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...
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John Rhodes
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August 16, 2023
Fleeting Medians
We define two sequences $S = \{S(1), S(2), …, S(n)\}$ and $S_2 = \{S_2(1), S_2(2), …, S_2(n)\}$: $S(k) = (p_k)^k \bmod 10007$ where $p_k$ is the $k$th...
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John Rhodes
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August 16, 2023
Factorial Trailing Digits 2
For any $N$, let $f(N)$ be the last twelve hexadecimal digits before the trailing zeroes in $N!$. For example, the hexadecimal representation of $20!$ is 21C3677C82B40000, so...
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John Rhodes
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August 16, 2023
Best Approximations by Quadratic Integers
Given a non-square integer $d$, any real $x$ can be approximated arbitrarily close by quadratic integers $a+b\sqrt{d}$, where $a,b$ are integers. For example, the following inequalities approximate...
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John Rhodes
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August 16, 2023
Sets with a Given Least Common Multiple
Let $H(n)$ denote the number of sets of positive integers such that the least common multiple of the integers in the set equals $n$. E.g.: The integers...
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John Rhodes
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August 16, 2023
Poohsticks Marathon
Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them to play for longer before one...