Expert Solutions | Senioritis

John Rhodes
August 16, 2023
Computational Mathematics

Distinct Rows and Columns

The complexity of an $n\times n$ binary matrix is the number of distinct rows and columns. For example, consider the $3\times 3$ matrices $$ \mathbf{A} = \begin{pmatrix}...

John Rhodes
August 16, 2023
Computational Mathematics

Feynman Diagrams

Let $F(n)$ be the number of connected graphs with blue edges (directed) and red edges (undirected) containing: two vertices of degree 1, one with a single outgoing...

John Rhodes
August 16, 2023
Computational Mathematics

Toriangulations

For positive real numbers $a,b$, an $a\times b$ torus is a rectangle of width $a$ and height $b$, with left and right sides identified, as well as...

John Rhodes
August 16, 2023
Computational Mathematics

Prime Factor and Exponent

For a positive integer $n \gt 1$, let $p(n)$ be the smallest prime dividing $n$, and let $\alpha(n)$ be its $p$-adic order, i.e. the largest integer such...

John Rhodes
August 16, 2023
Computational Mathematics

If $a,b$ are two nonnegative integers with decimal representations $a=(\dots a_2a_1a_0)$ and $b=(\dots b_2b_1b_0)$ respectively, then the freshman’s product of $a$ and $b$, denoted $a\boxtimes b$, is...

John Rhodes
August 16, 2023
Computational Mathematics

Digit Sum Division

For a positive integer $n$, $d(n)$ is defined to be the sum of the digits of $n$. For example, $d(12345)=15$. Let $\displaystyle F(N)=\sum_{n=1}^N \frac n{d(n)}$. You are...

John Rhodes
August 16, 2023
Computational Mathematics

Saving Paper

When wrapping several cubes in paper, it is more efficient to wrap them all together than to wrap each one individually. For example, with 10 cubes of...

John Rhodes
August 16, 2023
Computational Mathematics

Conjunctive Sequences

Let ‘$\&$’ denote the bitwise AND operation. For example, $10\,\&\, 12 = 1010_2\,\&\, 1100_2 = 1000_2 = 8$. We shall call a finite sequence of non-negative integers...