Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Divisibility Comparison Between Factorials

Let $f_5(n)$ be the largest integer $x$ for which $5^x$ divides $n$. For example, $f_5(625000) = 7$. Let $T_5(n)$ be the number of integers $i$ which satisfy...

John Rhodes
August 15, 2023
Computational Mathematics

Generating Polygons

A polygon is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit. A polygon consists of at least...

John Rhodes
August 15, 2023
Computational Mathematics

$(\text{prime}-k)$ Factorial

For a prime $p$ let $S(p) = (\sum (p-k)!) \bmod (p)$ for $1 \le k \le 5$. For example, if $p=7$, $(7-1)! + (7-2)! + (7-3)! +...

John Rhodes
August 15, 2023
Computational Mathematics

Amazing Mazes!

An $m \times n$ maze is an $m \times n$ rectangular grid with walls placed between grid cells such that there is exactly one path from the...

John Rhodes
August 15, 2023
Computational Mathematics

Least Common Multiple Count

Let $f(n)$ be the number of couples $(x, y)$ with $x$ and $y$ positive integers, $x \le y$ and the least common multiple of $x$ and $y$...

John Rhodes
August 15, 2023
Computational Mathematics

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...

John Rhodes
August 15, 2023
Computational Mathematics

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: $5$, $14$, $23$,...

John Rhodes
August 15, 2023
Computational Mathematics

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 $C$:...