The stepping stones puzzle is a placement puzzle on an infinite square grid. Start with a positive
integer 
and place 
stones of value 1 in distinct cells. Then place stones of values 2, 3, ... in order,
with each new stone occupying an empty cell whose eight neighboring cells, including
diagonal neighbors, have values adding to the new stone's value. Empty cells contribute
zero. Stones remain in place, and the neighboring values are checked only when a
stone is placed (Sloane 2023).
Let 
be the largest label attainable, maximizing over the initial placement and all subsequent
choices. Thus 
is the greatest number of stones added after the initial stones. The integer
sequence 
for 
,
2, ... begins 1, 16, 28, 38, 49, 60, ... (OEIS A337663,
Sloane 2023). In particular, 
because a single initial stone cannot supply the neighboring
values needed to place 2.
A counting argument establishes that 
is finite (OEIS A337663).
Suppose a placement reaches label 
. At the instant a stone of value 
is placed, its value is the sum
of at least two positive earlier neighboring values, since every earlier value is
less than 
.
At least one of these values is at most 
. Following such earlier neighbors for at most 11 steps reaches
either an initial stone or a value at most 
. There are at most 
such cells. Each has at most 
cells within 11 horizontal and 11 vertical steps. All
noninitial stones lie in the union of these neighborhoods,
so
 |
Since 
,
this bounds 
solely in terms of 
.
