Sequences with Nice Divisibility Properties
Let $Seq(n,k)$ be the number of positive-integer sequences $\{a_i\}_{1 \le i \le n}$ of length $n$ such that: $n$ is divisible by $a_i$ for $1 \le i...
-
John Rhodes
-
August 15, 2023
Tangent Circles
Circles $A$ and $B$ are tangent to each other and to line $L$ at three distinct points. Circle $C$ is inside the space between $A$, $B$ and...
-
John Rhodes
-
August 15, 2023
Divisor Nim
Anton and Bertrand love to play three pile Nim. However, after a lot of games of Nim they got bored and changed the rules somewhat. They may...
-
John Rhodes
-
August 15, 2023
Shortest Lattice Vector
Let $t_n$ be the tribonacci numbers defined as: $t_0 = t_1 = 0$; $t_2 = 1$; $t_n = t_{n-1} + t_{n-2} + t_{n-3}$ for $n \ge 3$...
-
John Rhodes
-
August 15, 2023
Clock Sequence
Consider the infinite repeating sequence of digits: 1234321234321234321… Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the...
-
John Rhodes
-
August 15, 2023
Bidirectional Recurrence
Let: $\begin{array}{ll} x(0)&=0 \\ x(1)&=1 \\ x(2k)&=(3x(k)+2x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \text {, where } \lfloor \text...
-
John Rhodes
-
August 15, 2023
Square on the Inside
Let $ABCD$ be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows: $A(a, 0)$, $B(0, b)$, $C(-c, 0)$, $D(0, -d)$, where $1...
-
John Rhodes
-
August 15, 2023
Compromise or Persist
Alice is playing a game with $n$ cards numbered $1$ to $n$. A game consists of iterations of the following steps. (1) Alice picks one of the...