IBM Research | Ponder This | March 2006 solutions
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March 2006

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We give the answers with a sketch of the proofs. We assume the unit square lies in the x-y plane with corners (0,0),(0,1), (1,1) and (1,0).

1. .25. R is bounded by the minimum and maximum x and y coordinates of the 3 random points. It is well known and easy to prove that the minimum of the 3 random numbers chosen in the unit interval has average value .25. Similarly the maximum has average value .75. So on average the distance between opposite sides of R is .5. Since the x and y coordinates of the points are chosen independently this means the average area of R is .25.

2. 1/3, 2/3. Let the sides of R have lengths a and b. For case A we have 2 choices for the diagonally opposite corners and the third point can occur anywhere in R so we have 2 degrees of freedom and area element 2*a*b. For case B we have 4 choices for the corner element with the remaining 2 points lying along line segments of length a and b. So again we have 2 degrees of freedom and area element 4*a*b. So case B is twice as likely as case A. Other configurations have fewer degrees of freedom hence occur with probability 0. So case A occurs with probability 1/3 and case B occurs with probability 2/3.

Alternatively as pointed out by several solvers note R is determined by the values of the x and y coordinates while the case is determined by how the x and y coordinates are paired together with case A occurring when the middle x coordinate is
paired with the middle y coordinate which will occur 1/3 of the time.

3. Case A - 1/6. Here the third point will lie in a triangle (determined by 3 corners of R) of area .5*R. It will divide this triangle into 3 triangles having equal area on average (by symmetry). So the area of the triangle including two opposite corners of R will average R/6.

Case B - 3/8. Here the triangle is what is left after triangles containing the 3 other corners of R are removed. The triangles containing the adjacent corners of R have average area R/4. The triangle containing the opposite corner of R has average area R/8. Hence the remaining triangular area in the middle determined by the 3 points has average area (3/8)*R.

4. 11/144. The computation is (1/4)*((1/3)*(1/6)+(2/3)*(3/8)) =(1/4)((1/18)+(1/4))=(1/4)*(11/36)=11/144.

This problem is closely related to Sylvester's four-point problem (1865) which asks for the probability that the convex hull of 4 points chosen at random is a quadrilateral (as opposed to a triangle). The first solution is attributed to Woolhouse (1867). I came up with the above derivation myself but it seems doubtful it is original. And in fact as a reader points out it can be found on the internet attributed to Robert Dawson.

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