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Einstein's Mathematical Problems in Frankfurt Newspapers
Fill in the numbers game

As shown in the figure below, the centers of several circles are the vertices of four small isosceles triangles and three large isosceles triangles respectively. Fill the circle with the number 1 ~ 9 to make the sum of the vertices of seven triangles equal.

Let's observe the figure first. The three zeros in the middle are used by three triangles respectively. Calculate the sum of three times; Among the six zeros around, each zero is used by two triangles and the sum is calculated twice. In this way, we know that the sum of the number of each vertex of seven triangles can be regarded as the sum of three zeros of the middle triangle after twice calculating the zeros in the figure.

The number of zeros to be filled in the figure is 1-9, which is calculated twice, and the sum is (1+2+3+4+5+6+7+8+9) × 2 = 90, that is, the sum of 90 plus 3 zeros in the middle triangle is equal to the sum of the number of each vertex of seven triangles. It is easy to think that 90 is six times the sum of the vertices of the middle triangle, so the sum of the vertices of the middle triangle is 15. After the experiment, we got two sets of answers: Figure 2 and Figure 3.