Richard K. Guy
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Numerical and Statistical Analysis of Aliquot Sequences Open
We present a variety of numerical data related to the growth of terms in aliquot sequences, iterations of the function $s(n) = σ(n) - n$. First, we compute the geometric mean of the ratio $s_k(n)/s_{k-1}(n)$ of $k$th iterates for $n \leq 2…
Difference Necklaces Open
An $(a,b)$-difference necklace of length $n$ is a circular arrangement of the integers $0, 1, 2, \ldots , n-1$ such that any two neighbours have absolute difference $a$ or $b$. We prove that, subject to certain conditions on $a$ and $b$, s…
Fibonacci Plays Billiards Open
A chain is an ordering of the integers 1 to n such that adjacent pairs have sums of a particular form, such as squares, cubes, triangular numbers, pentagonal numbers, or Fibonacci numbers. For example 4 1 2 3 5 form a Fibonacci chain while…
The Triangle. Open
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle. Stei…
The Triangle Open
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle. Stei…
The Outercoarseness of the n-cube Open
Guy and Nowakowski showed that the outercoarseness of the n-cube was, for sufficiently large n, at least 0.96 of its maximum possible value, n⋅2n−4. Here we give some exact results, including that the maximum is attained for all n≥24.