Combination

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Demian

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Apr 25, 2022
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This problem is about a combination lock which consists of 5 wheels with 4 possible symbols each, which turn one step in any direction ( like the bike combination locks), but what is special about this one is the fact that each wheel can't be rotated individually. Every time a wheel is turned, 2 more also turn always one step at a time. The following operations are the only ones possible:

We will call a clockwise wheel turn: +1 and a couterclockwise wheel turn: -1

1.. +1 -1 +1 0 0

2.. +1 +1 0 0 -1

3.. 0 -1 +1 -1 0

4.. -1 0 0 +1 +1

5.. 0 0 -1 +1 +1

My question is: How can one, given an initial combination of symbols calculate what sequence of the above operations is needed to get to a final combination of symbols?

Example: If the 4 symbols of each wheel are named A,B,C,D,E and let's say the initial combination is C,E,D,A,B how can we get to E,C,A,D,B ? or from any initial to any final combination. What is the math or formulas which allow for this calculations?. Thanks beforehand guys.
 
Demian said:
This problem is about a combination lock which consists of 5 wheels with 4 possible symbols each, which turn one step in any direction ( like the bike combination locks), but what is special about this one is the fact that each wheel can't be rotated individually. Every time a wheel is turned, 2 more also turn always one step at a time. The following operations are the only ones possible:
We will call a clockwise wheel turn: +1 and a couterclockwise wheel turn: -1
1.. +1 -1 +1 0 0
2.. +1 +1 0 0 -1
3.. 0 -1 +1 -1 0
4.. -1 0 0 +1 +1
5.. 0 0 -1 +1 +1
My question is: How can one, given an initial combination of symbols calculate what sequence of the above operations is needed to get to a final combination of symbols?
Example: If the 4 symbols of each wheel are named A,B,C,D,E and let's say the initial combination is C,E,D,A,B how can we get to E,C,A,D,B ? or from any initial to any final combination. What is the math or formulas which allow for this calculations?.
Click to expand...
This is one of the most confusing post in a long time.
As a long cyclist I gone through many bike locks.
But none of the post seems correct. You say there are four symbols: ABCDE, but that is five not four.
Moreover how does a cylinder lock have wheels that turn clockwise/counter-clockwise?
 
But none of the post seems correct. You say there are four symbols: ABCDE, but that is five not four.
Click to expand...
Could not resist remembering about three types of people, i.e., the ones who can count and the ones who cannot :)
Moreover how does a cylinder lock have wheels that turn clockwise/counter-clockwise?
Click to expand...
If you are asking whether the wheels can turn both ways, I have a very basic cable bike in which they do. But I don't know which way is clockwise :)
 
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Your problem is solvable if you can find a combination of your allowed operations for each of the "elementary" turns of one wheel. For example, to turn the first wheel counterclockwise by 1 you would apply operations 2,3 and 5. If you can now find the combinations for the remaining 4 wheels you can then combine those elementary turns to any situations.

The problem is indeed solvable. You can represent you set of operations as a matrix. It turns out that a) your matrix is invertible, and b) the result of the inversion has integer coefficients (which is equivalent to the fact that the matrix's determinant is 1).

Hope you can take it from here, but feel free to ask more questions if you get stuck.

Good luck.
 
blamocur said:
Your problem is solvable if you can find a combination of your allowed operations for each of the "elementary" turns of one wheel. For example, to turn the first wheel counterclockwise by 1 you would apply operations 2,3 and 5. If you can now find the combinations for the remaining 4 wheels you can then combine those elementary turns to any situations.

The problem is indeed solvable. You can represent you set of operations as a matrix. It turns out that a) your matrix is invertible, and b) the result of the inversion has integer coefficients (which is equivalent to the fact that the matrix's determinant is 1).

Hope you can take it from here, but feel free to ask more questions if you get stuck.

Good luck.
Click to expand...
Thank you, I know my post was not very clear but you understood anyway. I will work on it with the tips you gave me
 
pka said:
This is one of the most confusing post in a long time.
As a long cyclist I gone through many bike locks.
But none of the post seems correct. You say there are four symbols: ABCDE, but that is five not four.
Moreover how does a cylinder lock have wheels that turn clockwise/counter-clockwise?
Click to expand...
pka said:
This is one of the most confusing post in a long time.
As a long cyclist I gone through many bike locks.
But none of the post seems correct. You say there are four symbols: ABCDE, but that is five not four.
Moreover how does a cylinder lock have wheels that turn clockwise/counter-clockwise?
Click to expand...
There are four symbols, ABCD, but there are five wheels. So a combination can be ABABD.
 
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