This month we searched for a starting arrangement to an algorithm that
maximized its number of steps. We can generalize the active square’s
corner numbers from integers to reals, consider the four
numbers on the corners of the active square to be an element of
**R**^{4}, and the function representing a step to be

f((a,b,c,d)) = ( |
a-b |
, | b-c |
, | c-d |
, | d-a |
). |

Without loss of generality we can consider an input (*a*,*b*,*c*,*d*) to have
*a* largest and *b*>*d*. By simple case checking, the only such
arrangement that doesn’t lead to (0,0,0,0) in fewer than 10 steps
has *a*>*b*>*c*>*d*. We further can “normalize” our input by
subtracting *d* from everything, and then dividing by *a*-*d*, to get
a general input of the form (1,*x*,*y*,0) for 1>*x*>*y*>0. In order to
find arbitrarily long sequences of integer inputs for *f* we want to
search for a real input to *f* that never reaches
(0,0,0,0). This would be achieved if the normalized output of *f*
matches the input. In normalized space, we want either

*x* = (1-*x*-*y*)/(1-*y*) **AND**
*y* = (*x*-2*y*)/(1-*y*)

or

*x* = (*x*+*y*-1)/*x* **AND**
*y* = (2*x*-*y*-1)/*x*

The first set of equations resolves to

*x*^{3}-4*x*^{2}+6*x*-2=0

which happily has a zero at approximately *x*=0.456311…, and a
corresponding *y*=0.160713….

The second set of equations resolves to a cubic without a zero between
0 and 1, so there is a unique fixed point of this normalized
function. Now the only challenge is to find integer points
(*a*,*b*,*c*,0) that are as close as possible, when normalized, to
(1,*x*,*y*,0), and test them to see how large their *g* values
are. Searching over all *c* values between 1 and 10,000,000, choosing
a small set of *a* and *b* that are near to *c*/*y* and *cx*/*y*
respectively, will find the optimal quadruple
**8646064;3945294;1389537;0**, which has *g* value 44. Observant
solvers noticed some overlaps with these special input integers and the
Tribonacci sequence!

**Congrats to this month’s solvers that found the smallest maximal
input!**