Another solution is that right examples can be folded down flat onto one isosceles triangle while left examples cannot.

All examples in this Problem feature four isosceles triangles connected by corners and/or edges.

This was conceived as a false solution for BP898.

Adjacent-numbered pages:
BP892 BP893 BP894 BP895 BP896  *  BP898 BP899 BP900 BP901 BP902

precise, allsorted, notso, traditional, preciseworld

[smaller | same | bigger]

Molly C Klenzak, Aaron David Fairbanks

Which two sides are the long sides and which side is the short side, or equivalently which angles are the wider angles and which angle is the narrower angle, is the only relevant information to consider for each triangle. Triangles are all assumed isosceles and congruent to one another.

All examples in this Problem feature four of these triangles connected by corners and/or edges.

BP897 was conceived as a false solution for this.

Adjacent-numbered pages:
BP893 BP894 BP895 BP896 BP897  *  BP899 BP900 BP901 BP902 BP903

hard, precise, allsorted, notso, math, preciseworld

[smaller | same | bigger]

Molly C Klenzak

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:
BP964 BP965 BP966 BP967 BP968  *  BP970 BP971 BP972 BP973 BP974

stub

Aaron David Fairbanks

Adjacent-numbered pages:
BP965 BP966 BP967 BP968 BP969  *  BP971 BP972 BP973 BP974 BP975

stub

Aaron David Fairbanks

See BP1199 for the version with rectangles.

Adjacent-numbered pages:
BP1196 BP1197 BP1198 BP1199 BP1200  *  BP1202 BP1203 BP1204 BP1205 BP1206

precise, traditional, left-listable, right-listable

Aaron David Fairbanks

The triangles vary in size in order to make the solution clearly about quantity, not "total mass".

Adjacent-numbered pages:
BP1203 BP1204 BP1205 BP1206 BP1207  *  BP1209 BP1210 BP1211 BP1212 BP1213

stub, precise, spectrum, dual, handed, leftright, traditional

Aaron David Fairbanks

See BP6 for "triangle vs. quadrilateral".

Adjacent-numbered pages:
BP1206 BP1207 BP1208 BP1209 BP1210  *  BP1212 BP1213 BP1214 BP1215 BP1216

stub, notso, left-narrow, traditional

Aaron David Fairbanks

BP839 is about applying opposite transformations to a single object.

Adjacent-numbered pages:
BP1255 BP1256 BP1257 BP1258 BP1259  *  BP1261 BP1262 BP1263 BP1264 BP1265

easy, nice, abstract, arbitrary, anticomputer, left-null, structure, orderedtriplet, traditional, rules

Aaron David Fairbanks