# Diameter of the M24 Conway puzzle is 45

Submitted by Paul Timmons on Wed, 02/08/2023 - 11:21.

This is a well know puzzle in which there are two moves, one which rotates the central "clock" by one position clockwise or counter-clockwise.

The other switches or swaps each pair of numbers with matching colours.

I decided to plug these values into GAP to investigate God's number for the underlying group for this puzzle

as it still seemed be unknown or undocumented anywhere at least. Feeding these numbers into GAP we get:

S := (1,24)(2,23)(3,4)(5,22)(6,11)(7,8)(9,10)(12,21)(13,14)(15,20)(16,17)(18,19);

R := (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);

Note that we have the number 24 in place of 0 to be GAP compatible.

Table: Distribution of word count in the Cayley graph for M24 Conway puzzle

So there is exactly one node requiring a manouevre of length 45 in the Cayley graph for the group generated by R and S

assuming my results are correct. It would be nice to have independent verification of of the results in the table here

(even up to a certain length).

Expressed in term of the generators the single node w of length 45 is:

w := R^3*S*R^-4*S*R^2*S*R^-1*S*R*S*R^-2*S*R*(R^2*S*R^-4*S)^2*R^2*S*R^-3*S*R*S;

It is the permutation

original switch (S) in that 4 pairs of transpositions are omitted, and 8 remaining pairs of transpositions

(colours) get swapped around (see diagram above to visualise).

For the enthusiasts only! We may take in turn the group generated by w and R and we have a free of charge puzzle for the point stabiliser of M24

(namely M23) with the point 24 (or zero as originally put forward) omitted using the same diagram. This in turn has so-called

God's number 32 with 11 nodes at the diameter - the odd one out of which is the involution

R^3*w*(R*w*R)^2*R^4*(w*R^-1)^2*R^-1*w*R*(w*R^-1)^2*(w*R)^3*w

Here is a snippet of the GAP code I used to preform the M24 Conway puzzle Cayley graph calculations:

This is a well know puzzle in which there are two moves, one which rotates the central "clock" by one position clockwise or counter-clockwise.

The other switches or swaps each pair of numbers with matching colours.

I decided to plug these values into GAP to investigate God's number for the underlying group for this puzzle

as it still seemed be unknown or undocumented anywhere at least. Feeding these numbers into GAP we get:

S := (1,24)(2,23)(3,4)(5,22)(6,11)(7,8)(9,10)(12,21)(13,14)(15,20)(16,17)(18,19);

R := (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);

Note that we have the number 24 in place of 0 to be GAP compatible.

Table: Distribution of word count in the Cayley graph for M24 Conway puzzle

Len Count0 1 (the identity) 1 3 (R, R^-1 and S) 2 6 {R^2, R^-2, R*S, S*R, R^-1*S, S*R^-1} 3 10 4 16 5 26 6 42 7 68 8 110 9 178 10 288 11 466 12 752 13 1214 14 1958 15 3132 16 4980 17 7957 18 12738 19 20375 20 32563 21 52010 22 83000 23 132360 24 210896 25 335265 26 531855 27 842722 28 1332401 29 2098880 30 3291324 31 5121482 32 7872753 33 11885315 34 17455355 35 24565715 36 32401139 37 38652273 38 39596760 39 32213183 40 18556076 41 6407846 42 1041497 43 55472 44 577 45 1

So there is exactly one node requiring a manouevre of length 45 in the Cayley graph for the group generated by R and S

assuming my results are correct. It would be nice to have independent verification of of the results in the table here

(even up to a certain length).

Expressed in term of the generators the single node w of length 45 is:

w := R^3*S*R^-4*S*R^2*S*R^-1*S*R*S*R^-2*S*R*(R^2*S*R^-4*S)^2*R^2*S*R^-3*S*R*S;

It is the permutation

*. This differs from the*

**(2,19)(6,20)(7,16)(8,17)(11,15)(12,13)(14,21)(18,23)**original switch (S) in that 4 pairs of transpositions are omitted, and 8 remaining pairs of transpositions

(colours) get swapped around (see diagram above to visualise).

For the enthusiasts only! We may take in turn the group generated by w and R and we have a free of charge puzzle for the point stabiliser of M24

(namely M23) with the point 24 (or zero as originally put forward) omitted using the same diagram. This in turn has so-called

God's number 32 with 11 nodes at the diameter - the odd one out of which is the involution

R^3*w*(R*w*R)^2*R^4*(w*R^-1)^2*R^-1*w*R*(w*R^-1)^2*(w*R)^3*w

Here is a snippet of the GAP code I used to preform the M24 Conway puzzle Cayley graph calculations:

*M24:=Group(S,R);*

w:=(2,19)(6,20)(7,16)(8,17)(11,15)(12,13)(14,21)(18,23);

f:=Factorization(M24,w); /* Needed as this function can otherwise return suboptimal results */

arr:=[];

for i in [1..45] do; arr[i]:=0;od;

n:=Iterator(M24);

for i in [1..244823040] do;

a:=NextIterator(n);

f:=Factorization(M24,a);

l:=Length(f);

if l>0 then arr[l]:=arr[l]+1;fi;

if Length(f) > 43 then Print(f, "-", Length(f),"-",i,"\n");fi;od;

w:=(2,19)(6,20)(7,16)(8,17)(11,15)(12,13)(14,21)(18,23);

f:=Factorization(M24,w); /* Needed as this function can otherwise return suboptimal results */

arr:=[];

for i in [1..45] do; arr[i]:=0;od;

n:=Iterator(M24);

for i in [1..244823040] do;

a:=NextIterator(n);

f:=Factorization(M24,a);

l:=Length(f);

if l>0 then arr[l]:=arr[l]+1;fi;

if Length(f) > 43 then Print(f, "-", Length(f),"-",i,"\n");fi;od;