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,...
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John Rhodes
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August 15, 2023
Hopping Game
Let $s_k$ be the number of 1’s when writing the numbers from 0 to $k$ in binary. For example, writing 0 to 5 in binary, we have...
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John Rhodes
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August 15, 2023
Triangles with Non Rational Sides and Integral Area
Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$. It can be shown that this triangle has area $9$. $S(n)$ is the sum of...
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John Rhodes
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August 15, 2023
Platonic Dice
An unbiased single $4$-sided die is thrown and its value, $T$, is noted.$T$ unbiased $6$-sided dice are thrown and their scores are added together. The sum, $C$,...
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John Rhodes
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August 15, 2023
Distinct Lines
Consider all lattice points $(a,b,c)$ with $0 \le a,b,c \le N$. From the origin $O(0,0,0)$ all lines are drawn to the other lattice points. Let $D(N)$ be...
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John Rhodes
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August 15, 2023
Harshad Numbers
A Harshad or Niven number is a number that is divisible by the sum of its digits. $201$ is a Harshad number because it is divisible by...