Articles of problem solving

How many $a$-nary sequences of length $b$ never have $c$ consecutive occurrences of a digit?

Let $S(a,b,c): = \#\{a$-nary sequences of length $b$ without $c$ consecutive occurrences of a digit$\}$. For example, $S(2,n,3)$ would be the number of binary sequences of length $n$ without $3$ consecutive occurences of a $0$ or $1$. In particular, $S(2,n,3) = 2f_{n+1}$, twice the $(n+1)$st Fibonacci number. A bit more about this example can be […]

Greatest common divisor of $3$ numbers

Let $a,b, c$ belong to $\mathbb Z$ such that $(a,b,c) \neq (0,0,0)$. Define the [highest common factor] greatest common divisor ${\rm gcd}(a, b, c)$ to be the largest positive integer that divides $a, b$, and $c$. Prove that there are integers $s, t, u$, such that $${\rm gcd}(a, b, c) = sa + tb + […]

Determine the angle of 3 drawn lines from each corner of 3 congruent squares

Three squares are drawn next to each other. Three lines are drawn from a corner as illustrated. Determine the sum of the three angles exposed (the exact number of degrees or radians):

$0 = \left(\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right)^2 -k^2-m^2$ solving for $k$

This question is related to $\delta(f(k))$ concerning the Dirac-delta. OK I know this might seem trivial but the result is very very important to me so I want to check with you if my logic seems correct. The equation in $k$ is $0 = \left(\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right)^2 -k^2-m^2, (1)$ The parameters satisfy: $p,m>0, k≥0$ and […]

Accumulation points of $\{ \sqrt{n} – \sqrt{m}: m,n \in \mathbb{N} \}$

This is my first post on MSE, so, pardon me if I’m not used to the site’s rule yet. I’m trying to prepare myself for competitions in the future and I’m trying to improve my problem solving skills. I’ve come accross this problem from the book “Problems in mathematical analysis” by Witkowski and Piotr. It […]

“Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it’s acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and KH is the remainder. KHHR represents P times KMS and ALH is KRRA minus KHHR; the E is […]

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I’m looking for a “best fit” transformation consisting of a rotation and a translation. I.e. $$ \min_{A,b} \sum (A p_i + b – q_i)^2 , […]

For which $n, k$ is $S_{n,k}$ a basis? Fun algebra problem

Here it is a nice algebra problem I had some fun with Let $V$ be a vector space over $\mathbb R$ of finite dimension $\dim V = n$. Let $v = \{ v_1, \dots, v_n\}$ be a basis for $V$. Let $$S_{n,k} = \{ v_1 + v_2 + \dots + v_k, v_2 + \dots + […]

Calculating Non-Integer Exponent

I just wanted to directly calculate the value of the number $2^{3.1}$ as I was wondering how a computer would do it. I’ve done some higher mathematics, but I’m very unsure of what I would do to solve this algorithmically, without a simple trial and error. I noted that $$ 2^{3.1} = 2^{3} \times 2^{0.1} […]

Looking for a simple problem for math demonstration

I’m holding a 3-5 minute speech next week on mathematical problem solving, and how it makes me happy, to 15-20 non-mathematicians. As a part of it, I had thought about demonstrating two problems, but I can only come up with one. That would be the following: I shuffle a deck of cards and deal each […]