IBM Ponder This #123 · July 2008

This month's puzzle concerns polyominoes. A polyomino is a plane figure constructed by joining unit squares along their edges. Two polyominoes are considered distinct if one cannot be mapped to the other by translations, rotations and reflections. Polyominoes consisting of 5 unit squares are known as pentominoes. It is well known that there are exactly 12 distinct pentominoes. We say a polyomino A covers a polyomino B if A can be constructed by adding unit squares to B. We ask:

1. How many distinct pentominoes can a polyomino consisting of 6 unit squares (hexomino) cover? Give an example.
2. What is the smallest size (least number of unit squares) required for a polyomino which covers all 12 pentominoes? Give an example.
   As usual we ask that you only submit answers consisting of your original work.

Clarification: Question 1 intends to ask for the maximum number of distinct pentominoes which can be covered by a single hexomino.

Note: Some of the examples submitted appear to be getting lost or garbled on the way. Unfortunately I can not give credit for these. Please ensure your answer will make sense when read in a fixed width plain text editor. Pdf, jpg, gif, bmp, xls or docbin attachments also seem to work.

Solution

Best opened after a real attempt

To be added.