Adjacent-numbered pages:
BP849 BP850 BP851 BP852 BP853  *  BP855 BP856 BP857 BP858 BP859

left-finite, right-finite, left-full, right-full, finished, experimental, funny

nothing [same | bigger]
zoom in left (nothing) | zoom in right (nothing)

Aaron David Fairbanks

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.

⠀

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:

⠀

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.

Adjacent-numbered pages:
BP929 BP930 BP931 BP932 BP933  *  BP935 BP936 BP937 BP938 BP939

hard, allsorted, solved, left-finite, right-finite, perfect, pixelperfect, unorderedtriplet, finishedexamples

3_dots_on_square_grid [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP957 BP958 BP959 BP960 BP961  *  BP963 BP964 BP965 BP966 BP967

precise, allsorted, minimal, dual, blackwhite, gap, left-finite, right-finite, left-full, right-full, left-null, finished, preciseworld, unstableworld

[smaller | same | bigger]
zoom in left (blank_image) | zoom in right (black_image)

Leo Crabbe

Two kinds of "nothing".

See also BP1219, "blank image vs. image of blank square".

Adjacent-numbered pages:
BP1051 BP1052 BP1053 BP1054 BP1055  *  BP1057 BP1058 BP1059 BP1060 BP1061

left-finite, right-finite, left-full, right-full, left-null, finished, invalid, experimental, funny, finishedexamples

blank_image [smaller | same | bigger]
zoom in left (blank_image) | zoom in right (nothing)

Aaron David Fairbanks

Adjacent-numbered pages:
BP1092 BP1093 BP1094 BP1095 BP1096  *  BP1098 BP1099 BP1100 BP1101 BP1102

nice, precise, notso, left-finite, right-finite, preciseworld

Jago Collins

See also BP1056, "blank image vs. nothing".

BP1 is also a (less specific) solution to this.

BP1209 (flipped) is also a (less specific) solution to this.

Adjacent-numbered pages:
BP1214 BP1215 BP1216 BP1217 BP1218  *  BP1220 BP1221 BP1222 BP1223 BP1224

minimal, gap, left-narrow, right-narrow, left-finite, right-finite, left-full, right-full, left-null, funny, unstableworld

Aaron David Fairbanks

All examples are a three by three grid of black or white squares.

Adjacent-numbered pages:
BP1218 BP1219 BP1220 BP1221 BP1222  *  BP1224 BP1225 BP1226 BP1227 BP1228

stub, left-finite, right-finite, traditional, preciseworld

Aaron David Fairbanks