Integer-valued Polynomials
It can be shown that the polynomial $n^4 + 4n^3 + 2n^2 + 5n$ is a multiple of $6$ for every integer $n$. It can also be...
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John Rhodes
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August 15, 2023
Sum of Squares of Divisors
The divisors of $6$ are $1,2,3$ and $6$. The sum of the squares of these numbers is $1+4+9+36=50$. Let $\operatorname{sigma}_2(n)$ represent the sum of the squares of...
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John Rhodes
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August 15, 2023
Fibonacci Tree Game
A Fibonacci tree is a binary tree recursively defined as:$T(0)$ is the empty tree. $T(1)$ is the binary tree with only one node. $T(k)$ consists of a...
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John Rhodes
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August 15, 2023
Cutting Rope
Inside a rope of length $n$, $n – 1$ points are placed with distance $1$ from each other and from the endpoints. Among these points, we choose...
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John Rhodes
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August 15, 2023
Triangle on Parabola
On the parabola $y = x^2/k$, three points $A(a, a^2/k)$, $B(b, b^2/k)$ and $C(c, c^2/k)$ are chosen. Let $F(K, X)$ be the number of the integer quadruplets...
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John Rhodes
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August 15, 2023
Weak Goodstein Sequence
For any positive integer $n$, the $n$th weak Goodstein sequence $\{g_1, g_2, g_3, \dots\}$ is defined as: $g_1 = n$ for $k \gt 1$, $g_k$ is obtained...
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John Rhodes
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August 15, 2023
Pythagorean Tree
The Pythagorean tree is a fractal generated by the following procedure: Start with a unit square. Then, calling one of the sides its base (in the animation,...
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John Rhodes
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August 15, 2023
Migrating Ants
An $n \times n$ grid of squares contains $n^2$ ants, one ant per square. All ants decide to move simultaneously to an adjacent square (usually $4$ possibilities,...