[{"data":1,"prerenderedAt":949},["ShallowReactive",2],{"layer:patterns:extend":3},{"layer":4,"contentHash":926,"dependencyHashes":927,"approval":942,"releaseId":948},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":921,"reviewStatus":922,"authoring":923},1,"patterns","en","extend","Pattern hunters: puzzles, projects and open questions","Magic squares from Khajuraho, tessellations, figurate numbers, cycles, olympiad problems and unsolved mysteries","Take patterns into the wider world: Lo Shu, Khajuraho and Ramanujan magic squares, tessellations and symmetry, figurate numbers, cycles of last digits and weekdays, the chessboard legend and binary, olympiad problems, patterns in music and careers, projects, and open questions like Collatz.",[13,14,15,16,17],"Test and build magic squares, and explain what makes the Khajuraho square most-perfect.","Use the 360° rule to decide which shapes tessellate, and describe symmetry in rangoli and kolam.","Recognise figurate numbers and use cycles and remainders to answer “far-away” questions.","Solve olympiad-style problems with triangular numbers, pairing and double counting.","Explore open questions and plan a pattern project of your own.",60,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 60 minutes",{"label":29,"value":30},"Prior knowledge","Deepen: nth terms, sums, proof",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Sort games ×2, match pairs, pattern machine",{"label":38,"value":39},"Open problem","The Collatz conjecture",[41,45,51,57,60,89,104,109,112,117,173,178,181,243,248,251,256,293,296,335,340,343,353,364,367,372,375,380,383,460,465,473,482,491,500,513,524,542,547,607,610,615,643,648,653,656,661,665,670,726,873,879,884,888,892,907],{"id":42,"type":43,"markdown":44},"intro","prose","You now know how to spot, describe, predict and prove patterns. This last layer takes you **out into the wide world** of patterns: squares of numbers that are magic in every direction (one carved on a temple wall at Khajuraho in medieval India), floors and rangoli that tile the plane, number shapes beyond squares and triangles, patterns that go round in cycles, a legend about rice on a chessboard, olympiad-style puzzles, patterns in music, cricket and calendars, projects to try, and questions that nobody in the world has answered yet.\n\nDip in anywhere. Each chapter stands on its own.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to","callout","observation","How to use this layer","Treat this layer as a menu, not a march. Try the puzzles before reading the solutions, and pick at least one project to actually do. The best pattern-hunters keep a notebook of patterns they notice in everyday life.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","Magic squares","Chapter 01","1 Magic squares",{"id":58,"type":43,"markdown":59},"loshu","A **magic square** is a square grid of numbers in which every **row**, every **column** and both **main diagonals** add up to the same total, called the **magic constant**. Usually the numbers are 1, 2, 3, … up to the number of cells, each used once.\n\nThe oldest known is the Chinese **Lo Shu** square, 3 by 3, said in legend to have appeared on the back of a turtle:\n\n| | | |\n| --- | --- | --- |\n| 4 | 9 | 2 |\n| 3 | 5 | 7 |\n| 8 | 1 | 6 |\n\nEvery line adds to **15**: 4 + 9 + 2, 3 + 5 + 7, 4 + 5 + 6, and so on.\n\n**Why 15?** The numbers 1 to 9 add up to 45. The three rows share out all nine numbers, so each row must be 45 ÷ 3 = 15. In general, a square of side n using 1 to n² has magic constant **n(n² + 1) ÷ 2**: 15 for 3 × 3, 34 for 4 × 4, 65 for 5 × 5, 260 for 8 × 8.\n\nIn India, magic squares were studied in depth. Nārāyaṇa Paṇḍita’s *Gaṇita Kaumudī* (1356) gave general methods for building them of many sizes, and they appear in older texts too, sometimes linked to ideas of good luck or medicine.",{"id":61,"type":62,"title":63,"items":64},"steps-siamese","steps","Build any odd-sized magic square (the “staircase” method)",[65,69,73,77,81,85],{"title":66,"tag":67,"text":68},"Start","Top middle","Write 1 in the middle cell of the top row.",{"title":70,"tag":71,"text":72},"Move up and right","Diagonal step","Put each next number one row up and one column right.",{"title":74,"tag":75,"text":76},"Off the top?","Wrap to bottom","If that takes you above the top row, go to the bottom row of that column.",{"title":78,"tag":79,"text":80},"Off the right?","Wrap to left","If it takes you past the right edge, go to the left end of that row.",{"title":82,"tag":83,"text":84},"Blocked?","Drop down","If the cell is already filled (or you would leave by the top-right corner), put the number directly below the last one instead.",{"title":86,"tag":87,"text":88},"Result for 3 × 3","Magic 15","Rows 8 1 6 \u002F 3 5 7 \u002F 4 9 2. Every line adds to 15. It is the Lo Shu turned over.",{"id":90,"type":91,"itemId":92,"prompt":93,"check":94,"hints":98,"feedback":101},"pr-magic5","practice","patterns.extend-magic-constant","A 5 by 5 magic square uses the numbers 1 to 25 once each. What is its magic constant?",{"kind":95,"answer":96,"tolerance":97},"number",65,0,[99,100],"Add 1 + 2 + … + 25 first.","Share the total equally between the 5 rows.",{"correct":102,"incorrect":103},"Yes: 1 + … + 25 = 325, and 325 ÷ 5 = 65.","1 + 2 + … + 25 = 25 × 26 ÷ 2 = 325. The five rows share this equally: 325 ÷ 5 = 65.",{"id":105,"type":53,"title":106,"eyebrow":107,"navLabel":108},"ch02","The Khajuraho square and Ramanujan’s birthday","Chapter 02","2 Khajuraho",{"id":110,"type":43,"markdown":111},"khaj","At the **Parshvanath Jain temple in Khajuraho** (Madhya Pradesh) a 4 by 4 magic square is carved into the stone:\n\n| | | | |\n| --- | --- | --- | --- |\n| 7 | 12 | 1 | 14 |\n| 2 | 13 | 8 | 11 |\n| 16 | 3 | 10 | 5 |\n| 9 | 6 | 15 | 4 |\n\n**How old is it?** The temple itself was built about **950–970 CE**, but the inscription is usually dated a little later, to around the **12th century**. You will often see “10th century” quoted, from the date of the temple rather than of the carving.\n\nIt uses 1 to 16 once each, and every row, column and main diagonal adds to **34** — which is why it is known as the *Chautisa Yantra* (*chautisa* is Hindi for thirty-four). But it does far more than that:\n\n- The **broken diagonals** also add to 34. For example 12 + 8 + 5 + 9 (wrapping round the edge) = 34. A square like this is called **pandiagonal**.\n- **Every 2 by 2 block** of four neighbouring cells adds to 34, even blocks that wrap round the edges: 7 + 12 + 2 + 13 = 34, 10 + 5 + 15 + 4 = 34.\n- Numbers two steps apart along a diagonal add to 17: 7 + 10, 12 + 5, 13 + 4, …\n\nSquares with all these properties are called **most-perfect magic squares**. The Khajuraho square is one of the oldest surviving examples anywhere in the world, and studying it is what taught European mathematicians in the late 19th century to care about pandiagonal squares.",{"id":113,"type":47,"variant":114,"title":115,"markdown":116},"example-ramanujan","example","The birthday square named for Ramanujan","The great Indian mathematician **Srinivasa Ramanujan** was born on **22 December 1887**. A magic square that carries that date across its top row is famous in India:\n\n| | | | |\n| --- | --- | --- | --- |\n| 22 | 12 | 18 | 87 |\n| 88 | 17 | 9 | 25 |\n| 10 | 24 | 89 | 16 |\n| 19 | 86 | 23 | 11 |\n\nEvery row, column and main diagonal adds to **139**. So do the four corners (22 + 87 + 19 + 11), the middle 2 by 2 block (17 + 9 + 24 + 89) and each corner 2 by 2 block (22 + 12 + 88 + 17). It does not use 1 to 16, so its constant is not 34.\n\n**Did Ramanujan make it?** Probably not. It spread through India as an anonymous presentation titled “Ramanujan’s magic square”, and it is better described as a square *in his honour*: there is no record of it in his own work. The reason is the nicest part — once you know the method, **any** date can be turned into such a square, including yours. Try it (see chapter 9).",{"id":118,"type":119,"component":120,"componentVersion":5,"config":121,"objective":171,"textAlternative":172},"lab-magic-sort","interactive","sort-game",{"prompt":122,"bins":123,"items":130,"seconds":97},"Is each 3 by 3 grid (rows separated by \u002F) a magic square, with every row, column and diagonal adding to the same total?",[124,127],{"id":125,"label":126},"magic","Magic square",{"id":128,"label":129},"not","Not magic",[131,135,139,143,147,151,155,159,163,167],{"id":132,"label":133,"bin":125,"why":134},"m1","4 9 2 \u002F 3 5 7 \u002F 8 1 6","The Lo Shu: every row, column and diagonal adds to 15.",{"id":136,"label":137,"bin":125,"why":138},"m2","2 7 6 \u002F 9 5 1 \u002F 4 3 8","All eight lines add to 15; it is the Lo Shu reflected.",{"id":140,"label":141,"bin":128,"why":142},"m3","1 2 3 \u002F 4 5 6 \u002F 7 8 9","Rows add to 6, 15 and 24: not equal.",{"id":144,"label":145,"bin":125,"why":146},"m4","8 1 6 \u002F 3 5 7 \u002F 4 9 2","The staircase-method square: every line is 15.",{"id":148,"label":149,"bin":125,"why":150},"m5","2 9 4 \u002F 7 5 3 \u002F 6 1 8","Another turn of the Lo Shu; all lines are 15.",{"id":152,"label":153,"bin":128,"why":154},"m6","4 9 2 \u002F 3 5 7 \u002F 6 1 8","The rows work (15), but the columns give 13, 15, 17.",{"id":156,"label":157,"bin":125,"why":158},"m7","5 5 5 \u002F 5 5 5 \u002F 5 5 5","Every line adds to 15, though it repeats one number; a “trivial” magic square.",{"id":160,"label":161,"bin":125,"why":162},"m8","6 7 2 \u002F 1 5 9 \u002F 8 3 4","Rows 15, columns 15, diagonals 6 + 5 + 4 and 2 + 5 + 8 are 15.",{"id":164,"label":165,"bin":128,"why":166},"m9","5 1 9 \u002F 3 6 6 \u002F 7 8 0","Rows are 15 each, but the columns are 15, 15, 15 and the diagonals are 11 and 22.",{"id":168,"label":169,"bin":125,"why":170},"m10","10 3 8 \u002F 5 7 9 \u002F 6 11 4","Every line adds to 21: the Lo Shu with 2 added to every number.","Check rows, columns and diagonals to decide which grids are magic squares.","Ten 3 × 3 grids to sort (rows are separated by slashes).\n\nMagic: 4 9 2 \u002F 3 5 7 \u002F 8 1 6 (the Lo Shu); 2 7 6 \u002F 9 5 1 \u002F 4 3 8; 8 1 6 \u002F 3 5 7 \u002F 4 9 2; 2 9 4 \u002F 7 5 3 \u002F 6 1 8; 6 7 2 \u002F 1 5 9 \u002F 8 3 4 (all magic 15, all turns or reflections of the Lo Shu); 5 5 5 \u002F 5 5 5 \u002F 5 5 5 (trivially magic); 10 3 8 \u002F 5 7 9 \u002F 6 11 4 (magic 21: adding the same number to every cell keeps a square magic).\n\nNot magic: 1 2 3 \u002F 4 5 6 \u002F 7 8 9 (rows 6, 15, 24); 4 9 2 \u002F 3 5 7 \u002F 6 1 8 (columns 13, 15, 17); 5 1 9 \u002F 3 6 6 \u002F 7 8 0 (diagonals 11 and 22).\n\nAlways check all eight lines: three rows, three columns and two diagonals.",{"id":174,"type":53,"title":175,"eyebrow":176,"navLabel":177},"ch03","Tessellations and symmetry patterns","Chapter 03","3 Tessellations",{"id":179,"type":43,"markdown":180},"tess","A **tessellation** (or tiling) is a pattern of shapes that covers a flat surface with **no gaps and no overlaps**, and can go on for ever. Floor tiles, brick walls, honeycombs, jali screens, the grid of dots under a kolam, and the repeating designs on block-printed cloth are all tessellations.\n\nWhich **regular polygons** (all sides and angles equal) can tile a floor on their own? Where the corners meet, the angles must add up to exactly **360°**:\n\n| Shape | Each angle | How many fit round a point? |\n| --- | --- | --- |\n| Equilateral triangle | 60° | 360 ÷ 60 = **6** ✓ |\n| Square | 90° | 360 ÷ 90 = **4** ✓ |\n| Regular pentagon | 108° | 360 ÷ 108 = 3.33… ✗ |\n| Regular hexagon | 120° | 360 ÷ 120 = **3** ✓ |\n| Regular octagon | 135° | 360 ÷ 135 = 2.67 ✗ |\n\nOnly **three** regular polygons tile on their own: triangles, squares and hexagons. That is why bees use hexagons: of the three, hexagons enclose the most space for the least wall.\n\nMix shapes and more is possible. Octagons and squares together (135° + 135° + 90° = 360°) make a common bathroom-floor pattern. Triangles and hexagons (60° + 60° + 120° + 120°) make a pattern found in old Indian and Islamic jali work.",{"id":182,"type":119,"component":120,"componentVersion":5,"config":183,"objective":241,"textAlternative":242},"lab-tess",{"prompt":184,"bins":185,"items":192,"seconds":97},"Can copies of this single shape tile a flat floor with no gaps or overlaps?",[186,189],{"id":187,"label":188},"yes","Tiles on its own",{"id":190,"label":191},"no","Cannot tile on its own",[193,197,201,205,209,213,217,221,225,229,233,237],{"id":194,"label":195,"bin":187,"why":196},"t1","Equilateral triangle (60° corners)","Six corners of 60° make 360°.",{"id":198,"label":199,"bin":187,"why":200},"t2","Square (90°)","Four corners of 90° make 360°.",{"id":202,"label":203,"bin":190,"why":204},"t3","Regular pentagon (108°)","360 ÷ 108 is not a whole number: three leave a 36° gap, four overlap.",{"id":206,"label":207,"bin":187,"why":208},"t4","Regular hexagon (120°)","Three corners of 120° make 360°: the honeycomb.",{"id":210,"label":211,"bin":190,"why":212},"t5","Regular octagon (135°)","Two make 270° and three make 405°. Octagons need squares to fill the gaps.",{"id":214,"label":215,"bin":187,"why":216},"t6","Any rectangle","Four right angles meet at each corner, like bricks or books on a shelf.",{"id":218,"label":219,"bin":187,"why":220},"t7","Any triangle, even a scalene one","Two copies make a parallelogram; its three angles, used twice each, make 360° at a point.",{"id":222,"label":223,"bin":187,"why":224},"t8","Any quadrilateral (four-sided shape)","Its four angles add to 360°, so turned copies fit round every point.",{"id":226,"label":227,"bin":190,"why":228},"t9","Circle","Circles always leave curved gaps between them.",{"id":230,"label":231,"bin":190,"why":232},"t10","Regular 12-sided polygon (150°)","360 ÷ 150 = 2.4. It tiles only with triangles or other shapes filling gaps.",{"id":234,"label":235,"bin":187,"why":236},"t11","Parallelogram","Slide copies along its sides: a slanted brick pattern.",{"id":238,"label":239,"bin":190,"why":240},"t12","Regular hexagon with a triangle cut off one corner","Cutting off a corner makes a convex 7-sided shape, and no convex polygon with 7 or more sides can tile a flat surface.","Use the 360° rule to decide which shapes tessellate on their own.","Twelve shapes to sort.\n\nTiles on its own: equilateral triangle (6 × 60° = 360°), square (4 × 90°), regular hexagon (3 × 120°), any rectangle, any triangle, any quadrilateral, any parallelogram.\n\nCannot tile on its own: regular pentagon (108° does not divide 360°), regular octagon (135°), circle (curved gaps), regular 12-sided polygon (150°), and a hexagon with a corner cut off.\n\nThe test is the corners: the angles meeting at each point must add to exactly 360°.",{"id":244,"type":47,"variant":245,"title":246,"markdown":247},"nuance-pentagons","nuance","Some pentagons do tile","The *regular* pentagon cannot tile, but some irregular pentagons can. Mathematicians have found exactly **15 families** of pentagons that tile a flat surface, the last discovered in 2015, and in 2017 a computer-assisted proof showed there are no others. The story of pentagon tilings took over a century and involved an amateur mathematician, Marjorie Rice, who discovered four families from her kitchen table.",{"id":249,"type":43,"markdown":250},"symmetry","Rangoli, kolam and many textile patterns combine tessellation with **symmetry**:\n\n- **Reflection symmetry**: one half is a mirror image of the other.\n- **Rotational symmetry**: the design looks the same after a turn. A rangoli flower with 8 petals looks the same after 1\u002F8 of a turn (45°), so it has rotational symmetry of **order 8**.\n- **Translation symmetry**: a border pattern looks the same after sliding along by one unit. This is the repeating unit you met in Discover.\n\nKolam traditions have their own deep mathematics: some kolam are drawn as a **single unbroken line** that loops around every dot and returns to its start. Computer scientists have studied these as examples of patterns generated by simple rules, and designs like them are now used to teach coding.",{"id":252,"type":53,"title":253,"eyebrow":254,"navLabel":255},"ch04","Number shapes beyond squares and triangles","Chapter 04","4 Figurate numbers",{"id":257,"type":258,"caption":259,"columns":260,"rows":264},"table-figurate","table","Figurate numbers: dots arranged in shapes (first eight, computed)",[261,262,263],"Family","Formula","First eight",[265,269,273,277,281,285,289],[266,267,268],"Triangular","n(n + 1) ÷ 2","1, 3, 6, 10, 15, 21, 28, 36",[270,271,272],"Square","n²","1, 4, 9, 16, 25, 36, 49, 64",[274,275,276],"Pentagonal","n(3n − 1) ÷ 2","1, 5, 12, 22, 35, 51, 70, 92",[278,279,280],"Hexagonal","n(2n − 1)","1, 6, 15, 28, 45, 66, 91, 120",[282,283,284],"Centred hexagonal","3n(n − 1) + 1","1, 7, 19, 37, 61, 91, 127, 169",[286,287,288],"Tetrahedral (triangular pyramid)","n(n + 1)(n + 2) ÷ 6","1, 4, 10, 20, 35, 56, 84, 120",[290,291,292],"Square pyramidal","n(n + 1)(2n + 1) ÷ 6","1, 5, 14, 30, 55, 91, 140, 204",{"id":294,"type":43,"markdown":295},"figurate","Numbers that can be arranged as dots in a regular shape are called **figurate numbers**. The Greeks loved them, and they appear all over puzzles.\n\n- **Centred hexagonal numbers** 1, 7, 19, 37, 61, … are one dot surrounded by rings of 6, 12, 18, 24 dots. Look at a pack of pencils or a bundle of straws held together: 7 and 19 are the natural sizes. Their running totals are the cubes: 1 = 1, 1 + 7 = 8, 1 + 7 + 19 = 27!\n- **Tetrahedral numbers** 1, 4, 10, 20, 35, … count oranges stacked as a triangular pyramid at a fruit stall.\n- **Square pyramidal numbers** count balls stacked as a square pyramid. They also answer a classic puzzle: **how many squares of all sizes are on a chessboard?** There are 64 small ones, 49 of size 2 × 2, 36 of size 3 × 3, … down to 1 of size 8 × 8: 1 + 4 + 9 + … + 64 = **204**.",{"id":297,"type":119,"component":298,"componentVersion":5,"config":299,"objective":333,"textAlternative":334},"lab-match-figurate","match-pairs",{"prompt":300,"mode":301,"pairs":302},"Match each number family to its first five terms.","memory",[303,306,309,312,315,318,321,324,327,330],{"a":304,"b":305},"Triangular numbers","1, 3, 6, 10, 15",{"a":307,"b":308},"Square numbers","1, 4, 9, 16, 25",{"a":310,"b":311},"Pentagonal numbers","1, 5, 12, 22, 35",{"a":313,"b":314},"Hexagonal numbers","1, 6, 15, 28, 45",{"a":316,"b":317},"Centred hexagonal numbers","1, 7, 19, 37, 61",{"a":319,"b":320},"Tetrahedral numbers","1, 4, 10, 20, 35",{"a":322,"b":323},"Cube numbers","1, 8, 27, 64, 125",{"a":325,"b":326},"Fibonacci numbers","1, 1, 2, 3, 5",{"a":328,"b":329},"Powers of 2","1, 2, 4, 8, 16",{"a":331,"b":332},"Lucas numbers","2, 1, 3, 4, 7","Recognise the famous number families from their first terms.","A memory game with ten families and their first five terms:\n\ntriangular 1, 3, 6, 10, 15; square 1, 4, 9, 16, 25; pentagonal 1, 5, 12, 22, 35; hexagonal 1, 6, 15, 28, 45; centred hexagonal 1, 7, 19, 37, 61; tetrahedral 1, 4, 10, 20, 35; cubes 1, 8, 27, 64, 125; Fibonacci 1, 1, 2, 3, 5; powers of 2 1, 2, 4, 8, 16; Lucas 2, 1, 3, 4, 7.\n\nTip: the second term is often the giveaway: 3 triangular, 4 square or tetrahedral, 5 pentagonal, 6 hexagonal, 7 centred hexagonal, 8 cube.",{"id":336,"type":53,"title":337,"eyebrow":338,"navLabel":339},"ch05","Cycles: last digits, weekdays and clocks","Chapter 05","5 Cycles",{"id":341,"type":43,"markdown":342},"cycles","Some growing patterns hide a **repeating** pattern inside them. The powers of 7 grow enormous: 7, 49, 343, 2,401, 16,807, … but their **last digits** go 7, 9, 3, 1, 7, 9, 3, 1, … a cycle of length 4.\n\n| Powers of | Last digits | Cycle length |\n| --- | --- | --- |\n| 2 | 2, 4, 8, 6, 2, 4, 8, 6, … | 4 |\n| 3 | 3, 9, 7, 1, 3, 9, 7, 1, … | 4 |\n| 4 | 4, 6, 4, 6, 4, 6, 4, 6, … | 2 |\n| 7 | 7, 9, 3, 1, 7, 9, 3, 1, … | 4 |\n| 9 | 9, 1, 9, 1, 9, 1, 9, 1, … | 2 |\n| 5 | 5, 5, 5, 5, 5, 5, 5, 5, … | 1 |\n\nThis lets you answer questions that look impossible. **What is the last digit of 7²⁰²⁶?** The cycle has length 4, and 2026 ÷ 4 = 506 remainder 2, so 7²⁰²⁶ ends like 7², in **9**. It is exactly the “which bead is 20th?” trick from Discover, applied to a number with 1,713 digits.",{"id":344,"type":345,"title":346,"problem":347,"steps":348},"we-cycles","worked_example","Three cycle puzzles","(a) The word **PATTERN** is written again and again: PATTERNPATTERNPATTERN… What is the 2,026th letter? (b) Today is Sunday. What day will it be in 100 days? (c) It is 9 o’clock on a 12-hour clock. What time will it show after 50 hours?",[349,350,351,352],"(a) The unit PATTERN has 7 letters. 2,026 ÷ 7 = 289 remainder 3, so the 2,026th letter is the 3rd letter of the unit: **T**.","(b) Weekdays repeat every 7 days. 100 ÷ 7 = 14 remainder 2, so it is 2 days after Sunday: **Tuesday**.","(c) A 12-hour clock repeats every 12 hours. 50 ÷ 12 = 4 remainder 2, so it shows 9 + 2 = **11 o’clock**.","All three are the same idea: in a cycle, only the **remainder** matters. Mathematicians call this *modular arithmetic* or *clock arithmetic*.",{"id":354,"type":91,"itemId":355,"prompt":356,"check":357,"hints":358,"feedback":361},"pr-last-digit","patterns.extend-last-digit","What is the **last digit** of **3¹⁰⁰**?",{"kind":95,"answer":5,"tolerance":97},[359,360],"The last digits of powers of 3 go 3, 9, 7, 1 and repeat.","100 ÷ 4 leaves remainder 0.",{"correct":362,"incorrect":363},"Yes: 100 is a multiple of 4, so 3¹⁰⁰ ends like 3⁴ = 81, in 1.","The cycle 3, 9, 7, 1 has length 4. 100 ÷ 4 = 25 remainder 0, so 3¹⁰⁰ ends like the 4th in the cycle, 1.",{"id":365,"type":43,"markdown":366},"calendar-cycle","**The calendar's great cycle.** A year of 365 days is 52 weeks and 1 day, so the same date moves on one weekday each year (two after 29 February). Leap years follow a three-part rule: a year divisible by 4 is a leap year, **except** century years, which are leap years only if divisible by 400. So 2000 was a leap year, but 1900 was not and 2100 will not be.\n\nIn 400 years there are 97 leap years, giving 400 × 365 + 97 = **146,097 days**, which is **exactly 20,871 weeks**. So the Gregorian calendar repeats perfectly every 400 years: 1 January 2426 will fall on the same weekday as 1 January 2026, a Thursday.",{"id":368,"type":53,"title":369,"eyebrow":370,"navLabel":371},"ch06","The chessboard legend and binary","Chapter 06","6 Chessboard & binary",{"id":373,"type":43,"markdown":374},"chess","An old legend, told in India and Persia, says that the inventor of chess asked the king for a humble reward: **one grain of rice** on the first square of the chessboard, **two** on the second, **four** on the third, doubling on every square up to the 64th. The king laughed at such a small request, and then his treasurers started counting.\n\n- Square 10: 2⁹ = 512 grains.\n- Square 20: 2¹⁹ = 524,288 grains.\n- Square 64: 2⁶³ = **9,223,372,036,854,775,808** grains.\n- The whole board: 1 + 2 + 4 + … + 2⁶³ = 2⁶⁴ − 1 = **18,446,744,073,709,551,615** grains (using the doubling-sum trick from Deepen).\n\nA grain of rice weighs roughly 0.025 g (a thousand grains weigh about 20–30 g). At 0.025 g a grain, the whole board comes to about **461 thousand million tonnes** of rice. The world grows around 550 million tonnes of rice a year, so that is **more than 800 years** of the entire world’s rice harvest. The king could not possibly pay.",{"id":376,"type":47,"variant":377,"title":378,"markdown":379},"model-limit-legend","model_limit","A legend, not history","The chessboard story is a legend: there is no record of the inventor of chess or of such a reward. Its versions differ (sometimes it is wheat, sometimes the reward is a trick question), and the weight of a grain varies with the type of rice — 0.02 g instead of 0.025 g would drop the answer to about 370 thousand million tonnes, and 0.03 g would raise it to about 550. So treat “more than 800 years of harvest” as an order of magnitude, not a measurement. What is certain is the mathematics: 2⁶⁴ − 1 is a twenty-digit number, and doubling 63 times turns one grain into an impossible mountain.",{"id":381,"type":43,"markdown":382},"binary","Doubling also gives us **binary**, the number system inside every computer. In our everyday decimal system each place is worth **10 times** the place to its right (ones, tens, hundreds, thousands). In binary each place is worth **2 times** the place to its right: ones, twos, fours, eights, sixteens, …, and the only digits are 0 and 1.\n\n| Decimal | Made from powers of 2 | Binary |\n| --- | --- | --- |\n| 5 | 4 + 1 | 101 |\n| 13 | 8 + 4 + 1 | 1101 |\n| 20 | 16 + 4 | 10100 |\n| 100 | 64 + 32 + 4 | 1100100 |\n| 255 | 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 | 11111111 |\n\nCounting in binary makes its own pattern: 1, 10, 11, 100, 101, 110, 111, 1000, … The last digit alternates 1, 0, 1, 0 (odd, even); the second-last goes 0, 1, 1, 0, 0, 1, 1, … in pairs; the next in fours. Pingala’s rules for listing patterns of short and long syllables, over 2,000 years ago, work just like counting in binary with S and L instead of 0 and 1.",{"id":384,"type":119,"component":385,"componentVersion":5,"config":386,"objective":458,"textAlternative":459},"lab-extend-seq","pattern-machine",{"puzzles":387},[388,403,416,423,428,445,454],{"kind":95,"rule":389,"show":392,"ask":402},{"type":390,"terms":391,"ruleText":401},"list",[5,392,393,394,395,396,397,398,399,400],5,12,22,35,51,70,92,117,145,"Pentagonal numbers n(3n − 1) ÷ 2: differences 4, 7, 10, 13, …",2,{"kind":95,"rule":404,"show":392,"ask":402},{"type":390,"terms":405,"ruleText":415},[5,406,407,408,409,410,411,412,413,414],7,19,37,61,91,127,169,217,271,"Centred hexagonal numbers: add rings of 6, 12, 18, 24, …",{"kind":95,"rule":417,"show":422,"ask":402},{"type":390,"terms":418,"ruleText":421},[406,419,420,5,406,419,420,5,406,419,420,5],9,3,"Last digits of 7, 49, 343, 2401, …: the cycle 7, 9, 3, 1 repeats",6,{"kind":95,"rule":424,"show":392,"ask":402,"hint":427},{"type":425,"start":5,"factor":426},"multiply",-2,"The sign flips each time.",{"kind":95,"rule":429,"show":422,"ask":402,"hint":444},{"type":390,"terms":430,"ruleText":443},[406,394,431,432,433,434,435,436,437,438,439,392,440,441,442,402,5],11,34,17,52,26,13,40,20,10,16,8,4,"Collatz rule from 7: halve an even number; treble an odd number and add 1","Even or odd decides what happens next.",{"kind":95,"rule":446,"show":392,"ask":402},{"type":390,"terms":447,"ruleText":453},[5,442,439,438,395,448,449,450,451,452],56,84,120,165,220,"Tetrahedral numbers: running totals of the triangular numbers",{"kind":95,"rule":455,"show":406,"ask":420},{"type":390,"terms":456,"ruleText":457},[5,402,402,420,420,420,442,442,442,442,392,392,392,392,392],"Each number n is written n times","Extend unusual sequences: figurate numbers, digit cycles, sign-flipping, Collatz and self-describing lists.","Seven puzzles from beyond the syllabus.\n\n1. Pentagonal numbers 1, 5, 12, 22, 35, … next **51, 70** (differences 4, 7, 10, 13, 16, 19).\n2. Centred hexagonal numbers 1, 7, 19, 37, 61, … next **91, 127** (add 6, 12, 18, 24, 30, 36).\n3. Last digits of powers of 7: 7, 9, 3, 1, 7, 9, … next **3, 1**.\n4. Multiply by −2: 1, −2, 4, −8, 16, … next **−32, 64**. A geometric sequence with a negative ratio flips sign every step.\n5. Collatz from 7: 7, 22, 11, 34, 17, 52, … next **26, 13** (even → halve; odd → × 3 + 1). It reaches 1 after 16 steps.\n6. Tetrahedral numbers 1, 4, 10, 20, 35, … next **56, 84**.\n7. 1, 2, 2, 3, 3, 3, 4, … next **4, 4, 4** (each number n appears n times).",{"id":461,"type":53,"title":462,"eyebrow":463,"navLabel":464},"ch07","Olympiad corner","Chapter 07","7 Olympiad corner",{"id":466,"type":345,"title":467,"problem":468,"steps":469},"we-oly-1","Problem 1: the 100th term of 1, 2, 2, 3, 3, 3, …","In the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …, each whole number n appears n times. What is the **100th** term?",[470,471,472],"The last 1 is term 1, the last 2 is term 3, the last 3 is term 6, the last 4 is term 10: the positions of the last copies are the **triangular numbers**.","The last 13 is at position 13 × 14 ÷ 2 = 91. The last 14 is at position 14 × 15 ÷ 2 = 105.","Terms 92 to 105 are all 14, so the 100th term is **14**.",{"id":474,"type":345,"title":475,"problem":476,"steps":477},"we-oly-2","Problem 2: matchstick triangle grid","A big triangle of side n is divided into small equilateral triangles of side 1 using matchsticks (side 1: one triangle; side 2: four small triangles). How many matchsticks for **side 10**?",[478,479,480,481],"Sticks lie in three directions. Count the horizontal ones: the rows have 1, 2, 3, …, n sticks, a total of n(n + 1) ÷ 2.","By symmetry, each of the other two directions has the same number.","Total = 3 × n(n + 1) ÷ 2. Check: side 1 → 3 ✓; side 2 → 9 ✓ (count it!); side 3 → 18.","Side 10: 3 × 55 = **165** matchsticks.",{"id":483,"type":345,"title":484,"problem":485,"steps":486},"we-oly-3","Problem 3: a sum that nearly cancels","Find 1 − 2 + 3 − 4 + 5 − 6 + … + 99 − 100.",[487,488,489,490],"Group in pairs: (1 − 2) + (3 − 4) + … + (99 − 100).","Each pair is −1, and there are 50 pairs.","Total: **-50**.","Pairing again! The same trick Gauss used, in a new costume.",{"id":492,"type":345,"title":493,"problem":494,"steps":495},"we-oly-4","Problem 4: digits and handshakes","(a) How many digits do you write to number the pages of a 100-page book? (b) At a party of 30 children, everyone shakes hands with everyone else once. How many handshakes?",[496,497,498,499],"(a) Pages 1–9: 9 digits. Pages 10–99: 90 pages × 2 = 180 digits. Page 100: 3 digits.","Total: 9 + 180 + 3 = **192** digits.","(b) The 1st child shakes 29 hands, the 2nd shakes 28 new hands, …, the 29th shakes 1 new hand.","1 + 2 + … + 29 = 29 × 30 ÷ 2 = **435** handshakes. (Or: 30 × 29 ÷ 2, because each handshake involves two children.)",{"id":501,"type":91,"itemId":502,"prompt":503,"check":504,"hints":507,"feedback":510},"pr-oly-squares","patterns.extend-chessboard-squares","How many squares of **all sizes** are there on a **5 by 5** grid of small squares?",{"kind":95,"answer":505,"tolerance":97,"unit":506},55,"squares",[508,509],"Count 1 × 1 squares, then 2 × 2, then 3 × 3, …","You should be adding square numbers.",{"correct":511,"incorrect":512},"Yes: 25 + 16 + 9 + 4 + 1 = 55.","There are 25 squares of size 1, 16 of size 2, 9 of size 3, 4 of size 4 and 1 of size 5: 25 + 16 + 9 + 4 + 1 = 55.",{"id":514,"type":91,"itemId":515,"prompt":516,"check":517,"hints":518,"feedback":521},"pr-oly-diag","patterns.extend-polygon-diagonals","A **diagonal** joins two corners of a polygon that are not next to each other. How many diagonals does a **10-sided** polygon (decagon) have?",{"kind":95,"answer":395,"tolerance":97},[519,520],"From each corner, how many other corners can you join with a diagonal (not itself, not its two neighbours)?","Each diagonal gets counted from both ends.",{"correct":522,"incorrect":523},"Yes: 10 × 7 ÷ 2 = 35 diagonals.","Each of the 10 corners joins to 10 − 3 = 7 others. 10 × 7 = 70 counts every diagonal twice, so there are 35.",{"id":525,"type":526,"prompt":527,"options":528,"explanation":541},"predict-oly","prediction","The sequence of **rectangles** on a chessboard: how many rectangles of all sizes (squares included) are there on an 8 × 8 board? (Hint: a rectangle is fixed by choosing 2 of the 9 vertical grid lines and 2 of the 9 horizontal ones.)",[529,532,535,538],{"id":530,"label":531},"a","204",{"id":533,"label":534},"b","1,296",{"id":536,"label":537},"c","4,096",{"id":539,"label":540},"d","64","**1,296.** There are 9 × 8 ÷ 2 = 36 ways to choose two vertical lines and 36 ways to choose two horizontal lines, so 36 × 36 = **1,296** rectangles. Notice: 36 = 1 + 2 + … + 8, and 1,296 = 1³ + 2³ + … + 8³, the sum-of-cubes pattern from Deepen!",{"id":543,"type":53,"title":544,"eyebrow":545,"navLabel":546},"ch08","Patterns at work: music, sport, nature and careers","Chapter 08","8 Patterns at work",{"id":548,"type":549,"title":550,"prompt":551,"options":552},"explorer-work","explorer","Where patterns earn their living","Pick a field to see how patterns are used there.",[553,562,571,580,589,598],{"id":554,"label":555,"chain":556,"note":561},"music","Indian music",[557,558,559,560],"Tala: a cycle of beats","Teentaal 16 = 4+4+4+4","Rupak 7 = 3+2+2","Jhaptaal 10 = 2+3+2+3","A tala is a repeating rhythmic cycle. Tabla players count cycles and make patterns called tihai, a phrase played three times so that it lands exactly on sam, the first beat of the cycle. Musicians do real arithmetic with remainders to make tihais land on time. Carnatic Adi tala has 8 beats (4 + 2 + 2).",{"id":563,"label":564,"chain":565,"note":570},"cricket","Cricket",[566,567,568,569],"6 balls an over","T20: 20 overs","120 balls","Run rate × overs","An over is a repeating unit of 6 balls, so ball 45 of an innings is the 3rd ball of the 8th over (45 = 6 × 7 + 3). Required run rates, projected scores and the patterns of a batter’s scoring are all sequences that analysts study.",{"id":572,"label":573,"chain":574,"note":579},"textiles","Weaving and print",[575,576,577,578],"Repeating motif","Slide along the border","Turn around a centre","Loom follows a code","Block printers in Bagru and Sanganer stamp a carved block again and again: a translation pattern. Jacquard looms used punched cards to encode patterns, an idea that helped inspire early computers. Kanchipuram and Banarasi weavers still plan designs on squared paper.",{"id":581,"label":582,"chain":583,"note":588},"computing","Coding",[584,585,586,587],"Loop: repeat a rule","Recursion: use earlier terms","Binary counting","Patterns in data","A computer loop is a term-to-term rule: “add 3 and repeat”. Recursion, where a function uses its own earlier answers, is the Fibonacci idea. Programmers, data scientists and machine-learning engineers spend their days finding and using patterns.",{"id":590,"label":591,"chain":592,"note":597},"science","Science and weather",[593,594,595,596],"Collect data","Spot a trend","Guess a rule","Test with new data","Scientists find patterns in monsoon rainfall, the orbits of planets, and how diseases spread (which can grow like a geometric sequence at first). The detective method from Investigate is the scientific method in miniature.",{"id":599,"label":600,"chain":601,"note":606},"crypto","Secret codes",[602,603,604,605],"Message","Apply a rule","Coded text","Undo the rule","Codes hide patterns and code-breakers hunt for them. Modern encryption that protects UPI payments relies on number patterns that are easy to compute one way and extremely hard to undo, involving huge prime numbers and clock arithmetic.",{"id":608,"type":43,"markdown":609},"careers","Almost every career that uses mathematics is really about patterns: an **actuary** predicts future costs from past data, a **cryptographer** designs and breaks codes, an **architect** tiles floors and repeats structural units, a **textile designer** creates repeats that line up at the edges of fabric, a **composer** builds music from repeated and varied phrases, a **biologist** models how populations grow, and a **software engineer** writes loops and recursive functions all day. The habit you have practised in this topic, *notice, describe, predict, check, explain*, is useful in all of them.",{"id":611,"type":53,"title":612,"eyebrow":613,"navLabel":614},"ch09","Projects to try","Chapter 09","9 Projects",{"id":616,"type":62,"title":617,"items":618},"steps-projects","Six pattern projects",[619,623,627,631,635,639],{"title":620,"tag":621,"text":622},"Kolam with a rule","Art","Design a kolam or rangoli border from a repeating unit. Then design a growing version: size 1, 2, 3. Count the dots at each size and find the rule.",{"title":624,"tag":625,"text":626},"Matchstick report","Investigation","Invent your own growing matchstick shape (a row of houses, a ladder). Count 5 cases, find the nth term, explain it with the picture, and predict the 100th.",{"title":628,"tag":629,"text":630},"Birthday magic square","Puzzle","Put your birth date in the top row of a 4 × 4 square and find the other numbers so every row, column and diagonal adds to the same total, as in the square named for Ramanujan.",{"title":632,"tag":633,"text":634},"Fibonacci in plants","Nature","Count spirals on a pineapple, sunflower or pinecone, and petals on ten flowers. How many counts are Fibonacci numbers? Record your data in a table.",{"title":636,"tag":637,"text":638},"Calendar tricks","Performance","Prepare three calendar tricks (the 3 × 3 box, the 2 × 2 cross-products, a 4-in-a-column sum) and explain each one with algebra.",{"title":640,"tag":641,"text":642},"Tala patterns","Music","Clap Teentaal (16 beats) or Rupak (7 beats). Make a tihai: a phrase played three times that ends exactly on beat 1. What lengths of phrase work?",{"id":644,"type":47,"variant":645,"title":646,"markdown":647},"try-tihai","try_it","The maths of a tihai","A tihai is a phrase played **three times** (sometimes with short gaps) so that its very last beat lands exactly on **sam**, beat 1 of the next cycle.\n\n- In 16-beat Teentaal, three phrases of 5 beats with no gaps last 15 beats. Start on beat 3 and the last beat falls on beat 17, which is sam of the next cycle. ✓\n- Phrases of 4 beats with gaps of 2 last 4 + 2 + 4 + 2 + 4 = 16 beats. Start on beat 2 to land on sam.\n\nThe rule: start beat = 18 − (length of the tihai) in a 16-beat cycle. Try inventing tihais for Rupak (7 beats).",{"id":649,"type":53,"title":650,"eyebrow":651,"navLabel":652},"ch10","Open questions: patterns nobody has proved","Chapter 10","10 Open questions",{"id":654,"type":43,"markdown":655},"collatz","Here is a pattern a ten-year-old can try and no mathematician in the world can prove. Start with any whole number. If it is **even, halve it**. If it is **odd, multiply by 3 and add 1**. Repeat.\n\n- From 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.\n- From 7: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.\n- From 27: it takes **111 steps**, climbing as high as **9,232** before finally falling to 1.\n\nThe **Collatz conjecture** (1937) says that every starting number eventually reaches 1. Computers have checked every starting number up to more than 10²⁰ and all of them reach 1. Yet nobody has a proof. One of the most famous mathematicians of the 20th century, Paul Erdős, said that mathematics “is not yet ready for such problems”.",{"id":657,"type":47,"variant":658,"title":659,"markdown":660},"question-open","question","More open questions about patterns","Nobody knows the answers to these. Pick one and explore it.\n\n- **Collatz:** does every starting number reach 1?\n- **Fibonacci primes:** 2, 3, 5, 13, 89, 233, … are Fibonacci numbers that are prime. Are there infinitely many?\n- **Goldbach:** every even number from 4 up seems to be the sum of two primes (10 = 3 + 7 = 5 + 5). Checked up to 4 × 10¹⁸, still unproved.\n- **Magic squares of squares:** is there a 3 × 3 magic square made entirely of different square numbers? Nobody has found one, and nobody has proved it impossible.\n- **Perfect numbers:** 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 equal the sum of their smaller factors. Are there infinitely many? Is there an odd one?\n- **Your own:** invent a rule, run it, and see whether you can predict where it goes. Some of the best questions start that way.",{"id":662,"type":663,"prompt":664},"reflect-extend","reflection","Of all the patterns in this topic, which one surprised you most, and why? Describe one pattern you have noticed in your own life (music, sport, a game, a building, a festival) and write down a question about it that you could investigate.",{"id":666,"type":53,"title":667,"eyebrow":668,"navLabel":669},"ch11","Final challenge","Chapter 11","11 Final challenge",{"id":671,"type":672,"title":673,"terms":674},"glossary-extend","glossary","Words from the wider world of patterns",[675,679,683,687,691,695,699,703,707,711,714,718,722],{"term":676,"meaning":677,"example":678},"magic square","A square grid of numbers where every row, column and main diagonal has the same total.","The Lo Shu: every line adds to 15.",{"term":680,"meaning":681,"example":682},"magic constant","The total of each line in a magic square; n(n² + 1) ÷ 2 for the numbers 1 to n².","34 for a 4 × 4 square",{"term":684,"meaning":685,"example":686},"pandiagonal","A magic square whose broken (wrap-around) diagonals also add to the magic constant.","The Khajuraho square",{"term":688,"meaning":689,"example":690},"tessellation","A pattern of shapes covering a flat surface with no gaps or overlaps.","Hexagons in a honeycomb",{"term":692,"meaning":693,"example":694},"rotational symmetry","Looking the same after a turn of less than a full turn; the order is how many times in one full turn.","An 8-petal rangoli has order 8.",{"term":696,"meaning":697,"example":698},"figurate number","A number that can be arranged as dots in a regular shape.","Pentagonal numbers 1, 5, 12, 22",{"term":700,"meaning":701,"example":702},"centred hexagonal number","A dot in the middle surrounded by hexagonal rings of 6, 12, 18, … dots.","1, 7, 19, 37",{"term":704,"meaning":705,"example":706},"cycle","A part of a pattern that repeats exactly, such as the last digits of powers.","7, 9, 3, 1",{"term":708,"meaning":709,"example":710},"modular (clock) arithmetic","Arithmetic where only the remainder after dividing by a fixed number matters.","50 hours after 9 o’clock is 11 o’clock.",{"term":381,"meaning":712,"example":713},"The base-2 number system, with place values 1, 2, 4, 8, … and digits 0 and 1.","13 = 1101 in binary",{"term":715,"meaning":716,"example":717},"Collatz conjecture","The unproved claim that “halve if even, 3n + 1 if odd” always reaches 1.","6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1",{"term":719,"meaning":720,"example":721},"tala","A repeating cycle of beats in Indian classical music.","Teentaal has 16 beats.",{"term":723,"meaning":724,"example":725},"tihai","A rhythmic phrase played three times to land exactly on the first beat of a tala cycle.","Three phrases of 5 beats in Teentaal",{"id":727,"type":728,"title":729,"questions":730},"quiz-extend","quiz","Final pattern challenge",[731,744,757,770,783,796,809,822,835,848,860],{"itemId":732,"prompt":733,"options":734,"correct":533,"why":743},"patterns.extend-q-magic","What is the magic constant of a 4 × 4 magic square using 1 to 16?",[735,737,739,741],{"id":530,"label":736},"30",{"id":533,"label":738},"34",{"id":536,"label":740},"36",{"id":539,"label":742},"40","1 + … + 16 = 136, shared between 4 rows: 136 ÷ 4 = 34.",{"itemId":745,"prompt":746,"options":747,"correct":533,"why":756},"patterns.extend-q-khaj","What makes the Khajuraho square special beyond ordinary magic squares?",[748,750,752,754],{"id":530,"label":749},"It uses only even numbers",{"id":533,"label":751},"Its broken diagonals and 2 × 2 blocks also add to 34",{"id":536,"label":753},"It is 5 × 5",{"id":539,"label":755},"Its rows add to 15","It is pandiagonal and most-perfect: wrap-around diagonals and every 2 × 2 block also total 34.",{"itemId":758,"prompt":759,"options":760,"correct":536,"why":769},"patterns.extend-q-tess","Which regular polygon can tile a floor on its own?",[761,763,765,767],{"id":530,"label":762},"Pentagon",{"id":533,"label":764},"Octagon",{"id":536,"label":766},"Hexagon",{"id":539,"label":768},"Decagon","A hexagon’s angle is 120°, and three make 360°. The others leave gaps.",{"itemId":771,"prompt":772,"options":773,"correct":536,"why":782},"patterns.extend-q-pent","What comes next in the pentagonal numbers 1, 5, 12, 22, 35, …?",[774,776,778,780],{"id":530,"label":775},"48",{"id":533,"label":777},"50",{"id":536,"label":779},"51",{"id":539,"label":781},"53","Differences 4, 7, 10, 13 go up by 3, so next is +16: 35 + 16 = 51.",{"itemId":784,"prompt":785,"options":786,"correct":536,"why":795},"patterns.extend-q-digit","What is the last digit of 2¹⁰⁰?",[787,789,791,793],{"id":530,"label":788},"2",{"id":533,"label":790},"4",{"id":536,"label":792},"6",{"id":539,"label":794},"8","Last digits of powers of 2 cycle 2, 4, 8, 6. 100 ÷ 4 = 25 remainder 0, so it is the 4th in the cycle: 6.",{"itemId":797,"prompt":798,"options":799,"correct":536,"why":808},"patterns.extend-q-day","Today is Wednesday. What day is it in 30 days?",[800,802,804,806],{"id":530,"label":801},"Wednesday",{"id":533,"label":803},"Thursday",{"id":536,"label":805},"Friday",{"id":539,"label":807},"Saturday","30 ÷ 7 = 4 remainder 2, so it is 2 days after Wednesday: Friday.",{"itemId":810,"prompt":811,"options":812,"correct":530,"why":821},"patterns.extend-q-binary","What is 11 in binary?",[813,815,817,819],{"id":530,"label":814},"1011",{"id":533,"label":816},"1101",{"id":536,"label":818},"111",{"id":539,"label":820},"1001","11 = 8 + 2 + 1, so the eights, twos and ones places have 1s: 1011.",{"itemId":823,"prompt":824,"options":825,"correct":530,"why":834},"patterns.extend-q-chess","On the legendary chessboard, how many grains are on square 11?",[826,828,830,832],{"id":530,"label":827},"1,024",{"id":533,"label":829},"2,048",{"id":536,"label":831},"512",{"id":539,"label":833},"22","Square n has 2ⁿ⁻¹ grains: 2¹⁰ = 1,024.",{"itemId":836,"prompt":837,"options":838,"correct":536,"why":847},"patterns.extend-q-handshakes","How many handshakes if 12 people each shake hands once with everyone else?",[839,841,843,845],{"id":530,"label":840},"144",{"id":533,"label":842},"132",{"id":536,"label":844},"66",{"id":539,"label":846},"78","12 × 11 ÷ 2 = 66, the 11th triangular number.",{"itemId":849,"prompt":850,"options":851,"correct":533,"why":859},"patterns.extend-q-collatz","Using “halve if even, × 3 + 1 if odd”, what comes after 13?",[852,854,855,857],{"id":530,"label":853},"6.5",{"id":533,"label":742},{"id":536,"label":856},"39",{"id":539,"label":858},"26","13 is odd, so 3 × 13 + 1 = 40.",{"itemId":861,"prompt":862,"options":863,"correct":533,"why":872},"patterns.extend-q-leap","Which of these years was or will be a leap year?",[864,866,868,870],{"id":530,"label":865},"1900",{"id":533,"label":867},"2000",{"id":536,"label":869},"2100",{"id":539,"label":871},"2026","Century years are leap years only if divisible by 400. 2000 is; 1900 and 2100 are not; 2026 is not divisible by 4.",{"id":874,"type":875,"conceptId":876,"relation":877,"explanation":878},"conn-shape","connection","shape-and-space","applied_in","Tessellations and rangoli symmetry apply number patterns to shapes: angles meeting at a point must total 360°.",{"id":880,"type":875,"conceptId":881,"relation":882,"explanation":883},"conn-angles","angles","related_to","Whether a regular polygon tiles depends on its interior angle: 60°, 90° and 120° divide 360° exactly; 108° and 135° do not.",{"id":885,"type":875,"conceptId":886,"relation":882,"explanation":887},"conn-hcf-lcm","hcf-and-lcm","When two cycles run together (a 7-day week and a 4-year leap cycle, or two blinking lights), they line up again after the LCM of their lengths.",{"id":889,"type":875,"conceptId":890,"relation":882,"explanation":891},"conn-number-system","number-system","Binary is a place-value system with base 2 instead of 10: places are worth 1, 2, 4, 8, … instead of 1, 10, 100, 1,000.",{"id":893,"type":894,"title":895,"points":896},"cheat-extend","summary","Cheat sheet",[897,898,899,900,901,902,903,904,905,906],"**Magic squares:** every row, column and main diagonal has the same total, n(n² + 1) ÷ 2 for 1 to n². Lo Shu 15; the Khajuraho 4 × 4 is most-perfect with constant 34; the birthday square named for Ramanujan totals 139.","**Staircase method** builds any odd magic square: start top middle, go up-and-right, wrap round, drop down when blocked.","**Tessellations:** angles at a point must total 360°. Only regular triangles, squares and hexagons tile alone; any triangle or quadrilateral tiles.","**Figurate numbers:** pentagonal 1, 5, 12, 22, 35; hexagonal 1, 6, 15, 28; centred hexagonal 1, 7, 19, 37 (running totals are cubes); tetrahedral 1, 4, 10, 20.","**Cycles:** last digits of powers repeat (7, 9, 3, 1). Only the remainder matters: 7²⁰²⁶ ends in 9; 100 days after Sunday is Tuesday.","**Calendar:** leap years every 4, except centuries not divisible by 400. The calendar repeats every 400 years (146,097 days = 20,871 weeks).","**Chessboard legend:** 2⁶⁴ − 1 = 18,446,744,073,709,551,615 grains, about 461 thousand million tonnes at 0.025 g a grain. **Binary:** place values 1, 2, 4, 8, …; 13 = 1101.","**Olympiad tricks:** triangular positions, counting in directions, pairing, counting each object twice then halving.","**Patterns at work:** tala cycles and tihais, cricket overs, block printing, loops and recursion, codes, science.","**Open questions:** Collatz, Fibonacci primes, Goldbach, a magic square of squares, odd perfect numbers.",{"id":908,"type":909,"sourceIds":910},"sources-extend","sources",[911,912,913,914,915,916,917,918,919,920],"patterns-wiki-magic-square","patterns-mathsisfun-tessellation","patterns-wiki-kolam","patterns-wiki-pascal","patterns-mathsisfun-sequences","patterns-ncert-ganita-prakash-6","patterns-wiki-fibonacci","patterns-wiki-khajuraho-temple","patterns-fao-rice","patterns-apu-ramanujan-square",[911,912,913,914,915,916,917,918,919,920],"needs_review",{"generatedBy":924,"notes":925},"claude-code","Draft generated locally from a Python script with every number computed and asserted; pending owner review.","53a67371c9c08d6b90d7bd64450cfa396b812883088d3063bdd1614a890a8a44",{"logic:practice":928,"component:sort-game@1":929,"component:match-pairs@1":930,"component:pattern-machine@1":931,"source:patterns-apu-ramanujan-square":932,"source:patterns-fao-rice":933,"source:patterns-mathsisfun-sequences":934,"source:patterns-mathsisfun-tessellation":935,"source:patterns-ncert-ganita-prakash-6":936,"source:patterns-wiki-fibonacci":937,"source:patterns-wiki-khajuraho-temple":938,"source:patterns-wiki-kolam":939,"source:patterns-wiki-magic-square":940,"source:patterns-wiki-pascal":941},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b3d384b49b4d138a00726767a816928da91f44fad70d7d5ccfca6db297c45839","4330d2881cdc9a2b3d993fe429e73ba3fd67a6f86423bb9169edd9a561b1f594","fd163d772d23c86f2d9d8fd1ddee1b6d2f15a2fc4db7315e290102990c20823d","1c634ad2570a389d1bc96431d0a220aa359ad82b4d33f5b55eb0b12dab8bab4c","552d6990b0c9647be50d7bf06f7b19ad0522155428c1074c2fc676001333e421","36726d3ff16d73627216db999b62b46580dfaf514f313774e79669de57d7c595","2b7f65c540dd62068d0f516bd71c5de387803d7540246de06abc95456694188a","964964aea753b64fb590e980c7130fa008dc4c9d95ac0a206fc53eb50dbacdef","f04bc658d82c029f670482f92ee9b676d695909de4758c42114516748a798bdb","73b34d05e25e1a384421070a78b4e9263f7cab7309df8ca4978360e2d30306ae","8b958ff4286a612ef0e8589f7261e9e18d2878dcfc122f6277212421fcccfa92",{"state":943,"reviewer":944,"selfReview":945,"reviewedAt":946,"method":947},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598292]