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...
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John Rhodes
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August 15, 2023
Power Sets of Power Sets
Let $P(n)$ be the set of the first $n$ positive integers $\{1, 2, \dots, n\}$. Let $Q(n)$ be the set of all the non-empty subsets of $P(n)$....
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John Rhodes
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August 15, 2023
Chinese Leftovers II
Let $A_n$ be the smallest positive integer satisfying $A_n \bmod p_i = i$ for all $1 \le i \le n$, where $p_i$ is the $i$-th prime. For...
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John Rhodes
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August 15, 2023
Sum of Digits Sequence
Let $a_0, a_1, \dots$ be an integer sequence defined by: $a_0 = 1$; for $n \ge 1$, $a_n$ is the sum of the digits of all preceding...