Rolling Boxes

Notes on Investigation

 

It helps to invent a notation to describe each of the possible moves. Here is a possible notation.

The three sides are denoted by L (long)   M (medium) and S (short)

The direction of roll is given by r (right), l (left), u (up) and d (down)

 

So Lr indicates a roll about the long side in the right direction

 

 

 

 

 

Consider the translations that create a translation up and to the right ie a translation (a,b) in which a and b are positive or zero

 

(1,1)  Lr – Sd – Mr – Lu – Sl –Mu

(2,2)  Lr – Su – Ml – Lu – Sr –Md

(3,3)  Ll – Su – Mr – Ld  - Sr –Mu

(6,6)  Lr – Su – Mr – Lu – Sr –Mu

 

(1,5)  Ll – Su – Mr – Mr – Su –Ll

(7,5)  Lr – Su – Mr – Mr – Su –Lr

 

(0,10) Lr – Su – Su – Su – Su – Ll

 

(3,9)  Lr – Su – Su – Lr – Mu – Mu

(3,1)  Lr – Su – Su – Lr – Md – Md

 

(8,4)   Lr – Lr – Mu – Sr – Sr - Mu

(2,4)   Ll – Ll – Mu – Sr – Sr – Mu

 

(10,0) Mu – Sr – Sr – Sr – Sr – Mu

 

(5,7)   Mu – Sr – Lu – Lu – Sr - Mu

(5,7)   Mu – Sr – Ld – Ld – Sr - Mu

 

 

With each of these positive translations there are 3 others that move the box in other directions. For example

(1,1)     Lr – Sd – Mr – Lu – Sl –Mu        has these other three translations 

(-1,1)   Ll – Sd – Ml – Lu – Sr  - Mu

(-1,-1)  Ll – Su – Ml – Ld – Sr - Md

 (1,-1)  Lr _ Su – Mr – Ld – Sl - Md

 

 

It is interesting to see what happens when you reverse a set of rolling instructions. It creates an identical transformation

 

 

 

Investigation 2

 

Notice that all the vectors that are possible have a + b even.

Combining the translations already discovered it is possible to create a new translation (x,y) as long as x + y is even. This means that if we were rolling a the box on a chess board then if the left hand corner covers a black square it is possible to roll the box so that it will cover any other black square. If you simplify the problem with a card which covers just one square and is white on one side and black on the other then this is clearly the case.

 

It is also interesting to find how other transformations can be brought about. For example rotations of (0 or 270 can be created in 3 rolls:

90 degrees about top right corner   Mu – Sr – Ld

270 degrees about top right corner Lr – Su – Ml

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rotation of 180 degrees requires 6 rolls   Lr – Su – Mr - Ld – Sl - Mu 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rotation of 180 degrees requires 6 rolls   Lr – Su – Mr - Ld – Sl - Mu