Four friends - four houses

I have four friends who shares the same peculiarity - they are fanatics by fairness. When the go to the pub, or a restaurant they always spend hours just discuss who will pay for what. They can't ever go to a date because they want to share everything and everyone to absolute fairness. (They found a quadruplets once but the four sisters weren't interested of the boys.)

One of them wanted to build a house, so the others said they want a house too. So they planned to build a house each. But no one wanted to live further away from the others. So the distance of the four houses must be of equal relative to each other.

But they found out a way how to build these houses so the distance form one to another were exactly the same, no matter from which house to which house you measured.

Where or how were the houses built?

In short:
Place four houses totally equidistant from each outer. How?

My hint. It's not possible in the Midwest Plains. Figure out why, and you're one step closer to a solution.

Originally posted by geepamoogle
My hint. It's not possible in the Midwest Plains. Figure out why, and you're one step closer to a solution.

Build them all at the vertices of a tetrahedron shape inside a freely rotating sphere. Each house will be equidistant from the others, and no one person has to take the stairs every time they want to get inside - everyone just rotates the sphere until their house appears at the entrance (for fairness' sake).

EDIT: D'oh...instead of having this ungainly sphere contraption, why not build the houses at the vertices of a tetrahedron inscribed in the sphere that is the Earth? (This assumes that the Earth is perfectly spherical, of course.)

Or you could just build the houses in a square so they are all touching each other in the centre like this....

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Originally posted by uzless
Or you could just build the houses in a square so they are all touching each other in the centre like this....

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Technically, in this case the houses will be a little farther from the house in the opposite corner than the ones in the adjacent corners.

I like PB's edit. You can't build the tetrahedron on a mountain because then one would be higher up, which isn't fair...

"....how were the houses built?"




With tools? 😉

Originally posted by PBE6
EDIT: D'oh...instead of having this ungainly sphere contraption, why not build the houses at the vertices of a tetrahedron inscribed in the sphere that is the Earth? (This assumes that the Earth is perfectly spherical, of course.)

Or just find a suitable hill.

I think geepamoogle have a correct answer. Thanks for not reveal the solution too quickly.

uzless had also a solution, the trivial one, but well within the definition about distances. Zero distance is also a distance.

PBE6 suggested a tetrahedron, and this is the non-trivial solution. If you in any way can place the four houses at the corners of a tetrahedron the problem is solved. The new problem is how. PBE6 suggested further that a tetrahedron can be inscribe in a sphere, and if the Earth itself is a sufficiently close to a sphere then just build a hose in every place where the tetrahedron is touching the surface to the sphere. Would the four friends be happy if they were to live in four corners of the world? In different continents? PBE6 has definitely a solution at hand.

mtthw on the other hand proposed a rather closer solution, when he asked us to find a suitable hill. Would you elaborate this a little, mtthw?

(Have I forgot any other with a good solution?)

Originally posted by FabianFnas
I think geepamoogle have a correct answer. Thanks for not reveal the solution too quickly.

uzless had also a solution, the trivial one, but well within the definition about distances. Zero distance is also a distance.

PBE6 suggested a tetrahedron, and this is the non-trivial solution. If you in any way can place the four houses at the corners of a t ...[text shortened]... ll. Would you elaborate this a little, mtthw?

(Have I forgot any other with a good solution?)

The hill idea is that one's at the peak and the other three lower down at equal elevations to one another, thus making a tetrahedron. However this fails because the one at the top is different than the others.

Originally posted by AThousandYoung
The hill idea is that one's at the peak and the other three lower down at equal elevations to one another, thus making a tetrahedron. However this fails because the one at the top is different than the others.

You mean, it fails because it is not fair that one of the friends has downhill to others and the others not? Hm...

But if the fairness in this problem is defined only that the distances is the same between the four, is it then a solution?