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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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Some Thoughts Parts A through D | Comment 6 of 7 |
Part A: The U and Z pentominos

Part B: The W pentomino

Part C: The skew tetromino and F and N pentominos

The monomial, domino, both triominos, the other four tetrominos and the other seven pentominos all have a kernel of at least one full square.

Part D: The octomino pictured has a 2x2 square kernel:
+--+--+
|     |
+  +  +--+
|        |
+  +  +  +
|        |
+--+--+--+

  Posted by Brian Smith on 2016-03-28 23:18:44
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