote must come from either President or Secretary, but not both — they always vote oppositely on water tank.
Every member must vote for at least one initiative and against at least one. Treasurer already has a 'for' (painting) and an 'against' (water tank), so Treasurer's security vote is free. President already has a 'for' (painting) and an 'against' (security), so President's water-tank vote is free. Secretary has only an 'against' (painting) so far, so Secretary must vote 'for' on at least one of security or water tank to satisfy the 'at least one for' rule.
Testing the arrangement where President votes for painting only, against both security and water tank: then Secretary must supply the single 'for' vote on both security and water tank (since President contributes zero to each), and Treasurer stays against on both. This satisfies every count (2 painting, 1 security, 1 water tank) and gives every member both a 'for' and an 'against' vote. In this arrangement, President votes for exactly one initiative (painting) while Secretary votes for exactly two (security, water tank).
Checking the other statements: water tank always splits President and Secretary into opposite votes (one 'for', one 'against'), so they can never match on water tank. Painting always has Secretary against and Treasurer for, so they can never match on painting. Security always splits Secretary and Treasurer into opposite votes, so they can never match on security. Since those three statements are impossible in every valid arrangement, while the arrangement above is a fully valid one where President votes for one initiative and Secretary votes for two, that is the statement that could be true.