geometry-labs

2, 3.
4.
2 × 2
2 × 3
2 × 4
3 × 3
2 × 5
3 × 5
4 × 4
2 × 8
3 × 6
Discussion Answers
2 × 10
2 × 11
2 × 7
3 × 4
A. Answers will vary.
B. The 3 9 rectangle is impossible because its
area is 1 less than the combined area of the
allowed pieces. However, the piece with least
area is the domino, and omitting it allows a
maximum area of 26.
3 × 8
2 × 9
2 × 14
C. Two such staircases can be combined to
make an x (x 1) rectangle. So the area
of one must be x(x 1)/2.
Lab 4.4: Family Trees
2 × 13
2 × 12
3 × 9
5 × 5
Family trees and envelopes (see Lab 4.5) offer
ways to make the search for hexominoes more
systematic. In addition, they offer yet two more
methods for classifying polyominoes.
If your students have studied probability, you
could assign the following research project.
4 × 6
4 × 5
4 × 7
Imagine that a machine creates pentominoes
with this algorithm: Start with a monomino
and add squares randomly, with each possible
location having an equal chance of being
used.What is the probability of getting each
pentomino Which pentomino is the most
likely Which is the least likely
2 × 6
3 × 7
It is interesting to note that the P pentomino is
the most likely, which is perhaps related to the
fact that it is most often the one you need in
order to complete pentomino puzzles.
186 Notes and Answers Geometry Labs
© 1999 Henri Picciotto, www.MathEducationPage.org