Brain Teasers
Game of Darts
If playing darts, what is the lowest possible score you CAN'T get with three darts on a standard dart board?
Answer
163Scoring combinations to achieve 0 thru 162 points:
all three darts miss --- 0 points
1st misses (0 points), 2nd misses (0 points), 3rd hits single 1 thru 20 (1 thru 20 points) --- 1 thru 20 points
1st misses (0 points), 2nd hits single 20 (20 points), 3rd hits single 1 thru 20 (0 thru 20 points) --- 21 thru 40 points
1st misses (0 points), 2nd hits double 20 (40 points), 3rd hits single 1 thru 20 (1 thru 20 points) --- 41 thru 60 points
1st misses (0 points), 2nd hits triple 20 (60 points), 3rd hits single 1 thru 20 (1 thru 20 points) --- 61 thru 80 points
1st hits double 20 (40 points), 2nd hits double 20 (40 points), 3rd hits single 1 thru 20 (1 thru 20 points) --- 81 thru 100 points
1st hits double 20 (40 points), 2nd hits triple 20 (60 points), 3rd hits single 1 thru 20 (1 thru 20 points) --- 101 thru 120 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits single 1 thru 20 (1 thru 20 points) --- 121 thru 140 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits triple 7 (21 points) --- 60 + 60 + 21 = 141 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits double 11 (22 points) --- 60 + 60 + 22 = 142 points
1st hits triple 20 (60 points), 2nd hits triple 19 (57 points), 3rd hits double 13 (26 points) --- 60 + 57 + 26 = 143 points
1st hits triple 20 (60 points), 2nd hits triple 18 (54 points), 3rd hits double 15 (30 points) --- 60 + 54 + 30 = 144 points
1st hits triple 20 (60 points), 2nd hits triple 17 (51 points), 3rd hits double 17 (34 points) --- 60 + 51 + 34 = 145 points
1st hits triple 20 (60 points), 2nd hits triple 16 (48 points), 3rd hits double 19 (38 points) --- 60 + 48 + 38 = 146 points
1st hits triple 20 (60 points), 2nd hits triple 16 (48 points), 3rd hits triple 13 (39 points) --- 60 + 48 + 39 = 147 points
1st hits triple 20 (60 points), 2nd hits triple 16 (48 points), 3rd hits double 20 (40 points) --- 60 + 48 + 40 = 148 points
1st hits triple 20 (60 points), 2nd hits triple 19 (57 points), 3rd hits double 16 (32 points) --- 60 + 57 + 32 = 149 points
1st hits triple 20 (60 points), 2nd hits triple 18 (54 points), 3rd hits double 18 (36 points) --- 60 + 54 + 36 = 150 points
1st hits triple 20 (60 points), 2nd hits triple 17 (51 points), 3rd hits double 20 (40 points) --- 60 + 51 + 40 = 151 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits double 16 (32 points) --- 60 + 60 + 32 = 152 points
1st hits triple 20 (60 points), 2nd hits triple 19 (57 points), 3rd hits double 18 (36 points) --- 60 + 57 + 36 = 153 points
1st hits triple 20 (60 points), 2nd hits triple 18 (54 points), 3rd hits double 20 (40 points) --- 60 + 54 + 40 = 154 points
1st hits triple 20 (60 points), 2nd hits triple 19 (57 points), 3rd hits double 19 (38 points) --- 60 + 57 + 38 = 155 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits double 18 (36 points) --- 60 + 60 + 36 = 156 points
1st hits triple 20 (60 points), 2nd hits triple 19 (57 points), 3rd hits double 20 (40 points) --- 60 + 57 + 40 = 157 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits double 19 (38 points) --- 60 + 60 + 38 = 158 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits triple 13 (39 points) --- 60 + 60 + 39 = 159 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits double 20 (40 points) --- 60 + 60 + 40 = 160 points
1st hits triple 20 (60 points), 2nd hits triple 17 (51 points), 3rd hits center bull's eye (50 points) --- 60 + 51 + 50 = 161 points
1st hits triple 20 (60 points), 2nd hits triple 20 (60 points), 3rd hits triple 14 (42 points) --- 60 + 60 + 42 = 162 points
163 points can not be achieved for the following reasons:
If two darts each hit a triple 20 for 60 points each, that produces 120 points, which requires another 43 points to reach 163. 43, being prime, can not be achieved with any single, double, or triple score.
If one dart hits triple 20 and another hits the center bull's eye, that achieves 110 points, leaving the need for an additional 53 points. 53 is prime, so can not be reached by any hit.
If two darts both hit the center bull's eye for 100 points, that still leaves a need for 63 points. 63 is not prime, but can not be reached, since it would require a triple 21 (there is no 21 on a dart board).
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