Uphill Paths
Let $n$ be a positive integer. Suppose there are stations at the coordinates $(x, y) = (2^i \bmod n, 3^i \bmod n)$ for $0 \leq i \leq...
-
John Rhodes
-
August 15, 2023
Circle and Tangent Line
Let $C$ be the circle with radius $r$, $x^2 + y^2 = r^2$. We choose two points $P(a, b)$ and $Q(-a, c)$ so that the line passing...
-
John Rhodes
-
August 15, 2023
Nim Extreme
Let $n$ be a positive integer. Consider nim positions where:There are $n$ non-empty piles. Each pile has size less than $2^n$. No two piles have the same...
-
John Rhodes
-
August 15, 2023
Admissible Paths Through a Grid
Let’s call a lattice point $(x, y)$ inadmissible if $x, y$ and $x+y$ are all positive perfect squares. For example, $(9, 16)$ is inadmissible, while $(0, 4)$,...
-
John Rhodes
-
August 15, 2023
Idempotents
If we calculate $a^2 \bmod 6$ for $0 \leq a \leq 5$ we get: $0,1,4,3,4,1$. The largest value of $a$ such that $a^2 \equiv a \bmod 6$...
-
John Rhodes
-
August 15, 2023
A Rectangular Tiling
We wish to tile a rectangle whose length is twice its width. Let $T(0)$ be the tiling consisting of a single rectangle. For $n \gt 0$, let...
-
John Rhodes
-
August 15, 2023
Crisscross Ellipses
$E_a$ is an ellipse with an equation of the form $x^2 + 4y^2 = 4a^2$. $E_a^\prime$ is the rotated image of $E_a$ by $\theta$ degrees counterclockwise around...
-
John Rhodes
-
August 15, 2023
Lattice Points Enclosed by Parabola and Line
For integers $a$ and $b$, we define $D(a, b)$ as the domain enclosed by the parabola $y = x^2$ and the line $y = a\cdot x +...