Slow exact 
-Nim
is a two-player pile game in which a move removes exactly one counter from each of
exactly 
nonempty piles (Gurvich and Ho 2015, Gurvich and Naumova 2026). Under normal play,
a player with no legal move loses. This differs from Nim,
where a move can remove several counters from a single pile.
For 
and 
piles, Gurvich and Naumova (2026) gave an explicit formula for the nim-value 
of every position. Sort the nonnegative
pile sizes as 
, put 
, and define
 |
(1)
|
where 
is the floor function. Replace 
by 
, put 
, and let 
count the odd 
. The residues 
and 
give 
and 
, respectively. For every
other residue put 
. Then
 |
(2)
|
Thus the range is exactly 
for every 
. The truncation preserves
all legal move sequences, and 
is also the maximum length of play. The formula uses linearly
many arithmetic operations once the pile sizes are sorted.
Adding one counter to every pile except a smallest pile changes 
to 
. In particular,
adding this shift twice preserves the nim-value. For
slow exact 4-Nim with five piles, 
has nim-value 3 for
every 
, correcting an earlier reported value of 5 (Gurvich
and Naumova 2026). The authors report ChatGPT assistance with computation and development
of the mathematical arguments.
-Nim." 24 Aug 2015. 
-Nim with 
Piles." 27 Oct 2026.