Published:
Recall that a tree is an undirected network with no cycles or loops such that any node is reachable from any other. A labeled tree is one where we’ve given each node a unique label, by convention a number between $1$ and $n$ for $n$ the number of nodes in the tree. Two labeled trees are the same if they can be drawn identically, with the same labels and connections.
Published:
Thanks to you, Gryxlb now has a satisfactory picture of what that malfunctioning machine got up to, and is ready to prepare their analysis of the newly-coined Orion Object cOunting interceptiOn (OOOO) for the wider scientific community on Xorgmathrx. They copy the raw interception data from a (functioning) computer onto a USB drive (it’s not called “universal serial bus” for nothing) to take to the Tbibthglx University Library, whose workstations have the analysis software Gryxlb likes to use.
Published:
Gryxlb, an alien radio astronomer on planet Xorgmathrx, has started intercepting a peculiarly structured signal. There’s a stream of pulses originating from the Orion Arm counting up from zero in binary; it must be a deliberate message from an intelligent civilization! The telescope’s computers are recording the pulses, but one is malfunctioning: instead of writing the pulses into memory, each memory slot is the bitwise exclusive or (xor) of the incoming pulse and the previous memory slot.
Published:
Dr. Kaprekar has stumbled upon an interesting class of numbers. A number $N$ is called a “Kaprekar number” if $N^2$, written in base $10$, can be split into two parts, $A$ and $B$, such that $N = A + B$. Here are some examples:
9² = 81 → 8 + 1 = 9
45² = 2025 → 20 + 25 = 45
4879² = 23804641 → 238 + 04641 = 4879
Notice that the parts do not have to be equal in size and can include leading zeros; the only restriction is that neither part is totally zero. Let’s help Dr. Kaprekar understand more about his numbers.
Published:
Let $\mathrm{sf}(n)$ be the sum-of-factorials function: $0! + 1! + \dots + n!$. Its first few values are $1$, $2$, $4$, $10$, $34$, ….
Published:
Autumn and Yvonne love to play “falling leaves” this time of year. They play on a tree containing some number of leaves (possibly none) on its left and right sides. Starting with Autumn, each player takes turn choosing a leaf. All leaves at or below the chosen leaf on the same side of the tree fall to the ground. The game ends once the tree is out of leaves; whoever is the last to choose a leaf is the winner.
Published:
Let $G$ be a graph with $n$ vertices. We say a graph is coprimely labeled if its vertices are labeled using all the integers from 1 to $n$ in such a way where, if any two vertices have an edge between them, their labels are coprime; that is, they share no common factors.
Published:
Marvin is interested in numbers whose sum of their digits equals the product of their digits. Call a nonnegative integer a $b$-Marvin number if it has this property when written in base $b$. Marvin noticed already that all single digits in base $b$ are trivially $b$-Marvin numbers.
Published:
Carol really likes counting, but can only keep track of so many numbers at once. She’s devised a family of sequences that matches these aptitudes, in the following manner: Let $A_k(1) = 1$, then let $A_k(n+1)$ be the number of times $A_k(n)$ appears in the last $k$ terms of $A_k$.
Published:
Finn Factor just learned about a neat way to compute the prime numbers. Let $S$ be the list of known primes (initially empty), and start at $n = 2$. For each $n$, we ask if it is divisible by any elements of $S$. If it isn’t, we add it to $S$; otherwise, $n$ is composite. We increment $n$ and continue, which will place into $S$ every prime up until wherever we decide to stop.