Golden Triplets
For any integer $n$, consider the three functions \begin{align} f_{1, n}(x, y, z) &= x^{n + 1} + y^{n + 1} – z^{n + 1}\\ f_{2, n}(x,...
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John Rhodes
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August 15, 2023
Consecutive Positive Divisors
Find the number of integers $1 \lt n \lt 10^7$, for which $n$ and $n + 1$ have the same number of positive divisors. For example, $14$...
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John Rhodes
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August 15, 2023
Step Numbers
Consider the number $45656$. It can be seen that each pair of consecutive digits of $45656$ has a difference of one. A number for which every pair...
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John Rhodes
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August 15, 2023
Integer Angled Quadrilaterals
Let $ABCD$ be a convex quadrilateral, with diagonals $AC$ and $BD$. At each vertex the diagonal makes an angle with each of the two sides, creating eight...
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John Rhodes
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August 15, 2023
Common Cathetus Right-angled Triangles
The four right-angled triangles with sides $(9,12,15)$, $(12,16,20)$, $(5,12,13)$ and $(12,35,37)$ all have one of the shorter sides (catheti) equal to $12$. It can be shown that...
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John Rhodes
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August 15, 2023
Fractions and Sum of Powers of Two
Define $f(0)=1$ and $f(n)$ to be the number of ways to write $n$ as a sum of powers of $2$ where no power occurs more than twice....
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John Rhodes
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August 15, 2023
Hollow Square Laminae II
We shall define a square lamina to be a square outline with a square “hole” so that the shape possesses vertical and horizontal symmetry. Given eight tiles...
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John Rhodes
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August 15, 2023
Hollow Square Laminae I
We shall define a square lamina to be a square outline with a square “hole” so that the shape possesses vertical and horizontal symmetry. For example, using...