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...
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John Rhodes
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August 15, 2023
Robot Walks
A robot moves in a series of one-fifth circular arcs ($72^\circ$), with a free choice of a clockwise or an anticlockwise arc for each step, but no...
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John Rhodes
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August 15, 2023
Integer Partition Equations
For some positive integers $k$, there exists an integer partition of the form $4^t = 2^t + k$, where $4^t$, $2^t$, and $k$ are all positive integers...
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John Rhodes
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August 15, 2023
Concealed Square
Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit. This problem is challenging because it limits the number...