Triangle Triples
Let $T(n)$ be the nth triangle number, so $T(n) = \dfrac{n(n + 1)}{2}$. Let $dT(n)$ be the number of divisors of $T(n)$. E.g.: $T(7) = 28$ and...
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John Rhodes
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August 15, 2023
Sum of Digits – Experience #13
There are $16$ positive integers that do not have a zero in their digits and that have a digital sum equal to $5$, namely: $5$, $14$, $23$,...
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John Rhodes
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August 15, 2023
Nontransitive Sets of Dice
Consider the following set of dice with nonstandard pips: Die $A$: $1$ $4$ $4$ $4$ $4$ $4$ Die $B$: $2$ $2$ $2$ $5$ $5$ $5$ Die $C$:...
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John Rhodes
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August 15, 2023
Minimum of Subsequences
Let $S_n$ be an integer sequence produced with the following pseudo-random number generator: \begin{align} S_0 & = 290797 \\ S_{n+1} & = S_n^2 \bmod 50515093 \end{align} Let...
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John Rhodes
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August 15, 2023
Maximum Integer Partition Product
An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers. Partitions that differ only in the order of...
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John Rhodes
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August 15, 2023
Circumscribed Circles
Every triangle has a circumscribed circle that goes through the three vertices. Consider all integer sided triangles for which the radius of the circumscribed circle is integral...
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John Rhodes
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August 15, 2023
Pencils of Rays
Let $R(M, N)$ be the number of lattice points $(x, y)$ which satisfy $M\!\lt\!x\!\le\!N$, $M\!\lt\!y\!\le\!N$ and $\large\left\lfloor\!\frac{y^2}{x^2}\!\right\rfloor$ is odd. We can verify that $R(0, 100) = 3019$...
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John Rhodes
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August 15, 2023
Licence Plates
Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]). While driving to work Seth plays the following...