The Primality of $2n^2 – 1$
Consider numbers $t(n)$ of the form $t(n) = 2n^2 – 1$ with $n \gt 1$. The first such numbers are $7, 17, 31, 49, 71, 97, 127$...
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John Rhodes
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August 15, 2023
Crack-free Walls
Consider the problem of building a wall out of $2 \times 1$ and $3 \times 1$ bricks ($\text{horizontal} \times \text{vertical}$ dimensions) such that, for extra strength, the...
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John Rhodes
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August 15, 2023
Totient Chains
Let $\phi$ be Euler’s totient function, i.e. for a natural number $n$, $\phi(n)$ is the number of $k$, $1 \le k \le n$, for which $\gcd(k, n)...
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John Rhodes
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August 15, 2023
Flea Circus
A $30 \times 30$ grid of squares contains $900$ fleas, initially one flea per square. When a bell is rung, each flea jumps to an adjacent square...
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John Rhodes
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August 15, 2023
Combined Volume of Cuboids
An axis-aligned cuboid, specified by parameters $\{(x_0, y_0, z_0), (dx, dy, dz)\}$, consists of all points $(X,Y,Z)$ such that $x_0 \le X \le x_0 + dx$, $y_0...
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John Rhodes
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August 15, 2023
Divisor Square Sum
For a positive integer $n$, let $\sigma_2(n)$ be the sum of the squares of its divisors. For example, $$\sigma_2(10) = 1 + 4 + 25 + 100...
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John Rhodes
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August 15, 2023
Obtuse Angled Triangles
Consider the set $S(r)$ of points $(x,y)$ with integer coordinates satisfying $|x| + |y| \le r$. Let $O$ be the point $(0,0)$ and $C$ the point $(r/4,r/4)$....
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John Rhodes
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August 15, 2023
Circular Logic
A $k$-input binary truth table is a map from $k$ input bits (binary digits, $0$ [false] or $1$ [true]) to $1$ output bit. For example, the $2$-input...