John H. Conway key-rings puzzle

I tried to solve the third question in the pi-day pizzahut contest, but I'm not sure i understood the puzzle correctly, so I am interested in lg community thoughts about it:

http://blog.pizzahut.com/flavor-news/national-pi-day-math-problems-solved/

Remark: it is too late to get a pi-year free pizza.

Don't look at it if you managed to avoid knot theory.

1"They are all linked together in a strange jumble, so that try as I might, I can’t tell any pair from any other pair"

All the types of “pairs” have symmetry and exist more than once

2“However, I _can_ tell some triple from other triples”

Odd triple exists (start with that) and at least two matched triples

3“, even though I’ve never been able to distinguish left from right”

Mirrored pattern maybe

so maybe a triple (3) with two it’s rings connected to a triple each (3x2) and add a pair to each (one going left the other right) (2x2)

minimum answer 13

expanded answer 13+2n (i.e keep mirror pattern)

much lower value

join two triples together, the join is now an odd pair,

add a further ring to convert the odd pair into an odd triple = 9 rings

can be drawn using overlapping circles

Thanks.

I am not sure I understand your point about pairs  : for me a pair is any two rings, so for the pairs to be undistinguishable, all the rings must be links two by two, otherwise I can distinguish an unlinked pair from a linked one. Do you understand a pair to be only two rings that are linked together?

More exactly, I understood the problem like this:

I have a collection of n rings, n>0: r1...rn,

1. Whatever I,j,k,l in [1,n], (ri,rj) can not be distinguished from (rk,rl) when mirror symmetry is considered equal.

2. There exist I,j,k,l,m,n in [1,n] so that (ri,rj,rk) can be distinguished from (rl,rm,rn), same proviso with symmetry.

what is the least n allowing 1. and 2.?

I should add that distinguish two links mean that they do not have the same topological link representation.

Also for 1. i<>j and k<>l,

for 2. I,j,k different and l,m,n different.

7 rings not 9

Can you show a drawing?

are you using http://katlas.org/wiki/L6a5 links or http://katlas.org/wiki/L6n1 ?

I agree that the question is ambiguous. One interpretation of the first statement is that the automorphism group of the link acts transitively on the pairs; another is that either every pair is linked or every pair is unlinked. Conway is probably happy for people to just get confused by the ambiguity. You could ask on math.stackexchange, you might get more of a response there?

Reread puzzle and changed to 9 rings, this is to pervent any unique interlocking pair been identified

due to their solid nature triples can be assembled and then twisted to look like type L6n1

make three of these

join the odd triple to the other two using two joins

pick them up in your hand and they appear to be a mass of connected rings, only when laid flat do they solve the puzzle