The first diagram below shows a circle with 57 positions around its circumference, numbered 0 to 56. The numbers actually noted on the diagram have the property that, if you rotate their original positions clockwise one position at a time you can complete a 360 degree rotation through every position and at every single step rotation one of the originally marked positions will coincide / match with another one (and only one) of the other originally marked positions. The second diagram shows the first rotation, the third shows the second rotation etc.

Is there a name or description for this that I could type in TATE to find out more about it?

combinatorics

esselte said:

The first diagram below shows a circle with 57 positions around its circumference, numbered 0 to 56. The numbers actually noted on the diagram have the property that, if you rotate their original positions clockwise one position at a time you can complete a 360 degree rotation through every position and at every single step rotation one of the originally marked positions will coincide / match with another one (and only one) of the other originally marked positions. The second diagram shows the first rotation, the third shows the second rotation etc.

Is there a name or description for this that I could type in TATE to find out more about it?

It’s not something I’m familiar with. I’m really only familiar with pre-1985 mathematics.
eg. it’s not in a book on “unsolved problems in mathematics” that I used to like.

It reminds me of couple of ancient problems, one involving positions around a ring,
and another involving a similar puzzle on an open-ended translation along the positive integers similar to the weighing problem,
but neither close enough to this to make the search useful.

When did you first become aware of this problem and how? Perhaps we can track back from that?
Particularly if you can give me a year.

The circle with rotations is the cyclic group Z_p. In this case Z₅₇.
This is also written as C_57.

SCIENCE said:

combinatorics

A google scholar search for combinatorics + z57 gives 127 entries.
But I don’t see anything appropriate on the first two pages.

Hi esselte,

This is called a “Cyclic Difference Set”.

https://en.wikipedia.org/wiki/Difference_set

If you had just watched the full video the first time, properly, the presenter would have told you this, you knob head!

How does Dobble (Spot It) work?
https://www.youtube.com/watch?v=VTDKqW_GLkw

Regards,

esselte

esselte said:

Hi esselte,

This is called a “Cyclic Difference Set”.

https://en.wikipedia.org/wiki/Difference_set

In combinatorics, a {\displaystyle (v,k,\lambda )}(v,k,\lambda ) difference set is a subset {\displaystyle D}D of size {\displaystyle k}k of a group {\displaystyle G}G of order {\displaystyle v}v such that every nonidentity element of {\displaystyle G}G can be expressed as a product {\displaystyle d_{1}d_{2}^{-1}}d_{1}d_{2}^{{-1}} of elements of {\displaystyle D}D in exactly {\displaystyle \lambda }\lambda ways.