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3Blue1Brown poses a binary-encoding puzzle about a robot on a distant moon. The challenge is to find the most efficient encoding and prove it optimal, an introduction to Shannon entropy; the short does not provide the solution.
- 1Four directions, unequal frequenciesThe robot follows a biased random walk: up one half, down one quarter, and left and right one eighth each.
- 2Find an efficient codeEncode the instructions in binary while minimizing bits per instruction.
- 3Prove optimalityShow that no other encoding can improve on your answer; the video leaves the proof as a challenge.
Conclusiones
- The robot moves up half the time, down one quarter, and left or right one eighth each.
- The first challenge is to encode the instructions in binary using as few bits per instruction as possible.
- The second is to prove no other encoding can do better, leading toward Shannon entropy.
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A robot on a distant moon
The robot can move in four directions with unequal frequencies.
Imagine I have sent a robot to a far-away moon, and we want to send it movement instructions from Earth. Each instruction is simple: step up, down, left or right.
The instructions make the robot follow a biased random walk. Half its movements are up, one quarter are down, and one eighth each are left and right.
The encoding challenge
Find the most efficient binary code and prove that it cannot be beaten.
Here is the two-part puzzle. First, find the most efficient way to encode these instructions in binary, using as few bits per instruction as possible. For challenge mode, prove that your answer really is optimal: no other encoding can do better.
The connection to entropy
The puzzle is a first step toward Shannon entropy; the linked full video explains further.
If you can answer that, you have taken a first step toward reinventing Shannon entropy. The full video, the first in a series on the phrase “compression is intelligence,” explains the details. This short poses the question without giving the solution.
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