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Star-shaped polyominos (Posted on 2016-03-28) Difficulty: 3 of 5
A star-shaped polygon is a polygon that contains at least one point from which the entire polygon boundary is visible. The set of all such points is called the kernel.

A) Find the smallest polyomino that is not star-shaped.
B) Find the smallest polyomino whose kernel is a single point.
C) Find the smallest polyomino whose kernel is a line segment.
D) Find the smallest polyomino whose kernel is precisely half the area of the polyomino.

E) Prove or disprove: For every rational number, Q, where 0≤Q≤1 there is a polyomino whose kernel is Q times the area of the polyomino.

Note: smallest refers to the number of squares comprising the polyomino.

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Solution Part E Comment 7 of 7 |
Any rectangular polyomino has a Q value of 1.
Any polyomino from cases B and C has a Q value of 0.

A generalized L polyomino is depicted with dimensions as:
|           |
+           + y
|           |    z
+  +  +  +  +--+--+--+
|                 |
+           +      +
|                  |
+           +      + x
|                  |
+           +      +
|                 |

The L has an area of x^2+xy+xz.  Its kernel has an area of x^2.  Then its Q value is x/(x+y+z).

For any fraction whose denominator is at least two greater than the numerator it is trivial to choose x, y, and z such that the generalized L has a Q value equal to the fraction. x equals the numerator and y+z are any pair of positive integers summing to the difference between the numerator and denominator.  That just leaves fractions of the form x/(x+1),  To make these, just double the numerator and denominator first.

  Posted by Brian Smith on 2016-03-29 10:43:39
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