A322740 Least area of a grid rectangle from which n >= 3 grid points can be chosen such that the binomial(n,3) triangles formed by any 3 of the points have distinct areas > 0.
1, 4, 15, 30, 65, 120, 198, 342
Offset: 3
Examples
a(3) = 1 (trivial).
a(4) = 4: Choosing (0,0),(0,2),(1,0),(2,1) gives 4 triangles with areas (1/2)*{1 2 3 4}.
a(5) = 15: Choosing (0,0),(0,1),(1,5),(2,0),(3,2) from a 3 X 5 rectangle gives 10 triangles with areas (1/2)*{1 2 3 4 5 7 9 10 11 13}.
a(6) = 30: (0,5),(1,0),(2,6),(3,0),(5,3),(5,4) is a solution on the 5 X 6 rectangle.
a(7) = 65: (0,3),(1,5),(7,0),(11,1),(11,5),(12,4),(13,2) is a solution on the 13 X 5 rectangle.
a(8) = 120: There are two minimal solutions, (0,12),(1,0),(2,15),(3,0),(6,1),(6,4),(7,7),(8,14) on the 8 X 15 rectangle giving 56 triangles with areas (1/2)*{1 2 3 4 5 6 8 9 11 12 14 15 17 18 20 21 22 23 24 26 27 28 29 30 31 33 34 35 37 38 39 40 42 43 46 47 48 49 54 61 62 63 64 67 69 71 74 76 79 80 83 89 91 98 100 102}, and (0,7),(0,9),(1,1),(5,1),(6,10),(8,0),(9,12),(10,3) on the 10 X 12 rectangle with triangle areas {2 3 4 8 9 10 11 12 13 16 17 18 19 20 21 23 24 25 26 28 29 32 34 36 37 38 39 40 41 42 43 44 46 47 49 50 51 53 54 55 56 59 62 66 68 71 74 75 79 83 84 85 86 87 103 105}.
a(9) = 198: The unique solution, up to rotation and reflection, is (0,3),(1,9),(2,0),(3,10),(4,11),(12,2),(17,11),(18,1),(18,4) on an 18 X 11 rectangle. The list of the 84 corresponding triangle areas is given in A322741.
a(10) = 342: The unique solution, up to rotation and reflection, is (0,3),(1,9),(2,18),(5,0),(5,10),(12,17),(15,13),(17,4),(18,0),(19,5) on a 19 X 18 rectangle. The list of the 120 corresponding triangle areas is given in A322742.
Links
- IBM Research, Ponder This Challenge - October 2018.
- IBM Research, Triangle Game.
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