The solution to find the number of dwarfs with orange hats is based on the assumption that the dwarfs with orange hats alternate between telling the truth and lying every other time. However, without any additional information, it is not possible to be completely confident in this solution.
`The solution is based on the information given in the problem, but there could be other scenarios that would give the same results for the number of positive answers to the first and third questions.
This is why it's important to consider all the information available and make informed assumptions when solving a problem like this.`
Let's call: - The number of dwarfs with green hats "G". - The number of dwarfs with orange hats "O". - The number of dwarfs with red hats "R".
From the information given, we can create the following equations:
G = 34 (the number of positive answers to the first question)
O + O = 26 (the number of positive answers to the second question, since the dwarfs with orange hats alternate between telling the truth and lying)
R = 11 (the number of positive answers to the third question)
Adding up the number of dwarfs in each group, we have:
G + O + R = 43 (the total number of dwarfs)
Solving for O, we can substitute the values of G and R and solve for O:
O + O = 26
2O = 26
O = 13
So there were 13 dwarfs with orange hats at the party.