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# Four-legged stool

## Question

Since three points determine a plane, or for some such reason, we all know that a milking stool with three legs will always sit firmly without rocking on an uneven surface (barn floors included, especially.) Moreover, the stool can be placed anywhere on that surface. In other words, the center of the stool can be positioned (without rocking) above any point on the floor.

Is the same true for a four-legged stool?

You may assume the floor satisfies reasonable conditions of smoothness (no cracks or holes, etc.)

Edited by bonanova
Reworded for clarity.

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A key point about this problem is that, when the center of the stool lies above your desired point, you can rotate the stool however many degrees you want to still have it fit the conditions in the OP.

Color three of the legs blue and color the other leg red. Place the stool at any desired initial rotational angle, and let the three blue legs sit on the surface as if this were a three legged stool, while the red leg's position is wherever it gets forced to be by virtue of the positions of the three blue legs. The red leg might lie either above or below the surface that it's supposed to rest on, rendering the stool unstable.

Now start rotating the stool. By the time you rotate it 90 degrees, each leg will have exchanged positions with one of the neighboring legs. Suppose the red leg was initially above the surface: then by the time you rotate the stool 90 degrees, a blue leg will now lie in the position where the red leg used to be and will have to be moved down in order to make the three blue legs lie on the surface, while the other two blue legs still lie at the same spot as a blue leg in the initial orientation. It should be obvious that the red leg's position in the new orientation after rotating 90 degrees must be lower than the initial position of the blue leg whose spot was taken by the red leg -- two points of the initial square formed by the legs remain unchanged while one point is moved downward, so the other point must also be moved downward in order to form a square.

That means that, in the new orientation after rotating 90 degrees, the red leg went from initially lying above the surface to now lying below the surface while the blue legs all touch the surface. If the surface is smooth (not meaning flat, but meaning that it doesn't have sharp cliffs and ledges and such) then there must be some point along that rotation while keeping the blue legs on the surface where the red leg transitions from lying above the surface to lying below the surface. At that transition point, the red leg lies on the surface.

Edit: This assumes that the legs of the stool form a square. Determining whether it applies if the legs are in any other orientation is left as an exercise for the reader.

Edited by plasmid
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No math required.

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a 3 legged stool with vertically straight legs of heights 10 meter,1 meter and 1/2 meter will be very unstable. do u mean to say all three legs are equal? if so, then i do not understand the question

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I totally agree that stools with legs like that would be problematical.

Happily very few stools are made that way.

The set-up part of the question is intended to call to mind the generally well known

property of three-legged stools (long ago milking stools were made with three

legs for this very reason) that they do not rock, even if placed on uneven (e.g. barn) floors.

The question is the sentence with the question mark at the end of it. To elaborate:

Can a four-legged stool (also) be positioned (arbitrarily) over a mildly uneven surface in such a way that it is steady - i.e. will not rock? Equivalently, can a 4-legged card table (with nominally equal legs) be made to rest steady, without rocking, when positioned arbitrarily over a mildly uneven floor?

The question is not meant to be tricky or out of the box, only to inspire some analytical thinking.

Feel free to reply if something is still not clear.

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Are there 4 points on a surface of a cylinder or disk that defines the vertices of a square?

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Are there 4 points on a surface of a cylinder or disk that defines the vertices of a square?

That is an equivalent question. Nice.

Along with the requirement that the center of the square can be arbitrarily positioned.

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Is there a proof for either question?

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From experience, the 4 legged stool will be unstable. If you have a table one of whose legs is even slightly shorter (read surface is uneven), it will rock. A 3 legged stool on the other hand will be stable.

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Choose three of the four legs. They will rest firmly by themselves as known. However, if the fourth leg does not also rest on the surface, then the stool can rock. The layout of the surface will be the single deciding factor in whether the fourth leg rests on the surface. So my answer would be no, it doesn't work for a 4-legged stool.

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Plasmid's observation is a necessary condition. If the stool / table has legs of differing lengths it will rock on a planar surface / floor, and no amount of rotation will make it steady. But if the ends form a square, then, for an uneven floor, a rotation of less than 90 degrees will bring about a stable condition, without translation.

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