Metroville is an intriguing logic game for one person published by Smart Games.
There are nine pieces (see below) which are placed on a board.
Each piece can be rotated 90 degrees, 180 degrees, 270 degrees, and 360 degrees.
How many different combinations can be made?
(Be careful -- when you rotate some pieces, they look the same,
and therefore, should
not be considered different).
Here is an explanation of the game.
There are 8 cities with 8 different challenges to complete. Your goal is to rework
each city's rail network to take the metro through every station without going off the rails.
First, take a challenge card (one of the 8 levels from Starter to Junior to Expert to Master) for a
particular city and place it in the frame (see below).
Next, rearrange the track (the nine pieces shown earlier) to match the
starting layout on the card. (see below).
Twist the nine pieces of track so that the train can go through each station in the order shown on the challenge card
(see below for the solution to Challenge #1 for the city of London):
Here is the solution to level 8 for the city of London. Notice that the nine pieces are in the exact same positions,
but two pieces have been rotated. Note that only the initial station was changed from level 1 to level 8 (see below):
Solution to the Problem:
The answer is: 8,192 different combinations.Look at the nine pieces:
The first piece (upper left corner) can only be placed one way (rotations of 90 degree degrees do not change it).
The second piece has 4 positions.
The third piece has only 2 positions.
Similarly, the next six pieces have 2, 4, 4, 4, 2, and 4 different positions when rotating.
Hence, the number of different ways the nine pieces can be rotated is:
1 x 4 x 2 x 2 x 4 x 4 x 4 x 2 x 4 = 8,192.
Correctly solved by:
| 1. Bailey Lupher |
Mountain View High School, Mountain View, Wyoming |
| 2. Samantha Brailsford |
Mountain View High School, Mountain View, Wyoming |