For the Belgian Pentomino Competition's 50th entry, Kate designed a challenge of filling the number 50 with the 21 polyominoes sizes 1 through 5. The key requirement was that none of the smaller polyominoes, 1 through 4, should touch each other, not even at the corners. Here's the grid to fill:
Peter Jeuken from The Netherlands sent in two entries, both winners. One he found by hand. The other he derived by computer. Both solutions fulfill the condition of the challenge of separating all the smaller polyominoes, and in addition all the pentominoes within each number form a connected group, and each number contains 6 pentominoes.
The two puzzle solvers introduce themselves:
P. My name is Peet. I love solving pentomino puzzles. The only tools I use are paper, pencil and eraser.
J. My name is Jeek. I love writing computer programs in general, and puzzle solving programs in particular. One of the programs I have written is a tool for finding perfect-fit solutions of objects (such as polyominoes) in a predefined field.
Peet and Jeek start a dialogue:
P. Hi, Jeek. Recently I saw a polyomino puzzle presented by Kate Jones as part of an anniversary contest.
Fit all the polyominoes from 1 through 5 into the grid within the number 50, so that no two of the tetrominoes, trominoes, the domino and monomino touch each other, not even at the corners. I propose that we both solve the puzzle. I do it my way, you do it your way.
J. That is a good idea. Basically it is not too complicated for my computer program. However, the extra restriction that polyominoes 1 through 4 are not allowed to touch each other is not supported. An extension to handle this extra condition is required.
P. I propose we meet again next week. Hopefully you can show a computer solution then. My task is to find a solution by hand.
J. OK. I am going to implement the extension. See you next week.
Peet and Jeek meet again one week later:
P. This is my solution. Six pentominoes and three tetrominoes in the 5. All other polyominoes in the 0.
J. This is my solution. As a matter of fact, this is the first solution from a long list.
P. Lets send in these two solutions for the anniversary contest.
J. Do you think that one might guess which solution was found by a human and which solution was found by a computer?
P. I think so. Your fitting program must operate according to a fixed plan. For instance, from left to right and from top to bottom. And probably the pentominoes are fitted in alphabetical order. Note that the solution on the right side starts with F and I, which are the first two pentominoes in alphabetical order. So, this is a good candidate for the computer solution.
Congratulations, "Peet" and "Jeek"! And with your prize, you have a new challenge of separating all the smaller polyominoes while filling in all around the number 50. It is not completely solved in the set Peter is holding above. Happy puzzling!