31:T612,# The best opening guess in Wordle I was inspired by 3Blue1Brown's video on Solving [Wordle](https://www.nytimes.com/games/wordle/index.html) with information theory to have a go at writing a similar Wordle solver in python. Whilst I wrote my solver in python, I've been meaning to try out Observable notebooks some more lately, so here are a few things I learned. ## What's the best opening guess in Wordle? The goal in Wordle is to guess the word in as few tries as possible, so a good strategy (at least early on) is to try and guess something that when scored will rule out a lot of possible words. If we guess something like "jazzy", unless our word contains a j or a z (unlikely) then our score is unlikely to narrow down our options a lot. We can be more rigorous about how much a guess narrows down the space of possible guesses with some information theory. This is what the 3Blue1Brown video was about, so I recommend you watch that.
## Information Content of a Score We can quantify the degree to which a score narrows down our options using the number of possible answers before and after. The ratio of these is the probability of getting that score - scores with a high probability of occurring are those that many answers would scored our guess with. Choose a probability below: 32:[{"id":"55efbde5049a20dccce56723fdbe8c972860463f9f6b03addce4dca88158031314da4b628babf5a44427ad9c180eca4cafda9e7bbaf22594c164b174ca9c3478","href":"https://static.observableusercontent.com/files/55efbde5049a20dccce56723fdbe8c972860463f9f6b03addce4dca88158031314da4b628babf5a44427ad9c180eca4cafda9e7bbaf22594c164b174ca9c3478","name":"adieu-scores@1.json","type":"application/json","lastModified":1651871072564.211,"size":113765},{"id":"211b0761823e7d513255fdba26cce5c1bdc2b07ba34df9037e0e06dac47563e1c4350d374192375ebba58532680124c32dc1d96476096f36ee9fc4b83a710fb6","href":"https://static.observableusercontent.com/files/211b0761823e7d513255fdba26cce5c1bdc2b07ba34df9037e0e06dac47563e1c4350d374192375ebba58532680124c32dc1d96476096f36ee9fc4b83a710fb6","name":"guess_values.json","type":"application/json","lastModified":1651874122633.968,"size":699040}] 2f:{"title":"The best opening guess in Wordle","cells":[{"id":0,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":218,"value":"viewof prob = Inputs.range([0,1.0], {label: \"Score Probability\", format: d3.format(\".2f\")})","pinned":false,"hidden":false,"mode":"ojs"},{"id":203,"value":"{\n const cols = d3.schemeCategory10;\n return html``\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":488,"value":"Then, we can express the information content of that probability this as the number of bits of information (I) the score gives us. A lower probability means we'll eliminate more potential answers if we get that score, but of course it's also less likely to occur - in which case we'll get some other amount of information from a different score.\n\n ${tex`I(score) = log_{2}(\\frac{1}{p(score)})`}\n ${tex`= log_{2}(\\frac{1}{ ${d3.format('.2f')(prob)} }) = ${d3.format('.2f')(Math.log2(1/prob))} \\text{ bits of information} `}\n\nSo if we calculate a weighted average of _all_ the information values for each score (weighted by the probability of getting that score) a guess could receive, we get the expected information value of a guess as a whole.\n\n${ tex`E(guess) = \\sum p(score)\\times I(score) = \\sum p(score)\\times log_2(\\frac{1}{p(score)})` }\n\nHere's the probability of each possible score for guessing the word _'adieu'_ as our first guess, along with some of the actual answers that would result in that score.","pinned":false,"hidden":false,"mode":"md"},{"id":255,"value":"{\n const adieuScorePlot = Plot.plot({\n height: adieuScores.length * 15,\n width: width,\n marginLeft: 150,\n grid: true,\n title: \"Most Useful Scores after guessing \\\"adieu\\\" (smaller is better)\",\n x: {\n axis: \"top\",\n label: \"Probability\",\n labelAnchor: \"center\",\n domain: [0, 0.25]\n },\n y: {\n axis: null,\n domain: d3.sort(adieuScores, d => -d.probability).map(d => d.score)\n },\n marks: [\n Plot.barX(adieuScores, {\n y: \"score\",\n x: \"probability\",\n }),\n Plot.text(adieuScores, {\n y: \"score\",\n x: 0,\n text: \"score\",\n textAnchor: \"end\",\n }),\n Plot.text(adieuScores, {\n y: \"score\",\n x: \"probability\",\n text: d => `${d.possible.slice(0,10).join(', ')} ${d.possible.length > 10 ? '...' : ''}`,\n textAnchor: \"start\",\n dx: 10\n }),\n ]\n })\n\n return html`

${adieuScorePlot}

`\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":521,"value":"\nAnd on average that gives us an expected value of:\n\n${ tex`E(adieu) = ${d3.format('.2f')(d3.sum(adieuScores, d => d.probability * Math.log2(1/d.probability)))} \\text { bits of information}` }\n\nWhich means guessing _'adieu'_ narrows down our initial pool of guesses by around a factor of ${tex`2^5`} (${2**5}), assuming all guesses are equally likely.\n","pinned":false,"hidden":false,"mode":"md"},{"id":395,"value":"## Whats the best first guess then?\nThis brings us to the question of what is the \"best\" opening guess. Based on calculating the expected information value of _every_ possible guess, the distribution of guesses can be seen in the chart below. The \"best\" are likely to eliminate a large number of possibilities (cutting the space of remaining answers down by a factor of about ${tex`2^6=64`}). Our average expected information value is ${d3.format('.1f')(d3.mean(guessValues, d => d.expected_information))} bits.","pinned":false,"hidden":false,"mode":"md"},{"id":428,"value":"{\n const guessInfoPlot = Plot.plot(\n {\n height: 500,\n width: 600,\n marginLeft: 40,\n grid: true,\n x: {\n axis: \"bottom\",\n label: \"Guess Rank\",\n labelAnchor: \"center\",\n domain: [0, guessValues.length]\n },\n y: {\n axis: \"left\",\n label: \"Expected Information (bits)\",\n labelAnchor: \"center\",\n domain: [0, 7]\n },\n color: {\n domain: ['main', 'avg'],\n range: [\"black\", \"#aaa\"]\n },\n marks: [\n Plot.line(guessValues, {\n x: (d, i) => i,\n y: 'expected_information',\n stroke: () => 'main'\n }),\n Plot.line(Array(2).fill(d3.mean(guessValues, d => d.expected_information)), {\n x: [0, guessValues.length],\n y: (d, i) => d,\n stroke: () => 'avg'\n })\n ]\n }\n )\n return html`

${guessInfoPlot}

`\n}\n\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":445,"value":"And the top 10 guesses using this metric are:\n - ${guessValues.slice(0,10).map(d => `_${d.guess}_: ${d3.format('.2f')(d.expected_information)} bits`).join('\\n - ')}\n\nIf you watch 3Blue1Brown's video you'll see this is the list he gets at around 14:55. So tares with 6.19 bits is a little over twice as good as adieu in narrowing down the available options. \n\nInterestingly the _worst_ possible guesses still provide about 2 bits of information. These abound with repeated letters, reducing the amount of information the score is likely to give you. Most read like the words you're always trying to remember when playing Scrabble, though there aren't enough 'z's for 'pzazz' (or even 'fuzzy'). 'fluff' with three 'f's may be one of the worst guesses, but it still eliminates ${d3.format('.0%')(1 - 1/(2**2.35))} of the available options...\n\n - ${guessValues.slice(-10).map(d => `_${d.guess}_: ${d3.format('.2f')(d.expected_information)} bits`).join('\\n - ')}\n","pinned":false,"hidden":false,"mode":"md"},{"id":183,"value":"## Appendix:","pinned":false,"hidden":false,"mode":"md"},{"id":262,"value":"adieuScores = {\n const adieuScores = FileAttachment(\"adieu-scores@1.json\").json();\n return adieuScores.then(data => data.map((d,i,a) => {\n return {...d, probability: d.count / d3.sum(a, d => d.count)}\n }));\n};\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":423,"value":"guessValues = FileAttachment(\"guess_values.json\").json()","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}