Dr. Ecco Solution
Solution to "Maximum Lottery," DDJ, September 2005.
Following the idea of the sultan's daughters problem, we will express our protocols by three numbers in increasing order: x, y, and z. The idea is to reject the first x, then choose the next one better than those first x. Call the position of that one p1 (so p1>x).
If p1>y then choose the next one better than any seen before (at a position we'll call p2). Otherwise, reject until position y and then pick the next one better than any seen before (at a position we'll still call p2).
If p2>z, then choose the next one better than any seen before (at a position called p3). Otherwise, reject until position z and then pick the next one better than any seen before (at a position we'll call p3).
For 100 balls in total and three keeps, you win roughly 68 percent of the time if you set x, y, and z to be 14, 32, and 64, respectively. Here is one example win for the following sequence:
3 78 80 90 25 95 51 27 57 40 65 48 55 72 26 73 54 31 15 2 89 61 97 98 8 50 38 18
88 52 4 42 68 16 62 9 94 99 20 28 56 58 76 93 10 96 63 35 81 91 66 11 30 5 0 24
82 29 41 12 47 71 44 92 43 32 85 84 7 59 60 86 69 21 83 79 64 67 74 37 1 46 22
19 33 39 87 45 36 13 23 75 34 70 53 49 77 17 6 14
The p1 position value in this case would be 23 where the value 97 is found, because 97 is the first value larger than 90, which is the largest of the first 14 numbers. The p2 value would be 38 where 99 is found and the third keep then is irrelevant.
Reader Improvements Concerning "Treasure Arrow"
Alan Dragoo pointed out that the delta is 15 centimeters so the arrow and plank each weigh 75 kilograms. Carl Smotricz noted that because the arrow is not quite horizontal, the horizontal distance between the two ends of the arrow is less than 10 meters. In fact, it is (102-(0.6)2)= 9.98. So, the arrowhead has dipped 0.6 meters for each 9.98 meters of horizontal distance. Continuing this line (Carl used trigonometry, but let's do this in a more elementary way), there would be a further dip of some amount x for the next 10 meters of horizontal distance. So, x=0.6×(10/9.98). This gives an extra 0.601 meters. So the arrow, in fact, points to 2.201 meters below the ceiling.
DDJ
