Kevin Gomez

Kevin Gomez

he/him

Research Scientist

Teensy Problems: Alien Astronomer

1 minute read

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.

For example, if the last memory slot holds 4 (in decimal, as Gryxlb’s species also happens to have ten total digits across its forelimbs) and a pulse encoding 5 arrives, the value 4 ^ 5 = 100b ^ 101b = 001b = 1 is stored in the next slot.

Of course, Gryxlb will be reporting this Xorgmathrx-shattering discovery to the proper authorities, but they’re currently too fascinated by this malfunction. They want to figure out the structure of the resulting memory; luckily, they know that the very first memory slot held 0 before it started recording the pulses.

  1. Find the values of the first 32 malfunctioning memory slots.

  2. Find a pattern in the memory slots after many pulses have been recorded, then prove your pattern will hold indefinitely.

  3. How is the pattern altered if the first memory slot wasn’t 0?

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Teensy Problems: Kaprekar Numbers

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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.

Teensy Problems: Sum of Factorials

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Let $\mathrm{sf}(n)$ be the sum-of-factorials function: $0! + 1! + … n!$. Its first few values are $1$, $2$, $4$, $10$, $34$, ….

Teensy Problems: Falling Leaves

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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.

Teensy Problems: Coprime Labeling

1 minute read

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.