Strong Surround Numbers for Polyaboloes

A polyabolo is a plane figure formed by joining equal isosceles right triangles along equal edges. The strong surround number of a polyabolo is the fewest number of copies of the polyabolo that can surround it strongly; that is, including its corners. The polyaboloes must conform to the quadrille grid.

Strong surround numbers for polyominoes were proposed by Jaime Poniachik in Issue 8 of Puzzle Fun. He asked for the smallest polyominoes with a given strong surround number. In Issue 10, Rodolfo Kurchan extended the problem to polyiamonds, polyhexes, and polyaboloes. He also investigated the smallest polyforms that cannot surround themselves, and the smallest holeless such polyforms. However, his results were not complete.

Here I show minimal strong surrounds for small polyaboloes, the smallest polyaboloes with given surround numbers, and the smallest polyaboloes with no strong surrounds.

Minimal Strong Surrounds

Monabolo

Diaboloes

Triaboloes

Tetraboloes

Minimal Polyaboloes with Given Strong Surround Numbers

3 Copies, 16 Cells

These solutions were found by Juris Čerņenoks.

4 Copies, 6 Cells

The second solution was found by Rodolfo Kurchan.

5 Copies, 3 Cells

6 Copies, 3 Cells

7 Copies, 3 Cells

The second solution was found by Rodolfo Kurchan.

8 Copies, 2 Cells

The first solution was found by Rodolfo Kurchan.

9 Copies, 7 Cells

10 Copies, 1 Cell

This solution was found by Rodolfo Kurchan.

11 Copies, 9 Cells

12 Copies, 9 Cells

13 Copies, 11 Cells

14 Copies, 10 Cells

Minimal Polyaboloes that Cannot Surround Themselves Strongly

The fourth solution was found by Rodolfo Kurchan.

Last revised 2022-08-14.


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Col. George Sicherman [ HOME | MAIL ]