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October 2006

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Answer:

We give the answers to each part with a sketch of the proofs.

1. 5/14. It is easy to see that T is formed by removing a strip along each side of S and that each strip has expected area 1/4 of the area of S. However we have to add back the area near the corners of S which lies in two strips and is being counted twice. For the corner near 0 this area has expected value

Integral from 0 to 1 with respect to x (Integral from 0 to 1-x with respect to y ((1-x-y)**6)) = 1/56.

Dividing by the area of S this gives a ratio of 1/28. It can be shown the result for the other corners is the same. So the expected area of T is (1-3*1/4+3*1/28) = 5/14 of the area of S.

2. 3/5, 2/5. Let the sides of T have lengths a,b,c where a is the length of the vertical side, b is the length of the horizontal side and c is the length of the diagonal side. Let the sides have infinitesimal widths da,db,dc respectively. Then the hypervolume of a type A configuration with corner near 0 is (da*db)*(c*dc)*(.5*a*b)=.5*a*b*c*da*db*dc. Similarly the other two type A configurations have the same hypervolume. For example (da*dc/sin(r))*(b*db)*(.5*a*c*sin(r)) =.5*a*b*c*da*db*dc (where r is the angle between the vertical and diagonal sides of T) for the type A configurations with top corner. On the other hand type B configurations have hypervolume (a*da)*(b*db)*(c*db) = a*b*c*da*db*dc. So in total the type A configurations have hypervolume 3/2 the hypervolume of the type B configurations. The result follows.

3. 2/9, 1/4. For type A the corner point and the point on the opposite side divide T into 2 triangles. The third point lies in one of these triangles and divides it into 3 triangles. On average the third point lies in a triangle with 2/3 the area of T (since it is more likely to lie in big subtriangle) and divides it into equal parts. So the desired expected area ratio is (2/3)*(1/3)=2/9. For type B the expected area is what is left over when the corners of T are removed. Each corner has expected area 1/4 the area of T leaving expected area 1/4 of the area of T for the triangle determined by the three points.

4. 1/12. The calculation is (5/14)*((3/5)*(2/9)+(2/5)*(1/4)) = (5/14)*((2/15) + (1/10)) = (5/14)*(7/30) = 1/12.

This problem was asked by Watson (1865) and solved by Sylvester. I came up with the above derivation myself but it seems doubtful it is original.


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