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...
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John Rhodes
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August 16, 2023
Number of Lattice Points in a Hyperball
Let $T(r)$ be the number of integer quadruplets $x, y, z, t$ such that $x^2 + y^2 + z^2 + t^2 \le r^2$. In other words, $T(r)$...
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John Rhodes
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August 16, 2023
Incremental Random Sort
A deck of cards numbered from $1$ to $n$ is shuffled randomly such that each permutation is equally likely. The cards are to be sorted into ascending...
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John Rhodes
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August 16, 2023
Rhombus Tilings
For a polygon $P$, let $t(P)$ be the number of ways in which $P$ can be tiled using rhombi and squares with edge length 1. Distinct rotations...