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16 November
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We demonstrated a big transparent cube in class.
You can imitate a cube with dice.
Put one on the table, such that number 1 is on top, and number 2
faces you.
When you check, you will see number 6 at the bottom, and number 5
at the back.
The numbers 3 and 4 are on the side.
In class we marked the right hand side with an x and the color red, the
back (number 5) with a y and the color white, and the top (number
1) with a z and the color blue.
In class we answered the following question: In how many ways can you keep the
cube in a rotated position, while it is in the same space on the table as it was
in the beginning?
Answer: 24, because we can hold the field with number 1 in 6 directions,
after which we can still put the field with number 2 in 4 positions
around wherever we want to hold the field with number 1.
Total: 6 times 4 equals 24.
Your tasks:
One Suppose we handicap ourselves, by only permitting us to roll the cube
backward a quarter turn.
So a single quarter turn would move field 1 to where field 5 was,
and move field 2 to where field 1 was.
You are allowed to repeat this movement arbitrarily often.
Question: In which positions can you put the cube. and why?
How many positions do you see in total?
Two
Suppose we handicap ourselves by permitting two kinds of moves.
The first kind of move is rolling the cube half a turn, upside down.
A single such turn would move field 1 to where field 6 was,
and move field 2 to where field 5 was.
The second kind is rotating the cube from front to back.
A single such turn would leave field 1 in place, but would move field
2 to where field 5 was.
You are allowed to repeat these two moves as often as you want, repeatedly or
alternatingly.
Question: In which positions can you put the cube. and why?
How many positions do you see in total?
Three
Suppose we yet again handicap ourselves by permitting two kinds of moves.
The first kind of move is rolling the cube a quarter turn, as in task
one.
So a single quarter turn would move field 1 to where field 5 was,
and move field 2 to where field 1 was.
The second kind is rotating the cube one quarter clockwise.
A single such turn would leave field 1 in place, but would move field
2 to the left, and field 5 to the right.
You are allowed to repeat these two moves as often as you want, repeatedly or
alternatingly.
Question: In which positions can you put the cube. and why?
How many positions do you see in total?
Four
The really hard case.
Suppose we yet again handicap ourselves by permitting two kinds of moves.
The first kind rotates the cube one-third around the top front right corner.
A single such turn would move field 1 to the right, and field 2
to the top.
The second kind rotates the cube one-third around the top front left corner.
A single such turn would move field 1 to the left, and field 2
to the top.
You are allowed to repeat these two moves as often as you want, repeatedly or
alternatingly.
Question: In which positions can you put the cube. and why?
How many positions do you see in total?
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