Alex, Bert, and Carl are running for mayor in a town populated by knights and liars. 200 people from the town were gathered as a polling group. Each person in the group favors exactly one candidate.
The first surveyor asked each person "Will you vote for Alex?". The second surveyor asked each person "Will you vote for Bert?". The third surveyor asked each person "Will you vote for Carl?".
The results were as follows: 112 said they would vote for Alex, 82 said they would vote for Bert, and 64 said they would vote for Carl.
How many knights were in the group? At least how many knights said they would vote for Alex? For Bert? For Carl?
Knight = 142
Liars = 58
Let...
KA = Knights who vote for A
KB = Knights who vote for B
KC = Knights who vote for C
LA = Liars who vote for A
LB = Liars who vote for B
LC = Liars who vote for C
YES saying people in Survey#1 (=112) has 3 components
KA + LB + LC = 112 ---Eq#1
YES saying people in Survey#2 (=82) has 3 components
LA + KB + LC = 82 ---Eq#2
YES saying people in Survey#3 (=64) has 3 components
LA + LB + KC = 64 ---Eq#3
NO saying people in Survey#1 (=200-112) has 3 components
LA + KB + KC = 88 ---Eq#4
NO saying people in Survey#2 (=200-82) has 3 components
KA + LB + KC = 118 ---Eq#5
NO saying people in Survey#3 (=200-64) has 3 components
KA + KB + LC = 136 ---Eq#6
Eq#3 MINUS Eq#5 gives...
LA + LB + KC = 64 ---Eq#3
KA + LB + KC = 118 ---Eq#5
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LA - KA = -54 ---Eq#7
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Now, Eq#4 MINUS Eq#7 gives...
LA + KB + KC = 88 ---Eq#4
LA - KA = -54 ---Eq#7
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KA + KB + KC = 142
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... and the solution follows!!!!
[Knights = 142; Liars = 58]
NOTE: We haven't used Eq#1, Eq#2, Eq#6; but they are included for clarity.
A got ATLEAST 112-58=54 (knight) votes
B got ATLEAST 82-58=24 (knight) votes
C got ATLEAST 64-58=6 (knight) votes
Detailed Solution
