Convex Path in Square
Let $F(N)$ be the maximum number of lattice points in an axis-aligned $N\times N$ square that the graph of a single strictly convex increasing function can pass...
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John Rhodes
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August 16, 2023
Substring Sums of Prime Concatenations
Let $S(n)$ be the sum of all contiguous integer-substrings that can be formed from the integer $n$. The substrings need not be distinct. For example, $S(2024) =...
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John Rhodes
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August 16, 2023
Product of Head Counts
Alice enlists the help of some friends to generate a random number, using a single unfair coin. She and her friends sit around a table and, starting...
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John Rhodes
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August 16, 2023
Divisibility Streaks
For every positive number $n$ we define the function $\mathop{streak}(n)=k$ as the smallest positive integer $k$ such that $n+k$ is not divisible by $k+1$. E.g: $13$ is...
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John Rhodes
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August 16, 2023
Integer Sided Equiangular Hexagons
Let $H(n)$ be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding $n$. Hexagons are distinct if and only if they are not...
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John Rhodes
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August 16, 2023
Distinct Colourings of a Rubik’s Cube
The well-known Rubik’s Cube puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible faces, each with a coloured...
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John Rhodes
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August 16, 2023
Split Divisibilities
Consider the number $48$. There are five pairs of integers $a$ and $b$ ($a \leq b$) such that $a \times b=48$: $(1,48)$, $(2,24)$, $(3,16)$, $(4,12)$ and $(6,8)$....
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John Rhodes
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August 16, 2023
Torpids
The Torpids are rowing races held annually in Oxford, following some curious rules: A division consists of $n$ boats (typically 13), placed in order based on past...