ags@pucc-k (Seaman) (12/02/83)
At least two people have solved the Pirate Treasure problem, but no one found the simple solution, which uses complex numbers (see hint). The key observation is that multiplication by plus or minus i causes rotation through one right angle in the complex plane. Let P and M represent the (P)alm tree and the (M)ound of rocks, respectively. Let GAMMA represent the unknown position of the gallows (think of the shape of a capital GAMMA and you'll understand the choice). Let A and B represent the marked points, and T the Treasure. Then we have T = (A + B) / 2, where A - P = i * (P - GAMMA) and B - M = -i * (M - GAMMA) where the change of sign reflects the fact that you turn LEFT at the Palm tree, but RIGHT at the Mound of rocks. To make things simpler, let's choose our coordinate system so that P and M lie on the real axis at (P)lus one and (M)inus one, respectively. Then A - 1 = i * (1 - GAMMA) A = 1 + i - i * GAMMA and B + 1 = -i * (-1 - GAMMA) B = -1 + i + i * GAMMA. Finally the treasure is at T, where T = (A + B) / 2 = (1 + i - i * GAMMA - 1 + i + i * GAMMA) / 2 = (2 * i) / 2 = i. What could be simpler? (It's too bad the treasure is purely imaginary.) Dave Seaman ..!pur-ee!pucc-k!ags