Search: user:Leo Crabbe
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| BP934 |
| If "distance" is taken to be the sum of horizontal and vertical distances between points, the 3 points are equidistant from each other vs. not so. |
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COMMENTS
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In other words, we take the distance between points (a,b) and (c,d) to be equal to |c-a| + |d-b|, or, in other words, the distance of the shortest path between points that travels along grid lines. In mathematics, this way of measuring distance is called the 'taxicab' or 'Manhattan' metric. The points on the left hand side form equilateral triangles in this metric.
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An alternate (albeit more convoluted) solution that someone may arrive at for this Problem is as follows: The triangles formed by the points on the left have some two points diagonal to each other (in the sense of bishops in chess), and considering the corresponding edge as their base, they also have an equal height. However, this was proven to be equivalent to the Manhattan distance answer by Sridhar Ramesh. Here is the proof:
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An equilateral triangle amounts to points A, B, and C such that B and C lie on a circle of some radius centered at A, and the chord from B to C is as long as this radius.
A Manhattan circle of radius R is a turned square, ♢, where the Manhattan distance between any two points on opposite sides is 2R, and the Manhattan distance between any two points on adjacent sides is the larger distance from one of those points to the corner connecting those sides. Thus, to get two of these points to have Manhattan distance R, one of them must be a midpoint of one side of the ♢ (thus, bishop-diagonal from its center) and the other can then be any point on an adjacent side of the ♢ making an acute triangle with the aforementioned midpoint and center. |
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CROSSREFS
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Adjacent-numbered pages:
BP929 BP930 BP931 BP932 BP933 * BP935 BP936 BP937 BP938 BP939
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KEYWORD
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hard, allsorted, solved, left-finite, right-finite, perfect, pixelperfect, unorderedtriplet, finishedexamples
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CONCEPT
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triangle (info | search)
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WORLD
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3_dots_on_square_grid [smaller | same | bigger]
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AUTHOR
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Leo Crabbe
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| BP933 |
| Ball will reach edge of bounding box under gravity vs. not so. |
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COMMENTS
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Strictly this Problem's solution is not actually about gravity, it is about a constant downwards force (the ball's time-independent path does not depend on the magnitude of the force, only direction). The phrasing for the solution is a shorthand that takes advantage of human physical intuition. |
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CROSSREFS
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Adjacent-numbered pages:
BP928 BP929 BP930 BP931 BP932 * BP934 BP935 BP936 BP937 BP938
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KEYWORD
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physics
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CONCEPT
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bounding_box (info | search),
imagined_motion (info | search),
gravity (info | search)
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WORLD
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dot_with_lines_or_curves [smaller | same | bigger]
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AUTHOR
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Leo Crabbe
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| BP923 |
| Bongard Problem with solution relating to concept: permutation vs. Bongard Problem unrelated to this concept. |
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| BP922 |
| One row is rearranged to make the other by swapping an odd number of object pairs vs. one row is rearranged to make the other by swapping an even number of object pairs. |
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| BP921 |
| Bongard Problem with solution relating to concept: gravity vs. Bongard Problem unrelated to this concept. |
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| BP918 |
| Bongard Problem with solution relating to concept: function vs. Bongard Problem unrelated to this concept. |
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| BP917 |
| Reversible transformations vs. non-reversible transformations. |
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| BP916 |
| Bongard Problem with solution relating to concept: convey enough information vs. Bongard Problem unrelated to this concept. |
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