Maximal Perimeter
Construct triangle $ABC$ such that: Vertices $A$, $B$ and $C$ are lattice points inside or on the circle of radius $r$ centered at the origin; the triangle...
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John Rhodes
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August 15, 2023
Divisor Pairs
Let $S(n)$ be the number of pairs $(a,b)$ of distinct divisors of $n$ such that $a$ divides $b$. For $n=6$ we get the following pairs: $(1,2), (1,3),...
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John Rhodes
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August 15, 2023
Coprime Nim
Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is coprime with the...
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John Rhodes
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August 15, 2023
Irrational Base
Let $r$ be the real root of the equation $x^3 = x^2 + 1$. Every positive integer can be written as the sum of distinct increasing powers...
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John Rhodes
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August 15, 2023
Cutting Triangles
A triangle is cut into four pieces by two straight lines, each starting at one vertex and ending on the opposite edge. This results in forming three...
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John Rhodes
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August 15, 2023
Squarefree Gaussian Integers
A Gaussian integer is a number $z = a + bi$ where $a$, $b$ are integers and $i^2 = -1$. Gaussian integers are a subset of the...
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John Rhodes
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August 15, 2023
McCarthy 91 Function
The McCarthy 91 function is defined as follows: $$ M_{91}(n) = \begin{cases} n – 10 & \text{if } n > 100 \\ M_{91}(M_{91}(n+11)) & \text{if } 0...
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John Rhodes
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August 15, 2023
Centaurs on a Chess Board
On a chess board, a centaur moves like a king or a knight. The diagram below shows the valid moves of a centaur (represented by an inverted...