Kevin Gomez

Kevin Gomez

he/him

Research Scientist

Teensy Problems: Alien Astronomer (Part 2)

1 minute read

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.

However, upon loading the data, Gryxlb notices that their USB drive must be malfunctioning too, because instead of the signals counting up from 0, they’re all jumbled. Upon closer inspection of the underlying bits, they notice that every signal has had the transformation 01 -> 100 applied to its binary representation. For example, signal 5 = 101b became 1[01] -> 1[100] = 1100b = 12.

Intrigued by the chaos, Gryxlb decides to re-export the data onto the same drive, applying the bit transformation once again. For the same example, 12 stays at 12, since 01 doesn’t appear in its binary representation. After doing so a few more times, they’ve noticed that most of the earlier signals have settled down to a fixed point. Let’s figure out what’s going before Gryxlb does this for the rest of the day. Denote this transformation by $B(x)$.

  1. Prove that $B(x) \geq x$ for all $x \geq 0$.

  2. Prove that all $x \geq 0$ eventually stabilize to a fixed point under $B$.

  3. Mathematically describe the fixed points of $B$.

  4. Does $B$ have an inverse? Can the original signal data be recovered?

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Teensy Problems: Alien Astronomer (Part 1)

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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.

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: $0! + 1! + \dots + n!$. Its first few values are $1$, $2$, $4$, $10$, $34$, ….

Teensy Problems: Falling Leaves

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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.