Truthkins, Fibkins and Switchkins

[LittleMiss543]

Hey,

Sorry if this has been posted before i'm new to the site and don't really know my way round.

So i'm in school today and they give us this riddle to solve for the maths week challenge and all the form is totally confused by it. Could anyone give me the answer please??

Here's the riddle:
In the land of Zoz there live three types of person:
Truthkins, who live in hexagonal houses and always tell the truth;
Fibkins, who live in pentagonal houses and always tell lies;
Switchkins, who live in round houses and who turn into whatever they say.

One morning 90 of them gather in the city in three groups of 30. One group is all of one type; another group is made up evenly of two types; the third group evenly comprises three types.

Everyone in one of the groups says “We are all Truthkins”; everyone in another group says “We are all Fibkins”; and everyone in the remaining group says “We are all Switchkins”.

How many sleep in pentagon houses that night?

Thanks!

[flowerpetal]

We will not give you the answer as it is for school. We will however point you in the right direction to finding the answer yourself. Well someone will, I'm off to work.

[Steve]

Hi LittleMiss, welcome to the site.

Funny enough, I have posted many puzzles here over the years but recognised this one straight away. I know the answer because it was posted by me back in 2003 and solved by top puzzle solving member arcadedweller within 30 minutes. It is a great logic puzzle and can be solved by breaking it down into steps.

The puzzle states that Fibkins live in pentagon houses so what you are trying to work out is how many Fibkins are amongst the group of 90.

The 90 people are split into 3 groups of 30:
Group 1 is all of one type;
Group 2 is made up evenly of two types;
Group 3 comprises three types.

Each group makes one of the following statements:
We are all Truthkins
We are all Fibkins
We are all Switchkins

So far so good. Where to start?

Start with group 3, that is made up evenly of the three types. Why? because you know in this group will be 10 Truthkins, 10 Fibkins and 10 Switchkins. As there are Truthkins in this group the statement they make must be true, for Truthkins cannot lie. So which statement would this group of made and how many Fibkins would be in this group?

Now cross off group 3 from your list and the statement they made. Looking at the remaining two statements you can the 2 types of people in group 2, remembering that Truthkins cannot lie. Then, remembering that Fibkins cannot tell the truth will lead you to the statement group 2 made. Workout how many Fibkins are in this second group after the statement is made.

That will then leave group 1 and one statement. Again workout how many Fibkins would be in this group after making their statement.

Finally, add the number of Fibkins from all 3 groups together and you have your answer.

Let me know how you get on.

[Steve]

I guess the maths challenge must be over now and LittleMiss never came back here to follow this up. So for anyone else interested, here is the answer:

Group 3 had 3 types so had 10 Truthkins, 10 Fibkins, and 10 Switchkins. Since Truthkins always tell the truth, this group must have said “We are all Truthkins.” The Switchkins would have turned into Truthkins at that point.

= 10 Fibkins in group 3

Group 2 could not have any Truthkins in them, since the only statements left are “We are all Fibkins” and “We are all Switchkins" and Truthkins cannot lie. So group 2 must have 15 Fibkins and 15 Switchkins. Fibkins can't tell the truth and say “We are all Fibkins,” so group 2 must say “We are all Switchkins."

= 15 Fibkins in group 2.

That then leaves group 1 and the statement “We are all Fibkins.” Neither Fibkins or Truthkins could say that, so they must have been Switchkins. When the Switchkins said “We are all Fibkins,” they became Fibkins.

= 30 Fibkins in group 1.

That gives a total of 55 Fibkins, so 55 people sleep in pentagon houses that night.