Tatami
In ancient
The floor of a room has a rectangular shape with sides 5 and 6 yards. You have 15 mats of three colors (5 of each color). Each mat is a rectangle with sides 1 and 2 yards long. You want to create a cover such that,
(1) The pattern doesn't "split" into two rectangles.
(2) Mats with sides that touch must be of different colors.
To model the situation, cut from colored posterboard 15 rectangles (each 1 by 2 inches), five of each of three different colors. Then find a required pattern. These pieces can be put together to form a 5 by 6 rectangle.
How many different patterns are there?
When we consider all patterns that can be obtained from each other by symmetries to be equal, there are only two patterns which satisfy condition (1). But only one of the two can be colored according to condition (2). (It can be colored in three ways depending on which color is in the middle.)
Task.
Find both patterns that satisfy (1), and determine which one can formed according to rule (2).
Solutions (each piece is represented by two letters):
Condition:
| (1) & (2) | (1) |
| CCBBA | BBABB |
| ABCCA | CCACA |
| ABABB | ABBCA |
| CCACA | ACAAB |
| ABBCA | BCXXB piece XX must violate rule 2; it borders A, B, and C. |
| ACCBB | BAACC |
Optional question: Can you prove that there are only 2 patterns satisfying (1) and (2)?
Hint. Look at the floor as a grid of horizontal and vertical lines. The first condition assures that each line is straddled by at least one mat. And if a line splits the rectangle into two rectangles that have an even number of squares, it is straddled by at least two mats. Only later start considering colors.
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This puzzle satisfies both conditions (1) and (2) |
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This puzzle satisfies only condition (1) |
Based on a problem from the book Jinkoki
published in 2000 by the Wasan Institute 5-14-9-108-8
Sakurajousui, Satagaya-ku,