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Mathematical Games of Martin Gardner Part 3

April is Mathematics and Statistics Awareness Month (MSAM) and we are continuing with our weekly nod to recreational mathematics inspired by “Mathematical Games,” a monthly column that appeared in Scientific American between 1956 and 1981. This column was authored by Martin Gardner, a tireless advocate for mathematics education and mathematical games. This week’s puzzle features the polyomino.

A polyomino, according to the CRC Encyclopedia of Mathematics, is a “generalization of the domino to a collection of n squares of equal size arranged with coincident sides.” This basically means that two or more squares are attached, where n tells us how many squares are present. Another way of writing this out is n-polyomino or n-omino.

This week we are focusing on the pentomino, or 5-polyomino. Some examples of the pentomino are illustrated below to show the variety of shapes that can be formed:

A selection of 3 pentominoes.

Here we see just a sampling of the many pentominoes that can be formed.

This week’s puzzle: Divide the 6 x 10 rectangle of pentominoes into two parts that can be fitted together to create a 7 x 9 rectangle with 3 holes as shown. The pentominoes can not be broken up:

12 pentominoes arranged in a 6 x 10 rectangle.

12 pentominoes arranged in a 6 x 10 rectangle.

7 x 9 rectangle formed from the same pentominoes in the rectangle above.

This 7 x 9 rectangle can be formed by the same 12 pentominoes from the rectangle above.

These shapes may remind you of a certain arcade game that was released in the 80s. Tetris does in fact use polyominoes and the object of the game is to place each piece so that there are as few holes as possible. Toy building blocks of some kind may prove helpful in figuring out this week’s puzzle.

Last week’s puzzle dealt with contact numbers, sometimes also referred to as “kissing” numbers. The example given was an arrangement of 4 spheres where each sphere came into contact with the other 3. The puzzle was to arrange 6 rods in such a way that each rod is in contact with the other 5. The solution to the puzzle is below:

Solution showing how 6 rods can be arranged so that each rod "kisses" the other 5.

Solution to last week’s puzzle showing how each rod “kisses” the other 5.

As with previous weeks, this week’s solution will appear in next week’s post. If you take part in any mathematical activities during the month of April, be sure to use #MathStatMonth for posts on social media. Good luck and share your experience with us in the comments!

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