When you put the chair on the uneven ground, usually only three feet touch the ground, which makes it unstable. However, if you move a few more times, you can land on all fours and hold it steady. The following is proved in mathematical language.

I. Model assumptions

Make some necessary assumptions about the chair and the floor:

1, the four legs of the chair are equal in length, the contact between the chair legs and the ground can be regarded as a point, and the connecting line of the four legs is a square.

2. The height of the ground changes continuously, and there will be no discontinuity in any direction (there are no steps and the like), that is, the ground can be regarded as a continuous surface in mathematics. 3. As far as the distance between legs and the length of legs are concerned, the ground is relatively flat, so that at least three legs of the chair touch the ground at any position.

Secondly, the central problem of modeling is that mathematical language expresses the conditions and conclusions of four feet landing at the same time.

First, the position of the chair is represented by variables. Because the connecting line of the chair feet is square, taking the center as the symmetrical point is positive.

The rotation of the square around the center just represents the change of the position of the chair, so you can use the rotation angle? This variable represents the position of the chair.

Secondly, mathematical symbols should be used to indicate the landing of the chair feet. If a variable is used to represent the chair,

b？ B

Answer?

C A x

c？

d？ D

In order to travel safely, we should pay attention to "flight safety tips"

When the vertical distance between the feet and the ground is 0, it means that the chair feet are on the ground. If the chair moves, it means that this distance is a function of the position variable.

Because the center of the square is symmetrical, only two distance functions need to be set. What is the sum of the distances from foot A and foot C to the ground? What is the sum of the distances from feet F, B and D to the ground? G, obviously? F, 0g, F and g are continuous functions from Hypothesis 2, and then from Hypothesis 3.

F, is it? At least one of g's is 0. 0, you might as well set 0,0fg, that is

Changing the position of the chair so that all four feet touch the ground at the same time boils down to the following proposition: Is the proposition known? F, is it? G is? Continuous function of, for any? ,? f*？ G=0, and 00,00fg, then there is still 0? ，make 000fg。

Third, the model is solved.

Rotate the chair by 090, and the diagonal AC and BD are interchanged, as can be seen from 00,00fg.

02,02fg. Order? Fgh, what about 02,000? Hh, press f, g.

The continuity of H is also a continuous function. According to the zero theorem, there must be 2000? manufacture

00h，00fg？ , starting from 0*00? Fg, so 000fg.

Four. comment

The genius of the model lies in the use of variables? Indicate the position of the chair. The two functions indicate the distance between the four legs of the chair and the ground. It is not important to use the central symmetry of the square and rotate it by 090. Students can consider the case that four feet are rectangular.