45:["$","title",null,{"children":"Appreciation Post: 3Blue1Brown | Michael Deagen | Observable"}] 4d:Tb1d,## Why Now? Like many others on the Internet, I was intrigued by Wordle and curious about its underlying roots in probability and linguistics. About a month ago, I made [this notebook](https://observablehq.com/@mdeagen/guessing-wordles-96) which achieves a 96% win rate (in hard mode) by simply picking the highest-frequency available word, using a sorted list of 5-letter words curated from the [Datamuse API](https://observablehq.com/@mdeagen/datamuse-api-example). (At the time, I had not explored the Wordle source code and did not know the full list of solutions was readily available, so I made some assumptions for evaluation that turned out to be pretty representative—the algorithm turns out to have a 96.5% success rate on the actual list of Wordle solutions, with an average of 4.40 guesses.) If you use the same "select-highest-frequency-available-word" approach but begin with my preferred starting word ("AROSE"), the algorithm achieves a 98.6% success rate, with an average of 3.96 guesses. Although the earlier notebook/algorithm had a high success rate, I still found it highly unsatisfying since it completely ignores letter frequencies, letter + position frequencies, and the information content contained within the **gray**, **yellow**, and **green** squares returned after each guess. To make things worse, my news feed algorithms seem to think that, despite all else going on in the world, all I want to read about is Wordle-related news stories and blog entries. (I accept some responsibility by having clicked on a few of them.) Despite the abundance of analyses out there, those that truly resonated were [Ali Lloyd's post on Aperiodical](https://aperiodical.com/2022/02/a-mathematicians-guide-to-wordle/), which references [Andrew Taylor's code](https://twitter.com/Andrew_Taylor/status/1480911016582127620) for guaranteeing a 100% success rate by aiming for "how few solutions would give the same clues in the worst case." It was great to see a solution out there, but it still felt like there was something missing in terms of information theory and bits. Enter the [3Blue1Brown video](https://www.youtube.com/watch?v=v68zYyaEmEA). This video is on the long side compared to some of his other videos, but it really has everything from an explanation of the game to an approach based on Shannon entropy and word frequencies. I was blown away by the production value (those familiar with 3Blue1Brown know that Grant makes excellent supporting visuals in his videos) as well as the clarity of presentation and thought. If this were a movie, I would definitely rate it 6 bags of popcorn (🍿🍿🍿🍿🍿🍿) and 2 sodas (🥤🥤). If you like Wordle, math, or educational visualization, check it out.4e:T7f5,## Distilling Complexity Grant Sanderson discusses his philosophies on math and teaching in his [appearance on the Lex Fridman Podcast](https://www.youtube.com/watch?v=U_6AYX42gkU), and discusses his creative process and development of custom code for the various supporting visualizations and animations featured in his videos. One does not need to be a practitioner in information visualization to appreciate the quality of 3Blue1Brown's explanations, but having a little background in data visualization and interaction certainly helps one to appreciate the many design choices that go into structuring and delivering such high-quality educational content. Grant's videos seamlessly mix visuals, language (audio and text), and linked representations to give a sense of interactivity even though the consumption is largely passive by watching the video. In a sense, these videos do a great job capturing Bruner's three types of representations (symbolic, iconic, enactive) referenced in Bret Victor's keynote on [The Humane Representation of Thought](https://vimeo.com/115154289). The ability to mix-and-match representations of data and information is one of the reasons I love the Observable platform. Back to 3Blue1Brown... Grant achieves something with narrated videos which is difficult to capture with a standalone interactive visualization. In an interactive data visualization, the onus is entirely on the user to actively explore the data. Depending on the task at hand, providing an educational scaffold (narrated recording, interlaced Markdown text, etc.) helps to contextualize the problem and assist the user in developing their perspective accordingly. On an inverse note, sometimes a visual aid that hones our perspective becomes instrumental in solving really challenging problems (such as Einstein's notion of a [Gedankenexperiment](https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments)). Anyway, **thank you** Grant Sanderson and 3Blue1Brown for being a prime example of compelling educational content!4f:T466,## Additional Reflections on Wordle Grant's video laid to raid many open questions that I had about Wordle. He covered and explained entropy, bits, and game strategy better than I could have imagined. At [one point about 19.5 minutes into the video](https://youtu.be/v68zYyaEmEA?t=1170) he discusses normalizing word frequencies using a sigmoidal function. As a materials scientist, I had previously considered using a variation on the [Fermi–Dirac distribution](https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics) as a way to accomplish this normalization. The distribution looks at how energy states are filled at a given temperature (where absolute zero looks like a step function), but there are parallels when you are talking about [curating a list of *n* valid Wordle solutions](https://www.nytimes.com/2022/01/03/technology/wordle-word-game-creator.html) which is biased towards more recognizable words. From early analysis of past Wordle solutions, it was clear that the list was not a strict cutoff of the 2300 most common words, but being more frequent certainly improved its chances of being selected.50:T529,VLspec = ({ "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "data": {"values": words_freq_norm}, "width": width*0.75, "transform": [ {"window": [{"op": "row_number", "as": "row_number"}]}, {"calculate": "datum.index + 1", "as": "rank"}, {"filter": "datum.rank <= 10000"}, {"filter": "datum.freq > 0"} ], "layer": [ { "mark": {"type": "bar", "opacity": 0.7}, "encoding": { "x": {"field": "index", "type": "quantitative", "scale": {"type": "linear"}, "bin": {"step": 100}, "title": "Rank of Word (binned)"}, "y": {"aggregate": "count", "scale": {"type": "linear"}}, "color": {"field": "is_valid", "type": "nominal"} } }, { "mark": {"type": "line", "color": "white"}, "encoding": { "x": {"field": "index", "type": "quantitative", "scale": {"type": "linear"}}, "y": {"field": "FermiDirac", "type": "quantitative", "scale": {"type": "linear", "domain": [0,1]}, "title": "Probability (Fermi–Dirac)"}, "tooltip": [ {"field": "word", "type": "nominal"}, {"field": "freq", "type": "quantitative"}, {"field": "rank", "type": "quantitative"}, {"field": "is_valid", "type": "nominal"} ] } } ], "resolve": {"scale": {"y": "independent"}} })51:[{"id":"46a14b0cb85829d046ef4c4ce6e856bd089511e74d8ebeaa207137f177e3be17d8419e50cb59d6333e53a18a7910d0281d11893ca70e19875e317ebf92257393","href":"https://static.observableusercontent.com/files/46a14b0cb85829d046ef4c4ce6e856bd089511e74d8ebeaa207137f177e3be17d8419e50cb59d6333e53a18a7910d0281d11893ca70e19875e317ebf92257393","name":"wordles_valid.json","type":"application/json","lastModified":1644414372880.13,"size":20835}] 49:{"id":"aeda679928020d62","slug":"appreciation-post-3blue1brown","title":"Appreciation Post: 3Blue1Brown","thumbnail":"eed702823a5a3e83a15c2defbae12c0cf831ea758cd3c5f5fbac45eb38d1f0e4","access_level":"public","create_time":"$D2022-02-08T11:48:12.492Z","update_time":"$D2022-02-10T11:14:07.264Z","view_time":null,"first_public_time":"$D2022-02-09T14:57:10.356Z","has_unpublished_edits":null,"fork":false,"stars":18,"starred":false,"owner":{"type":"team","id":"5ca8a3812ea4a326","login":"mdeagen","name":"Michael Deagen","avatar_url":"https://avatars-next.observableusercontent.com/0b8e876e2eac1058faa6c1a2624346f3e72350b8a7e6c9acaec6a9ad920b0476","tier":"starter_2024","bio":"Materials science // data interaction","followed_by_you":false,"follows_you":false},"version":153,"publish_version":153,"role":null,"authored":false,"authors":[{"type":"individual","id":"d3611c0325cb99e9","login":"mdeagen","name":"Michael Deagen","avatar_url":"https://avatars-next.observableusercontent.com/0b8e876e2eac1058faa6c1a2624346f3e72350b8a7e6c9acaec6a9ad920b0476","tier":null,"bio":"Materials science // data interaction","followed_by_you":false,"follows_you":false}],"members":[{"type":"individual","id":"d3611c0325cb99e9","login":"mdeagen","name":"Michael Deagen","avatar_url":"https://avatars-next.observableusercontent.com/0b8e876e2eac1058faa6c1a2624346f3e72350b8a7e6c9acaec6a9ad920b0476","tier":null,"role":"editor"}],"comments":[{"id":"45f5b559881be7b8","reply_to_id":null,"content":"Typo? \"laid to raid\"","create_time":"$D2022-02-10T07:39:58.582Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"07362516b5994994","login":"mootari","name":"Fabian Iwand","avatar_url":"https://avatars-next.observableusercontent.com/0d7defa821f38094c03bad23b9b360a5364e6e97e21fc238c39ddc48db7994ad","tier":null}},{"id":"dd5744704d52f4f6","reply_to_id":"45f5b559881be7b8","content":"Thank you for catching!","create_time":"$D2022-02-10T11:14:25.233Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"d3611c0325cb99e9","login":"mdeagen","name":"Michael Deagen","avatar_url":"https://avatars-next.observableusercontent.com/0b8e876e2eac1058faa6c1a2624346f3e72350b8a7e6c9acaec6a9ad920b0476","tier":null}}],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Appreciation Post: 3Blue1Brown","cells":[{"id":0,"value":"# Appreciation Post: 3Blue1Brown\n\nThe YouTube channel [3Blue1Brown](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw), produced by [Grant Sanderson](https://twitter.com/3blue1brown), delivers fun and informative educational content around mathematical concepts. I have been a subscriber for several years now, and after watching [his most recent video](https://www.youtube.com/watch?v=v68zYyaEmEA) about finding the best approach to win the online word game [Wordle](https://www.powerlanguage.co.uk/wordle/) I felt compelled to express my appreciation for 3Blue1Brown and acknowledge its significant contributions over the years to education and information visualization.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":40,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"$4d","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":10,"value":"$4e","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":14,"value":"$4f","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":95,"value":"F(\\varepsilon) = \\frac{1}{e^{(\\varepsilon - \\mu) / k_{\\rm B}T} + 1}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":84,"value":"The 3Blue1Brown video does not dive super deeply into the parameters of the sigmoidal function, but below is what I used while playing around with this idea a few weeks ago, where the constant *k* describes the shape of the curve. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":100,"value":"F(\\varepsilon) = \\frac{1}{e^{(1 - array\\_length)*k} + 1}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":114,"value":"Turns out, you can model the distribution quite well if your constant is around 0.0007. Try out different constants by adjusting the slider!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":67,"value":"viewof k = Inputs.range([0,0.003], {label: \"k\", \"step\": 0.0001, value: 0.0007})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":75,"value":"vl.render({spec: VLspec})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":130,"value":"Anyway, I thought it was neat that Grant had a similar idea for approaching the word frequency problem, and I wanted to share these findings as an addendum. For a much more thorough treatment of the Wordle problem, I highly recommend watching Grant's video!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":80,"value":"**P.S.** I am happy for Josh Wardle for his [deal with the New York Times to acquire Wordle](https://www.nytimes.com/2022/01/31/business/media/new-york-times-wordle.html), capturing some value from a fun, freely released game that captured such widespread attention. It has been a really fun problem to ponder over the past month, and I am super pleased to see such a high-quality deep dive into the fundamental mathematical concepts by 3Blue1Brown.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":55,"value":"---\n\n## Appendix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"import {word_array} from '@mdeagen/datamuse-api-example'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":61,"value":"words_freq = word_array.map(x => ({word: x.word, freq: Number(x.f)})).sort((a, b) => (a.freq < b.freq) ? 1 : -1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":63,"value":"wordles_valid = await FileAttachment(\"wordles_valid.json\").json()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":64,"value":"words_freq_norm = words_freq.map((x,i) => ({word: x.word, freq: x.freq, index: i, FermiDirac: 1 / (1 + Math.exp((i - wordles_valid.length)*k)), is_valid: wordles_valid.some(y => y === x.word)}))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":73,"value":"$50","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W51","stdlib":"1"}} 44:null