Where on the circular highway should Stand B be located in order to get the most business and income?

Source: B.B. (former student)

**Hint:** Draw a circle and place Stand A anywhere (WLOG assume it is at 12 o'clock position). Ask: what happens if Stand B is placed here...e.g. at the 6 o'clock position or the 3 c'clock position. That is, what will owners of Stands C and D probably do given possible placements of Stand A and B?

**Solution Commentary:** Draw pictures as you follow this solution commentary from person B.B. who submitted this problem: "The placement of Stand B is the one that really controls the arrangement of the stands. It wants as much of the business possible with the least risk; that is, when Stand C and Stand D open up, Stand B wants to control more than a quarter of the circle's business.

If after A opens any place on the circle and B opens at the 1/2 mark directly opposite A, then C and D would probably place themselves at the respective midpoints of the half-arcs, and each of the four stands would control 1/4 of the total distance. [Note: If C and D wanted to hurt B, they could place themselves very close to B and still ensure themselves of 1/4 the business but definitely hurt B.]

If B places itself at the 1/3 position from A, stand C would pick the remaining 2/3 position, hoping to get one-half of the business between Stand's A and B (i.e. equal to 1/3). Stand D would then have to locate at the 5/6, 1/2, or 1/6 mark (viewed clockwise).

If D located at the 1/6 or 1/2 mark, B would gain 1/4 of the business (i.e. 1/12 + 1/6). But, if D located at the 5/6 mark, B would get 1/3 of the business. Note: In all cases, D gets the same business. Thus, as D's three possible positions are random, the average business expected for B is (1/4+1/4+1/3)/3 = 5/18, which is greater than the eralier case of 1/4.

A more creative answer would be to place B slightly closer to A than the 1/3 mark, making arc AB slightly smaller than arcs AC or CB. Then, to maximize its business, D would avoid arc AB and locate at the midpoint of arc AC or CB. If D locates at the midpoint of AC then B gets 1/3 of the business, while if D locates at the midpoint of BC, then B gets 1/4 of the business. Thus, again the average is (1/4+1/3)/2 = 7/24 of the business, which is greater than the previous values of 5/18 or 1/4.

Do you agree with this reasoning?