Eisele, P.; Hadeler, K.P.
Game of cards, dynamical systems, and a characterization of the 
floor and ceiling functions.
Am. Math. Mon. 97, No.6, 466-477 (1990).

Zbl 724.58036
Consider a deck of N cards, numbered 1 to N from top, face down. They are
distributed into b packs, face up. After reversing the packs, p cards are
put on top of an indicated pack and the remaining cards are put underneath.
The new position $x\sb 1$ of card $x\sb 0$ is then $x\sb 1=p+\lceil 
\frac{x\sb 0}{b}\rceil$, where for any real x, $\lceil x\rceil$ denotes the
smallest integer not less than x.  Thus the parameter N is not essential.
We thus have a dynamical system $$ x\sb{n+1}=f(x\sb n),\quad n=0,1,2,...,\quad 
f: {\bbfZ}\to {\bbfZ},\quad f(x)=a+\lceil \frac{x}{b}\rceil, $$ for given 
$a\in {\bbfZ}$ and $b\in {\bbfN}\setminus \{1\}$. 
A variant of the problem, in which we start with a face up deck and never 
reverse the packs, leads to a dynamical system 
$$ x\sb{n+1}=f(x\sb n),\quad n=0,1,2,...,\quad f(x)=a-\lceil \frac{x}{b}\rceil. $$
The authors study these dynamical systems and show that for the first one there
are at most two stationary points, and every trajectory converges to one of
these. They give an explicit expression for the stop time and as a consequence it
is known how many steps are necessary to determine the unknown card in the
general game of cards. For the variant system they distinguish the cases with
stationary points from those with periodic orbits of order two.


Redish, K.A.; Smyth, W.F.
Closed form expressions for the iterated floor function. (English)
[J] Discrete Math. 91, No.3, 317-321 (1991). [ISSN 0012-365X]

Zbl 757.05066
For positive integers $k$, $r$, and $n\ge k+1$, the iterated floor function
$f\sb{k,r}$ is defined by $$f\sb{k,r}(k+1)=r;\quad
f\sb{k,r}(n)=\left\lfloor{n\over n-k}f\sb{k,r}(n-1)\right\rfloor,\quad n<k+1.$$
A special case $(k=r=3)$ of this function occurs as an upper bound on the
number of 3-subsets, excluding tetrahedra, of an $n$-set (Turan's problem).
For certain values of $k$ and $r$, this note establishes closed form expressions
for $f\sb{k,r}$, then uses them to prove some interesting properties.


Haland, Inger Johanne; Knuth, Donald E.
Polynomials involving the floor function. (English)
[J] Math. Scand. 76, No.2, 194-200 (1995). [ISSN 0025-5521]

Zbl 843.11005
Some identities are presented that generalize the formula $$x\sp 3= 3x
\bigl\lfloor x\lfloor x\rfloor \bigr \rfloor- 3\lfloor x\rfloor \bigl \lfloor x\lfloor
x\rfloor \bigr \rfloor+ \lfloor x\rfloor\sp 3+ 3\{ x\} \{x\lfloor x\rfloor\}+ \{x \}\sp
3$$ to a representation of the product $x\sb 0 x\sb 1 \dots x\sb{n-1}$. For
example, $$\align xyz &= x\bigl \lfloor y \lfloor z\rfloor \bigr\rfloor+ y\bigl \lfloor
z\lfloor x\rfloor \bigr\rfloor+ z\bigl \lfloor x\lfloor y\rfloor \bigr\rfloor - \lfloor
x\rfloor \bigl \lfloor y\lfloor z\rfloor \bigr\rfloor- \lfloor y\rfloor \bigl \lfloor z\lfloor
x\rfloor \bigr\rfloor- z\bigl \lfloor x\lfloor y\rfloor \bigr \rfloor\\ &+ \lfloor x\rfloor
\lfloor y\rfloor \lfloor z\rfloor + \{x\} \{y\lfloor z\rfloor\}+ \{y\} \{z\lfloor
x\rfloor\}+ \{z\} \{x\lfloor y\rfloor\} + \{x\} \{y\} \{z\}. \endalign$$ These
identities make it possible to answer questions about the distribution $\text
{mod }1$ of sequences having the form $\alpha\sb 1 n\lfloor \alpha\sb 2 n\dots
\lfloor \alpha\sb{k-1} n\lfloor \alpha\sb k n\rfloor \rfloor \dots \rfloor$, $n=1, 2,
\dots\ $. Such sequences are known to be uniformly distributed $\text {mod }
1$ if the real numbers 1, $\alpha\sb 1, \dots, \alpha\sb k$ are rationally
independent; we prove that they are uniformly distributed in the special case
$\alpha\sb 1= \alpha\sb 2= \cdots= \alpha\sb k= \alpha$ if and only if
$\alpha\sp k$ is irrational, when $k$ is prime.


Fraenkel, Aviezri S.
Iterated floor function, algebraic numbers, discrete chaos, Beatty
subsequences, semigroups. (English)
Trans. Am. Math. Soc. 341, No.2, 639-664 (1994).

Zbl 808.05008
For a real number $\alpha$, the floor function $\lfloor \alpha \rfloor$ is the
integer part of $\alpha$. The sequence $\{\lfloor m \alpha \rfloor : m = 1, 2,
3,\dots \}$ is the Beatty sequence of $\alpha$. Identities are proved which
express the sum of the iterated floor functional $A\sp i$ for $1 \le i \le n$,
operating on a nonzero algebraic number $\alpha$ of degree $\le n$, in
terms of only $A\sp 1 = \lfloor m \alpha \rfloor$, $m$ and a bounded term.
Applications include discrete chaos (discrete dynamical systems), explicit
construction of infinite non chaotic subsequences of chaotic sequences,
discrete order (identities), explicit construction of nontrivial Beatty
subsequences, and certain arithmetical semigroups. Beatty sequences have
a large literature in combinatorics. They have also been used in nonperiodic
tilings (quasicrystallography), periodic scheduling computer vision (digital
lines), and formal language theory.