A Ducci sequence is obtained by starting with an 
-tuple of integers and repeatedly
replacing it by the cyclic tuple of the absolute values
of successive differences,
 |
For example,
 |
The sequence is named for the Italian mathematician Enrico Ducci (1864-1940), whom Ciamberlini and Marengoni (1937) credited with the four-number problem of determining
whether this process always reaches zero. Ciamberlini and Marengoni (1937) also gave
the first published proof that it does so for 4-tuples.
The process is consequently also called the Ducci map, the four-number game, or,
for general 
,
the 
-number
game (Chamberland and Thomas 2004, Brockman and Zerr 2007).
When 
is a power of 2, every Ducci sequence of integer 
-tuples eventually reaches the zero tuple.
For other lengths, nonzero cycles can occur (Breuer 2007).
-Number Ducci Game." J. Difference Equ. Appl. 10,
339-342, 2004.