2d:null 2e:["$","title",null,{"children":"Royal Game of Ur positional risk calculator | Max Bo | Observable"}] 33:T824,# *Royal Game of Ur* positional risk calculator **The Royal Game of Ur**, also known as the **Game of Twenty Squares**, is a two-player strategy race board game that was first played in ancient Mesopotamia during the early third millennium BC. The game was popular across the Middle East among people of all social strata and boards for playing it have been found at locations as far away from Mesopotamia as Crete and Sri Lanka. At the height of its popularity, the game acquired spiritual significance, and events in the game were believed to reflect a player's future and convey messages from deities or other supernatural beings. The Game of Ur remained popular until late antiquity, when it stopped being played, possibly evolving into, or being displaced by, a later form of tables game. It was eventually forgotten everywhere except among the Jewish population of the Indian city of Kochi, who continued playing a version of it called 'Asha' until the 1950s when they began emigrating to Israel. The Game of Ur received its name because it was first rediscovered by the English archaeologist [Sir Leonard Woolley](https://en.wikipedia.org/wiki/Leonard_Woolley) during his excavations of the [Royal Cemetery at Ur](https://en.wikipedia.org/wiki/Royal_Cemetery_at_Ur) between 1922 and 1934. Copies of the game have since been found by other archaeologists across the Middle East. The rules of the Game of Ur as it was played in the second century BC have been preserved on a Babylonian clay tablet written by the scribe Itti-Marduk-balāṭu. Based on this tablet and the shape of the gameboard, British Museum curator [Irving Finkel](https://en.wikipedia.org/wiki/Irving_Finkel) reconstructed the basic rules of how the game might have been played. The object of the game is to run the course of the board and bear all one's pieces off before one's opponent. Like modern [backgammon](https://en.wikipedia.org/wiki/Backgammon), the game combines elements of both strategy and luck. **Source**: [*Royal Game of Ur - Wikipedia*](https://en.wikipedia.org/wiki/Royal_Game_of_Ur)34:T8c3,{ const data = [...board, ...allJumpProbabilities]; return Plot.plot({ x: { domain: d3.range(0, 8) }, y: { domain: [-1, 1] }, color: { label: "Jumping from", domain: board.map((d) => [d.x, d.y].toString()) // make colors stable }, caption: "Probabilities of both white and black being able to jump to a given square. White is up, and black is down. Bars of the same color indicate the piece that contributed to the probability density.", padding: 0.03, width, height: width / 3, facet: { data, y: "y" }, fy: { domain: [0, 1, 2] }, marks: [ // Plot.frame(), Plot.cell(data, { x: "x", fill: "lightgrey", filter: (d) => d.type && ["standard", "rosette"].includes(d.type), stroke: "black" // facet: "exclude" }), Plot.barY( data, Plot.stackY({ x: "x", filter: (d) => d.by, y: "probability", stroke: "black", strokeWidth: 0.5, fill: (d) => d.by && [d.by?.x, d.by?.y].toString() }) ), Plot.text( data, Plot.groupX( { text: "sum" }, { x: "x", fill: "black", filter: (d) => d.by?.occupant === WHITE, stroke: "white", text: "probability", dy: -20 } ) ), Plot.text( data, Plot.groupX( { text: "sum" }, { x: "x", fill: "black", filter: (d) => d.by?.occupant === BLACK, stroke: "white", text: "probability", dy: 20 } ) ), Plot.text(data, { x: "x", stroke: (d) => [d.x, d.y].toString(), strokeWidth: 4, fontSize: PIECE_SIZE, text: (d) => d.occupant }), Plot.text(data, { x: "x", stroke: (d) => [d.x, d.y].toString(), strokeWidth: 6, fontSize: PIECE_SIZE - 20, text: (d) => d.occupant && "⬤" }) ].map((mark) => on(mark, { click: (event, { datum }) => handleCellClick(event, datum) }) ) }); }35:T464,function jumpProbabilities({ path, positions }) { // If there's nothing on the board the probability of landing on every space is 0 const base = _.tail(path).map((position) => ({ ...position, probability: 0 })); const jumps = positions.flatMap((position) => { // From the current postiton, get the current position and the rest of the path. const remainingPath = _.dropWhile( path, (pathPosition) => pathPosition.x !== position.x || pathPosition.y !== position.y ); // Accumulate the probability distribution of landing on the positions // of the path. const pathProbabilities = _.zip(remainingPath, probabilityDistribution) .flatMap(([pathPosition, probability]) => ({ ...pathPosition, // which position are we jumping from by: position, probability: probability || 0 })) // don't overflow the path .filter( (position) => position.x !== undefined && position.y !== undefined ); return include0Rolls ? pathProbabilities : _.tail(pathProbabilities); }); return [...base, ...jumps]; }36:T6df,function probabilityMap({ path, positions, legend, yDomain, caption }) { const probabilities = jumpProbabilities({ path, positions }); return Plot.plot({ height: width * (2 / 8), width, caption, padding: 0.02, color: { scheme: "viridis", type: "linear", legend, label: "Probability →", domain: [0, 1] }, y: { domain: yDomain }, marks: [ on( Plot.cell( probabilities, Plot.group( { fill: "sum" }, { x: "x", y: "y", fill: "probability" } ) ), { click: handleCellClick } ), on( Plot.text( probabilities, Plot.group( { text: "sum" }, { x: "x", y: "y", fill: "black", stroke: "white", text: "probability" } ) ), { click: handleCellClick } ), on( Plot.text(board, { x: "x", y: "y", dy: -10, fontSize: PIECE_SIZE, text: "occupant" }), { click: handleCellClick } ), Plot.text([_.last(path)], { x: "x", y: "y", dy: -10, fill: "black", stroke: "white", text: (d) => "End" }), on( Plot.text([_.first(WHITE_PATH), _.first(BLACK_PATH)], { x: "x", y: "y", dy: -10, fill: "black", stroke: "white", text: (d) => "Start" }), { click: handleCellClick } ) ] }); }37:[{"id":"857ce9b7827d505fe0191fbeab71ab49297ca1c87e679880c0e25ccb08570ea8c0d54e11c9baea8cfca0b7ce9f4aec18c12f0f921e39170c93bcf32defb054a7","href":"https://static.observableusercontent.com/files/857ce9b7827d505fe0191fbeab71ab49297ca1c87e679880c0e25ccb08570ea8c0d54e11c9baea8cfca0b7ce9f4aec18c12f0f921e39170c93bcf32defb054a7","name":"dice.jpg","type":"image/jpeg","lastModified":1646611320603.282,"size":801434}] 31:{"id":"5ccadb9e0667887c","slug":"royal-game-of-ur-calculator","title":"Royal Game of Ur positional risk calculator","thumbnail":"30e89aa9aeb040bbed6c591592c23f1a370311e4ad7356855927bfb7bd048528","access_level":"public","create_time":"$D2022-03-06T05:07:39.740Z","update_time":"$D2023-05-21T11:56:21.910Z","view_time":null,"first_public_time":"$D2022-03-06T05:08:46.194Z","fork":false,"stars":5,"starred":false,"owner":{"type":"team","id":"cc48459e4293f942","login":"mjbo","name":"Max Bo","avatar_url":"https://avatars-next.observableusercontent.com/05ae9baba40510a4a22fa7d179f29dd4ffd85fcff859c54a31d42e166ae61dfa","tier":"starter_2024"},"version":1128,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"5bb83a6e9493ea1d","login":"mjbo","name":"Max Bo","avatar_url":"https://avatars-next.observableusercontent.com/05ae9baba40510a4a22fa7d179f29dd4ffd85fcff859c54a31d42e166ae61dfa","tier":null},{"type":"individual","id":"6c3841713e9a43b3","login":"shupes","name":"David","avatar_url":"https://avatars-next.observableusercontent.com/e76e0fd4140e787f93e9fe7d82238ed81c53ef426e99c2014144321bb7a85b8c","tier":null}],"members":[{"type":"individual","id":"5bb83a6e9493ea1d","login":"mjbo","name":"Max Bo","avatar_url":"https://avatars-next.observableusercontent.com/05ae9baba40510a4a22fa7d179f29dd4ffd85fcff859c54a31d42e166ae61dfa","tier":null,"role":"editor"},{"type":"individual","id":"6c3841713e9a43b3","login":"shupes","name":"David","avatar_url":"https://avatars-next.observableusercontent.com/e76e0fd4140e787f93e9fe7d82238ed81c53ef426e99c2014144321bb7a85b8c","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":{"id":"a5615d262872a60a","slug":"misc","title":"Misc","owner":{"id":"cc48459e4293f942","login":"mjbo","name":"Max Bo"}},"type":"notebook","body":{"title":"Royal Game of Ur positional risk calculator","cells":[{"id":16,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6,"value":"

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":421,"value":"There are two primary variants of the Royal Game of Ur and a series of further complex modifications to serve a wide range of purposes. For simplicity's sake, we will narrow our view to the most widely-known Mesopotamian variant of the game, one consisting of two single short laps. \n\nThe game board is composed of two 4-square *'wings'*, a central 8-square *'aisle'*, and two 2-square *'bridges'*, giving us a total of 20 game squares. Two players each posses 7 tokens, divided between black and white tokens. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":684,"value":"## Board overview\n\nAccording to [Finkel](https://en.wikipedia.org/wiki/Irving_Finkel)'s reconstruction, if a piece is located on the space with the rosette, it is safe from capture. Finkel also states that when a piece lands on any of the three rosettes, the player gets an extra roll. In order to remove a piece from the board, a player must roll exactly the number of spaces remaining until the end of the course plus one. If the player rolls a number any higher or lower than this number, they may not remove the piece from the board.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":396,"value":"viewof showWhitePath = Inputs.toggle({\n label: \"Show path of white pieces\",\n default: false\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":403,"value":"viewof showBlackPath = Inputs.toggle({\n label: \"Show path of black pieces\",\n value: true\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":509,"value":"viewof reset = Inputs.button(\"Reset\", {\n reduce: () => {\n mutable board = DEFAULT_BOARD;\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"overviewPlot = Plot.plot({\n height: 275,\n caption: \"Click a tile to cycle through placed pieces\",\n padding: 0.02,\n color: {\n legend: true,\n scheme: \"set2\",\n domain: [\"standard\", \"rosette\"]\n },\n marks: [\n Plot.cell(board, {\n x: \"x\",\n y: \"y\",\n fill: \"type\"\n }),\n Plot.text(board, {\n x: \"x\",\n y: \"y\",\n fontSize: 24,\n text: \"occupant\"\n }),\n Plot.line(showWhitePath ? WHITE_PATH : [], {\n x: \"x\",\n y: \"y\",\n markerEnd: \"arrow\",\n stroke: \"#AAAAAA\"\n }),\n Plot.line(showBlackPath ? BLACK_PATH : [], {\n x: \"x\",\n y: \"y\",\n markerEnd: \"arrow\"\n }),\n Plot.text([_.last(WHITE_PATH), _.last(BLACK_PATH)], {\n x: \"x\",\n y: \"y\",\n dy: -10,\n text: (d) => \"End\"\n }),\n Plot.text([_.first(WHITE_PATH), _.first(BLACK_PATH)], {\n x: \"x\",\n y: \"y\",\n dy: -10,\n text: (d) => \"Start\"\n })\n ].map((mark) => on(mark, { click: handleCellClick }))\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":227,"value":"WHITE = \"🔘\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"BLACK = \"⚫\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":135,"value":"WHITE_PATH = [\n { x: 4, y: 0 },\n { x: 3, y: 0 },\n { x: 2, y: 0 },\n { x: 1, y: 0 },\n { x: 0, y: 0 },\n { x: 0, y: 1 },\n { x: 1, y: 1 },\n { x: 2, y: 1 },\n { x: 3, y: 1 },\n { x: 4, y: 1 },\n { x: 5, y: 1 },\n { x: 6, y: 1 },\n { x: 7, y: 1 },\n { x: 7, y: 0 },\n { x: 6, y: 0 }, \n { x: 5, y: 0 }\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":137,"value":"BLACK_PATH = [\n { x: 4, y: 2 },\n { x: 3, y: 2 },\n { x: 2, y: 2 },\n { x: 1, y: 2 },\n { x: 0, y: 2 },\n { x: 0, y: 1 },\n { x: 1, y: 1 },\n { x: 2, y: 1 },\n { x: 3, y: 1 },\n { x: 4, y: 1 },\n { x: 5, y: 1 },\n { x: 6, y: 1 },\n { x: 7, y: 1 },\n { x: 7, y: 2 },\n { x: 6, y: 2 },\n { x: 5, y: 2 }\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":504,"value":"DEFAULT_BOARD = [\n { x: 0, y: 0, type: \"rosette\" },\n { x: 1, y: 0, type: \"standard\" },\n { x: 2, y: 0, type: \"standard\" },\n { x: 3, y: 0, type: \"standard\" },\n { x: 4, y: 0, type: \"off board\", occupant: WHITE },\n { x: 5, y: 0, type: \"finish\" },\n { x: 6, y: 0, type: \"rosette\" },\n { x: 7, y: 0, type: \"standard\" },\n { x: 0, y: 1, type: \"standard\" },\n { x: 1, y: 1, type: \"standard\" },\n { x: 2, y: 1, type: \"standard\" },\n { x: 3, y: 1, type: \"rosette\" },\n { x: 4, y: 1, type: \"standard\" },\n { x: 5, y: 1, type: \"standard\" },\n { x: 6, y: 1, type: \"standard\" },\n { x: 7, y: 1, type: \"standard\" },\n { x: 0, y: 2, type: \"rosette\" },\n { x: 1, y: 2, type: \"standard\" },\n { x: 2, y: 2, type: \"standard\" },\n { x: 3, y: 2, type: \"standard\" },\n { x: 4, y: 2, type: \"off board\", occupant: BLACK },\n { x: 5, y: 2, type: \"finish\" },\n { x: 6, y: 2, type: \"rosette\" },\n { x: 7, y: 2, type: \"standard\" }\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"mutable board = DEFAULT_BOARD","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":201,"value":"import { on } from \"@fil/plot-onclick-experimental-plugin\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":204,"value":"handleCellClick = (event, datum) => {\n console.debug(event, datum);\n const newBoard = board.map((cell) => {\n // Don't modify a cell unless it's the one that's clicked\n if (cell.x !== datum.x || cell.y !== datum.y) {\n return cell;\n }\n\n // Different rows have different toggling behaviours\n const lookup = {\n // top row\n 0: {\n undefined: WHITE,\n [WHITE]: undefined\n },\n // middle row\n 1: {\n undefined: WHITE,\n [WHITE]: BLACK,\n [BLACK]: undefined\n },\n // bottom row\n 2: {\n undefined: BLACK,\n [BLACK]: undefined\n }\n };\n\n return { ...cell, occupant: lookup[cell.y][cell.occupant] };\n });\n\n mutable board = newBoard;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":389,"value":"## Roll probability","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":665,"value":"

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":660,"value":"Movements are determined by rolling a set of [four-sided, tetrahedron-shaped dice](https://en.wikipedia.org/wiki/Four-sided_die). Two of the four corners of each die are marked and the other two are not, giving each die an equal chance of landing with a marked or unmarked corner facing up. The number of marked ends facing upwards after a roll of the dice indicates how many spaces a player may move during that turn.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":539,"value":"probabilityPlot = {\n const data = probabilityDistribution.map((d, i) => ({\n probability: d,\n roll: i\n }));\n\n return Plot.plot({\n color: {\n scheme: \"viridis\",\n type: \"linear\",\n legend: true,\n label: \"Probability →\",\n domain: [0, 1]\n },\n y: {\n label: \"↑ Probability\",\n domain: [0, 1]\n },\n x: {\n label: \"Roll\"\n },\n grid: true,\n marks: [\n Plot.barY(data, {\n x: \"roll\",\n y: \"probability\",\n fill: \"probability\"\n }),\n Plot.text(data, {\n x: \"roll\",\n y: \"probability\",\n dy: 10,\n fill: \"#FFFFFF\",\n text: \"probability\"\n })\n ]\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"pascalsTriangle5thRow = [1, 4, 6, 4, 1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":115,"value":"probabilityDistribution = pascalsTriangle5thRow.map(\n (v) => v / _.sum(pascalsTriangle5thRow)\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":443,"value":"## Probability of a jump to a given square","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":588,"value":"viewof include0Rolls = Inputs.toggle({\n label: \"Include 0-rolls in cumulative probability\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":522,"value":"Inputs.button(\"Reset\", {\n reduce: () => {\n mutable board = DEFAULT_BOARD;\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":717,"value":"*Click a tile to cycle through placed pieces*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":333,"value":"whiteMap = probabilityMap({\n path: WHITE_PATH,\n caption: \"Probability of white being able to jump to a given square\",\n positions: whitePositions,\n legend: true,\n yDomain: [0, 1]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":769,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":357,"value":"blackMap = probabilityMap({\n path: BLACK_PATH,\n caption: \"Probability of black being able to jump to a given square\",\n positions: blackPositions,\n legend: false,\n yDomain: [1, 2]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1002,"value":"PIECE_SIZE = 25","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":237,"value":"whitePositions = board.filter((cell) => cell.occupant === WHITE)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":339,"value":"blackPositions = board.filter((cell) => cell.occupant === BLACK)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":239,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":755,"value":"allJumpProbabilities = [\n ...jumpProbabilities({ path: WHITE_PATH, positions: whitePositions })\n .filter((d) => d.probability)\n .map((d) => ({ ...d })),\n ...jumpProbabilities({ path: BLACK_PATH, positions: blackPositions })\n .filter((d) => d.probability)\n .map((d) => ({ ...d, probability: -d.probability }))\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":331,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W37","stdlib":"1","titling":"auto"}} 32:"$31:body"