1. When the coin is as close to the center as possible, the endpoints of a diameter of the coin must be touching the jar rim. There is a right triangle then from the center of the jar to the center of the coin to one of the end points where the jar rim and coin touch with sides Rcoin, Rcoin, and Rjar. Thus the radius of a coin in this configuration is Rjar/sqrt(2). If Rjar is 10, Rcoin is 5*sqrt(2). 2. If N coins are placed around the jar with their centers on the rim and tangent to each other, the angle made by the two jar radii tangent to one coin must be 360/N. The angle made by one of these radii and a jar radius drawn to the center of the coin will be half this, 180/N. The line from the center of the coin to the tangency point (Rcoin) is just the sine of this angle times the jar radius. So Rcoin is any value that satisifies Rcoin = Rjar * sin (180/N) for any whole number N (>1). coins Radii (Rjar = 10) 2 10 3 8.66 4 7.07 5 5.88 6 5 7 4.34 8 3.83 9 3.42 10 3.09 3. This is this same as 2) except the coin centers are on an inner circle with radius Jinner. Rcoin = RJinner * sin (180/N). The right triangle from the center of the jar to the center of the coin to the point where the edges of jar and coin intersect has sides of lengths RJinner, Rcoin, Rjar. So RJinner is sqrt (Rjar^2 - Rcoin^2). Solving gives Rcoin = Rjar/sqrt(1+1/sin(180/N)). coins Radii (Rjar = 10) 2 7.07 3 6.55 4 5.77 5 5.07 6 4.47 7 3.98 8 3.57 9 3.24 10 2.95 4) Solving these two equations together and trying the first few values shows that the only radius that works for both part 2 and 3 is achieved when you have 2 coins in the part 3 arrangement, or 4 coins the part 2 arrangement. For Rjar = 10, this is 5*sqrt(2) or 7.07.
This asks: Can we find different values of N for the answers to parts 2 and 3 such that the radii R are equal? Substituing M for N in part 3, we need: 10 sin(pi/N) = 10 sin(pi/M) / sqrt(1 + sin^2(pi/M)) or sqrt(1 + sin^2(pi/M)) = sin(pi/M) / sin(pi/N). for M, N integers >= 3 (or >= 2 if you accept the degenerate case). Squaring both sides, we get 1 + sin^2(pi/M) = sin^2(pi/M) / sin^2(pi/N). or, dividing by sin^2(pi/M): 1 + 1/sin^2(pi/M) = 1/sin^2(pi/N) We need two different values of X for which the values of 1/sin^2(pi/X) differ by 1. By inspection, the only possible answers are X=2 and 4; for higher values of X, the values of 1/sin^2(pi/X) differ by more than 1 for consecutive values of X. The coins with R = 5 sqrt(2) fit 4 around the jar when they are placed eith their centers on the rim, and 2 across the jar when they are placed with the centers in as close as possible to the center without falling in.