[{"data":1,"prerenderedAt":1043},["ShallowReactive",2],{"questions:patterns":3},{"bank":4,"contentHash":1031,"dependencyHashes":1032,"releaseId":1042},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":1018,"reviewStatus":1027,"authoring":1028},1,"patterns","Number and shape patterns: question bank","Practise every skill in this topic: repeating patterns and remainders, continuing sequences, finding rules with differences and ratios, arithmetic and geometric sequences, square, cube, triangular and Fibonacci numbers, digit patterns, matchstick and dot patterns, nth-term rules, and puzzles from Gauss’s trick to patterns that break. Questions go from foundation to challenge, and every one has a full worked solution. Keep a pencil handy: drawing the first few cases often cracks a problem.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"repeating","Repeating patterns","Finding the repeating unit and using remainders to predict any item: beads, bangles, kolam borders, words and signals.",{"id":15,"title":16,"description":17},"next-term","What comes next?","Continuing number sequences of every kind: adding, subtracting, doubling, squares, triangular, alternating and Fibonacci.",{"id":19,"title":20,"description":21},"finding-rule","Finding the rule","Using differences and ratios, checking a rule on every term, and seeing that a few terms can fit more than one rule.",{"id":23,"title":24,"description":25},"arithmetic","Arithmetic sequences","Adding a constant: far terms, money and seating problems, working backwards and first negative terms.",{"id":27,"title":28,"description":29},"geometric","Geometric sequences","Multiplying by a constant: doubling, halving, powers, growth races and sums of powers of 2.",{"id":31,"title":32,"description":33},"special","Special numbers and digit patterns","Square, cube, triangular and Fibonacci numbers, odd-number sums, and digit patterns such as 111 × 111.",{"id":35,"title":36,"description":37},"shapes","Shape patterns and matchsticks","Counting sticks, dots, tiles and blocks in growing shape patterns and predicting far pictures.",{"id":39,"title":40,"description":41},"nth-term","The nth term","Writing and using position-to-term rules, equivalent expressions, and spotting mistakes in rules.",{"id":43,"title":44,"description":45},"puzzles","Puzzles and reasoning","Gauss’s pairing, handshakes, calendar and hundred-square tricks, magic squares, cycles, Meru Prastara and patterns that break.",[47,72,85,99,113,124,140,150,159,168,177,186,195,204,213,232,250,269,288,306,324,334,344,353,364,375,384,393,403,420,430,439,449,459,470,480,497,507,517,536,546,555,566,584,594,603,612,622,631,642,652,661,670,681,690,709,719,728,736,754,771,790,808,817,836,845,864,874,884,893,903,912,921,931,939,950,960,970,981,990,999],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":67,"solution":69,"skills":70},"patterns.q001","foundation","What is the repeating unit of this pattern? **sun, moon, star, sun, moon, star, sun, moon, star**",{"kind":52,"options":53,"correct":66},"choice",[54,57,60,63],{"id":55,"label":56},"a","sun, moon",{"id":58,"label":59},"b","sun, moon, star",{"id":61,"label":62},"c","moon, star",{"id":64,"label":65},"d","sun, moon, star, sun",[58],[68],"Where does the pattern start again?","The group **sun, moon, star** comes round again and again, so it is the repeating unit. Writing it three times, sun, moon, star, sun, moon, star, sun, moon, star, gives back the whole pattern exactly.",[71],"repeating unit",{"id":73,"section":11,"level":49,"prompt":74,"check":75,"hints":79,"solution":82,"skills":83},"patterns.q002","A necklace repeats the beads **red, blue, blue, red, blue, blue, …**. What colour is the **17th** bead?",{"kind":76,"accept":77},"text",[78],"blue",[80,81],"How many beads are in the repeating unit?","Divide 17 by the unit length and look at the remainder.","The unit is red, blue, blue: 3 beads. 17 ÷ 3 = 5 remainder 2. After 5 whole units (15 beads), bead 16 is the 1st of a unit (red) and bead 17 is the 2nd (**blue**).",[71,84],"remainders",{"id":86,"section":11,"level":87,"prompt":88,"check":89,"hints":94,"solution":97,"skills":98},"patterns.q003","core","A kolam border repeats four motifs: **loop, dot, loop, line**. How many **loops** are there in the first **30** motifs?",{"kind":90,"answer":91,"tolerance":92,"unit":93},"number",15,0,"loops",[95,96],"How many loops are in one unit?","Count the whole units first, then the leftover motifs.","Each unit of 4 motifs has 2 loops. 30 ÷ 4 = 7 remainder 2, so there are 7 whole units (28 motifs) with 7 × 2 = 14 loops. The 2 extra motifs are loop, dot: one more loop. Total **15** loops.",[71,84],{"id":100,"section":11,"level":87,"prompt":101,"check":102,"hints":109,"solution":111,"skills":112},"patterns.q004","Bangles are stacked in the pattern **2 red, then 3 gold**, again and again. What colour is the **23rd** bangle?",{"kind":52,"options":103,"correct":108},[104,106],{"id":55,"label":105},"Red",{"id":58,"label":107},"Gold",[58],[110],"The unit has 5 bangles.","The unit is red, red, gold, gold, gold: 5 bangles. 23 ÷ 5 = 4 remainder 3. After 4 whole units (20 bangles), bangle 23 is the 3rd of a unit, which is **gold**.",[71,84],{"id":114,"section":11,"level":115,"prompt":116,"check":117,"hints":120,"solution":122,"skills":123},"patterns.q005","stretch","The word **INDIA** is written again and again with no spaces: INDIAINDIAINDIA… What is the **2,026th** letter?",{"kind":76,"accept":118},[119],"I",[121],"Only the remainder after dividing by 5 matters.","The unit INDIA has 5 letters. 2,026 ÷ 5 = 405 remainder 1. After 405 whole words (2,025 letters), the 2,026th letter is the 1st letter of the word: **I**.",[71,84],{"id":125,"section":11,"level":115,"prompt":126,"check":127,"hints":135,"solution":137,"skills":138},"patterns.q006","A traffic signal shows **green for 45 seconds, amber for 5 seconds, red for 40 seconds**, and then repeats. It turns green at 0 seconds. What colour is it at **200 seconds**?",{"kind":52,"options":128,"correct":134},[129,131,133],{"id":55,"label":130},"Green",{"id":58,"label":132},"Amber",{"id":61,"label":105},[55],[136],"How long is one full cycle?","One full cycle lasts 45 + 5 + 40 = 90 seconds. 200 ÷ 90 = 2 remainder 20. After 2 full cycles (180 s), the signal is 20 seconds into a new cycle. Green lasts from 0 to 45 seconds, so it is **green**.",[139,84],"cycles",{"id":141,"section":15,"level":49,"prompt":142,"check":143,"hints":145,"solution":147,"skills":148},"patterns.q007","What is the next term? **12, 17, 22, 27, …**",{"kind":90,"answer":144,"tolerance":92},32,[146],"Subtract neighbouring terms.","The difference is 5 each time (17 − 12 = 5), so the next term is 27 + 5 = **32**.",[149],"next term",{"id":151,"section":15,"level":49,"prompt":152,"check":153,"hints":155,"solution":157,"skills":158},"patterns.q008","What is the next term? **90, 81, 72, …**",{"kind":90,"answer":154,"tolerance":92},63,[156],"The numbers are getting smaller.","Each term is 9 less than the one before, so the next term is 72 − 9 = **63**. (These are the multiples of 9 counting back.)",[149],{"id":160,"section":15,"level":49,"prompt":161,"check":162,"hints":164,"solution":166,"skills":167},"patterns.q009","What is the next term? **1, 2, 4, 8, …**",{"kind":90,"answer":163,"tolerance":92},16,[165],"Try dividing a term by the one before it.","Each term is double the one before (× 2), so the next term is 8 × 2 = **16**.",[149],{"id":169,"section":15,"level":49,"prompt":170,"check":171,"hints":173,"solution":175,"skills":176},"patterns.q010","What is the next term? **1, 4, 9, 16, 25, …**",{"kind":90,"answer":172,"tolerance":92},36,[174],"Think of dot squares.","These are the square numbers 1 × 1, 2 × 2, …, 5 × 5. Next is 6 × 6 = **36**. (Or: the differences 3, 5, 7, 9 are odd numbers, so add 11.)",[149],{"id":178,"section":15,"level":87,"prompt":179,"check":180,"hints":182,"solution":184,"skills":185},"patterns.q011","What is the next term? **1, 3, 6, 10, 15, …**",{"kind":90,"answer":181,"tolerance":92},21,[183],"Look at how the differences change.","The differences are 2, 3, 4, 5, going up by 1. The next difference is 6, so the next term is 15 + 6 = **21**. These are the triangular numbers.",[149],{"id":187,"section":15,"level":87,"prompt":188,"check":189,"hints":191,"solution":193,"skills":194},"patterns.q012","What is the next term? **2, 5, 4, 7, 6, 9, …**",{"kind":90,"answer":190,"tolerance":92},8,[192],"Write the steps between terms, including signs.","The steps alternate: +3, −1, +3, −1, +3. The next step is −1, so the next term is 9 − 1 = **8**.",[149],{"id":196,"section":15,"level":87,"prompt":197,"check":198,"hints":200,"solution":202,"skills":203},"patterns.q013","What is the next term? **3, 5, 8, 13, 21, …**",{"kind":90,"answer":199,"tolerance":92},34,[201],"Add two neighbouring terms.","Each term is the sum of the two before it: 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21. Next: 13 + 21 = **34**. This is part of the Fibonacci sequence.",[149],{"id":205,"section":15,"level":115,"prompt":206,"check":207,"hints":209,"solution":211,"skills":212},"patterns.q014","What is the next term? **2, 6, 12, 20, 30, …**",{"kind":90,"answer":208,"tolerance":92},42,[210],"Try writing each term as a product of two neighbouring numbers.","Differences are 4, 6, 8, 10, going up by 2, so the next difference is 12 and the next term is 30 + 12 = **42**. Another way: the terms are 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, so next is 6 × 7 = 42.",[149],{"id":214,"section":19,"level":49,"prompt":215,"check":216,"hints":227,"solution":229,"skills":230},"patterns.q015","Which rule makes the sequence **5, 10, 15, 20, 25, …**?",{"kind":52,"options":217,"correct":226},[218,220,222,224],{"id":55,"label":219},"Multiply by 2",{"id":58,"label":221},"Add 5 each time",{"id":61,"label":223},"Add 10 each time",{"id":64,"label":225},"Multiply by 5",[58],[228],"Check the rule on every pair of neighbours.","The differences are all 5: 10 − 5 = 5, 15 − 10 = 5, and so on. So the rule is **add 5 each time** (start at 5). “Multiply by 2” works for 5 → 10 but not 10 → 15.",[231],"finding rules",{"id":233,"section":19,"level":49,"prompt":234,"check":235,"hints":246,"solution":248,"skills":249},"patterns.q016","Which rule makes the sequence **3, 9, 27, 81, …**?",{"kind":52,"options":236,"correct":245},[237,239,241,243],{"id":55,"label":238},"Add 6 each time",{"id":58,"label":240},"Multiply by 3 each time",{"id":61,"label":242},"Add 3, then 6, then 9",{"id":64,"label":244},"Square the term",[58],[247],"Divide each term by the one before it.","9 ÷ 3 = 3, 27 ÷ 9 = 3, 81 ÷ 27 = 3. Each term is **3 times** the one before, so this is a geometric sequence with ratio 3.",[231,27],{"id":251,"section":19,"level":87,"prompt":252,"check":253,"hints":264,"solution":266,"skills":267},"patterns.q017","Which rule fits **all** the terms of **1, 2, 4, 7, 11, 16, …**?",{"kind":52,"options":254,"correct":263},[255,257,259,261],{"id":55,"label":256},"Double each time",{"id":58,"label":258},"Add 1, then 2, then 3, then 4, …",{"id":61,"label":260},"Add 3 each time",{"id":64,"label":262},"Square numbers",[58],[265],"Write down the differences.","The differences are 1, 2, 3, 4, 5: each time you add one more than last time. Doubling fits 1, 2, 4 but then gives 8, not 7. So the rule is **add 1, then 2, then 3, …**, and the next term is 16 + 6 = 22.",[231,268],"differences",{"id":270,"section":19,"level":87,"prompt":271,"check":272,"hints":283,"solution":285,"skills":286},"patterns.q018","For the sequence **4, 8, 16, 32**, Rahul says the rule is “add 4 each time”. Ananya says it is “double each time”. Who is right?",{"kind":52,"options":273,"correct":282},[274,276,278,280],{"id":55,"label":275},"Rahul",{"id":58,"label":277},"Ananya",{"id":61,"label":279},"Both are right",{"id":64,"label":281},"Neither is right",[58],[284],"Test each rule on the second and third gaps too.","“Add 4” works for 4 → 8, but 8 + 4 = 12, not 16. “Double” works for every step: 4 × 2 = 8, 8 × 2 = 16, 16 × 2 = 32. So **Ananya** is right. Rahul only checked the first gap.",[231,287],"checking",{"id":289,"section":19,"level":87,"prompt":290,"check":291,"hints":302,"solution":304,"skills":305},"patterns.q019","Which of these sequences is **not** arithmetic (does not add the same number each time)?",{"kind":52,"options":292,"correct":301},[293,295,297,299],{"id":55,"label":294},"7, 10, 13, 16",{"id":58,"label":296},"50, 45, 40, 35",{"id":61,"label":298},"2, 4, 8, 16",{"id":64,"label":300},"0.5, 1.5, 2.5, 3.5",[61],[303],"Work out the differences for each one.","An arithmetic sequence has equal differences. (a) adds 3, (b) adds −5, (d) adds 1. In (c) the differences are 2, 4, 8, which are not equal: it multiplies by 2, so it is **geometric**, not arithmetic.",[23,27],{"id":307,"section":19,"level":115,"prompt":308,"check":309,"hints":319,"solution":321,"skills":322},"patterns.q020","The first three terms of a sequence are **1, 2, 4**. Which of these rules fit **all three** terms? (Choose every one that fits.)",{"kind":52,"options":310,"correct":318},[311,312,314,316],{"id":55,"label":256},{"id":58,"label":313},"Add 1, then 2, then 3, …",{"id":61,"label":315},"Add 1 each time",{"id":64,"label":317},"(n × n − n + 2) ÷ 2, where n is the position",[55,58,64],[320],"Test each rule on n = 1, 2 and 3.","Double: 1, 2, 4 ✓. Add 1, 2, 3: 1, 2, 4 ✓. Add 1 each time: 1, 2, 3 ✗. The formula: n = 1 gives (1 − 1 + 2) ÷ 2 = 1, n = 2 gives (4 − 2 + 2) ÷ 2 = 2, n = 3 gives (9 − 3 + 2) ÷ 2 = 4 ✓. So **three different rules** fit, and they predict different 4th terms (8, 7 and 7). Three terms are not enough to be sure of a rule.",[231,323],"reasoning",{"id":325,"section":19,"level":115,"prompt":326,"check":327,"hints":329,"solution":331,"skills":332},"patterns.q021","Find the next term of **3, 4, 7, 12, 19, …**",{"kind":90,"answer":328,"tolerance":92},28,[330],"Look at the differences, then at the differences of the differences.","Differences: 1, 3, 5, 7, the odd numbers. The next difference is 9, so the next term is 19 + 9 = **28**. (The second differences are all 2, so the formula has n² in it: it is (n − 1)² + 3.)",[268,333],"quadratic",{"id":335,"section":23,"level":49,"prompt":336,"check":337,"hints":339,"solution":341,"skills":342},"patterns.q022","What is the **10th** term of **4, 7, 10, 13, …**?",{"kind":90,"answer":338,"tolerance":92},31,[340],"How many jumps from the 1st term to the 10th?","The difference is 3. From the 1st term to the 10th there are 9 jumps of 3: 4 + 9 × 3 = 4 + 27 = **31**. (Or keep adding 3: 16, 19, 22, 25, 28, 31.)",[23,343],"far terms",{"id":345,"section":23,"level":87,"prompt":346,"check":347,"hints":349,"solution":351,"skills":352},"patterns.q023","What is the **20th** term of **5, 9, 13, 17, …**?",{"kind":90,"answer":348,"tolerance":92},81,[350],"There are 19 jumps, not 20.","Difference 4; from the 1st to the 20th term there are 19 jumps. 5 + 19 × 4 = 5 + 76 = **81**. Check with the formula 4n + 1: 4 × 20 + 1 = 81 ✓.",[23,343],{"id":354,"section":23,"level":87,"prompt":355,"check":356,"hints":359,"solution":361,"skills":362},"patterns.q024","An auto-rickshaw charges **₹25 for the first kilometre** and **₹12 for every extra kilometre**. How much is a **10 km** ride?",{"kind":90,"answer":357,"tolerance":92,"unit":358},133,"₹",[360],"The first kilometre costs ₹25; how many extra kilometres are there?","The fares go 25, 37, 49, … (add ₹12 per km). A 10 km ride has 9 extra kilometres: 25 + 9 × 12 = 25 + 108 = **₹133**.",[23,363],"money",{"id":365,"section":23,"level":87,"prompt":366,"check":367,"hints":370,"solution":372,"skills":373},"patterns.q025","Meena has **₹150** in her piggy bank and adds **₹35** every week. After how many weeks will she have exactly **₹500**?",{"kind":90,"answer":368,"tolerance":92,"unit":369},10,"weeks",[371],"How much more money does she need?","She needs 500 − 150 = ₹350 more. At ₹35 a week that takes 350 ÷ 35 = **10 weeks**. Check: 150 + 10 × 35 = 150 + 350 = 500 ✓.",[23,374],"working backwards",{"id":376,"section":23,"level":87,"prompt":377,"check":378,"hints":379,"solution":381,"skills":382},"patterns.q026","What is the **15th** term of **100, 94, 88, 82, …**?",{"kind":90,"answer":163,"tolerance":92},[380],"Each jump subtracts 6.","The difference is −6. There are 14 jumps: 100 − 14 × 6 = 100 − 84 = **16**.",[23,383],"decreasing",{"id":385,"section":23,"level":87,"prompt":386,"check":387,"hints":389,"solution":391,"skills":392},"patterns.q027","In a school hall, the front row has **20 seats**, and each row behind has **2 more** seats than the row in front. How many seats are in the **12th** row?",{"kind":90,"answer":208,"tolerance":92,"unit":388},"seats",[390],"There are 11 jumps from row 1 to row 12.","Rows: 20, 22, 24, … From row 1 to row 12 there are 11 jumps of 2: 20 + 11 × 2 = **42** seats.",[23,343],{"id":394,"section":23,"level":115,"prompt":395,"check":396,"hints":398,"solution":401,"skills":402},"patterns.q028","Which term of **7, 12, 17, 22, …** is **102**?",{"kind":90,"answer":397,"tolerance":92},20,[399,400],"Find the nth-term formula first.","Undo the + 2, then divide by 5.","The nth term is 5n + 2 (step 5; 5 × 1 + 2 = 7). Solve 5n + 2 = 102: 5n = 100, so n = **20**. It is the 20th term. Check: 7 + 19 × 5 = 102 ✓.",[23,374],{"id":404,"section":23,"level":115,"prompt":405,"check":406,"hints":415,"solution":417,"skills":418},"patterns.q029","Is **250** a term of the sequence **4, 7, 10, 13, …** (rule 3n + 1)?",{"kind":52,"options":407,"correct":414},[408,410,412],{"id":55,"label":409},"Yes, it is the 83rd term",{"id":58,"label":411},"Yes, it is the 84th term",{"id":61,"label":413},"No, it is not a term",[55],[416],"Solve 3n + 1 = 250 and see whether n is a whole number.","Solve 3n + 1 = 250: 3n = 249, n = 249 ÷ 3 = 83, a whole number. So **250 is the 83rd term**. Check: 3 × 83 + 1 = 249 + 1 = 250 ✓.",[23,419],"is it a term",{"id":421,"section":23,"level":115,"prompt":422,"check":423,"hints":425,"solution":427,"skills":428},"patterns.q030","The sequence **60, 53, 46, 39, …** keeps subtracting 7. What is the **first negative term**?",{"kind":90,"answer":424,"tolerance":92},-3,[426],"Keep subtracting 7 until you go below zero.","Terms: 60, 53, 46, 39, 32, 25, 18, 11, 4, then 4 − 7 = **−3**. It is the 10th term: 60 − 9 × 7 = 60 − 63 = −3.",[23,429],"negative numbers",{"id":431,"section":27,"level":49,"prompt":432,"check":433,"hints":435,"solution":437,"skills":438},"patterns.q031","What is the next term? **3, 6, 12, 24, …**",{"kind":90,"answer":434,"tolerance":92},48,[436],"Divide a term by the one before.","Each term is double the one before: 24 × 2 = **48**.",[27],{"id":440,"section":27,"level":87,"prompt":441,"check":442,"hints":444,"solution":446,"skills":447},"patterns.q032","What is the **7th** term of **1, 3, 9, 27, …**?",{"kind":90,"answer":443,"tolerance":92},729,[445],"There are 6 multiplications from the 1st term to the 7th.","Multiply by 3 each time: 1, 3, 9, 27, 81, 243, **729**. With the formula: 1 × 3⁶ = 729 (six multiplications to get from the 1st to the 7th term).",[27,448],"powers",{"id":450,"section":27,"level":87,"prompt":451,"check":452,"hints":454,"solution":456,"skills":457},"patterns.q033","What comes next? **800, 400, 200, 100, …**",{"kind":90,"answer":453,"tolerance":92},50,[455],"Is it subtracting or halving?","Each term is half the one before (ratio ½): 100 ÷ 2 = **50**.",[27,458],"halving",{"id":460,"section":27,"level":87,"prompt":461,"check":462,"hints":465,"solution":467,"skills":468},"patterns.q034","A single bacterium splits into 2 every **20 minutes**, and every new one does the same. How many bacteria are there after **3 hours**?",{"kind":90,"answer":463,"tolerance":92,"unit":464},512,"bacteria",[466],"How many 20-minute periods are in 3 hours?","3 hours = 180 minutes = 9 periods of 20 minutes, so the number doubles 9 times: 1 → 2 → 4 → 8 → 16 → 32 → 64 → 128 → 256 → **512**. That is 2⁹ = 512.",[27,469],"doubling",{"id":471,"section":27,"level":115,"prompt":472,"check":473,"hints":475,"solution":478,"skills":479},"patterns.q035","Which term of **5, 15, 45, 135, …** is **3,645**?",{"kind":90,"answer":474,"tolerance":92},7,[476,477],"Divide by 5 first.","Which power of 3 is 729?","Each term is 5 × a power of 3. 3,645 ÷ 5 = 729 = 3⁶. The 1st term has 3⁰, the 2nd 3¹, …, so 3⁶ belongs to the **7th** term. Check: 5, 15, 45, 135, 405, 1,215, 3,645 ✓.",[27,374],{"id":481,"section":27,"level":115,"prompt":482,"check":483,"hints":492,"solution":494,"skills":495},"patterns.q036","Sequence A is 1,000, 2,000, 3,000, … (add 1,000). Sequence B is 1, 2, 4, 8, … (double). Which has the larger **15th** term?",{"kind":52,"options":484,"correct":491},[485,487,489],{"id":55,"label":486},"Sequence A",{"id":58,"label":488},"Sequence B",{"id":61,"label":490},"They are equal",[58],[493],"The 15th term of B is 2 multiplied by itself 14 times.","A: 15th term = 1,000 × 15 = 15,000. B: 15th term = 2¹⁴ = 16,384. So **B** is larger, having overtaken A at the 15th term (the 14th terms are 14,000 and 8,192). Doubling always wins in the end.",[27,496],"comparing growth",{"id":498,"section":27,"level":115,"prompt":499,"check":500,"hints":502,"solution":504,"skills":505},"patterns.q037","Find **1 + 2 + 4 + 8 + … + 256** (powers of 2).",{"kind":90,"answer":501,"tolerance":92},511,[503],"Try 1 + 2, then 1 + 2 + 4, then 1 + 2 + 4 + 8. What do you notice?","Adding powers of 2 always gives one less than the next power: 1 = 2 − 1, 1 + 2 = 4 − 1, 1 + 2 + 4 = 8 − 1. The next power after 256 is 512, so the sum is 512 − 1 = **511**. (Double the sum and subtract the original to see why: everything cancels except 512 − 1.)",[27,506],"sums",{"id":508,"section":31,"level":49,"prompt":509,"check":510,"hints":512,"solution":514,"skills":515},"patterns.q038","What is the **7th** square number?",{"kind":90,"answer":511,"tolerance":92},49,[513],"n × n","Square numbers are 1 × 1, 2 × 2, 3 × 3, … The 7th is 7 × 7 = **49**.",[516],"square numbers",{"id":518,"section":31,"level":49,"prompt":519,"check":520,"hints":531,"solution":533,"skills":534},"patterns.q039","Which of these is a **triangular number**?",{"kind":52,"options":521,"correct":530},[522,524,526,528],{"id":55,"label":523},"12",{"id":58,"label":525},"18",{"id":61,"label":527},"21",{"id":64,"label":529},"25",[61],[532],"List them: 1, 3, 6, 10, …","Triangular numbers: 1, 3, 6, 10, 15, **21**, 28, … (add 2, 3, 4, 5, 6, 7). 21 = 1 + 2 + 3 + 4 + 5 + 6. None of 12, 18 or 25 is on the list.",[535],"triangular numbers",{"id":537,"section":31,"level":49,"prompt":538,"check":539,"hints":541,"solution":543,"skills":544},"patterns.q040","What is the **4th** cube number?",{"kind":90,"answer":540,"tolerance":92},64,[542],"4 × 4 × 4","Cube numbers are n × n × n. The 4th is 4 × 4 × 4 = 16 × 4 = **64**.",[545],"cube numbers",{"id":547,"section":31,"level":87,"prompt":548,"check":549,"hints":551,"solution":553,"skills":554},"patterns.q041","What is the **12th** triangular number?",{"kind":90,"answer":550,"tolerance":92},78,[552],"Use n × (n + 1) ÷ 2.","The nth triangular number is n × (n + 1) ÷ 2. For n = 12: 12 × 13 ÷ 2 = 156 ÷ 2 = **78**. (Or 1 + 2 + … + 12, pairing 1 + 12, 2 + 11, …: 6 pairs of 13 = 78.)",[535],{"id":556,"section":31,"level":87,"prompt":557,"check":558,"hints":560,"solution":563,"skills":564},"patterns.q042","Find **1 + 3 + 5 + 7 + … + 39** (all the odd numbers up to 39).",{"kind":90,"answer":559,"tolerance":92},400,[561,562],"How many odd numbers are there from 1 to 39?","The first n odd numbers add up to a square number.","The odd numbers up to 39 are the first 20 odd numbers (the 20th is 2 × 20 − 1 = 39). The first n odd numbers add up to n × n, so the total is 20 × 20 = **400**.",[565,516],"odd numbers",{"id":567,"section":31,"level":87,"prompt":568,"check":569,"hints":580,"solution":582,"skills":583},"patterns.q043","Which number is **both** a square number and a cube number?",{"kind":52,"options":570,"correct":579},[571,573,575,577],{"id":55,"label":572},"16",{"id":58,"label":574},"27",{"id":61,"label":576},"64",{"id":64,"label":578},"81",[61],[581],"List the cubes: 1, 8, 27, 64, 125.","**64** = 8 × 8 = 4 × 4 × 4, so it is both. 16 and 81 are squares but not cubes; 27 is a cube but not a square.",[516,545],{"id":585,"section":31,"level":87,"prompt":586,"check":587,"hints":589,"solution":591,"skills":592},"patterns.q044","In the Fibonacci sequence **1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …**, what comes after 89?",{"kind":90,"answer":588,"tolerance":92},144,[590],"Add the last two terms.","Each term is the sum of the two before: 55 + 89 = **144**. (144 is also 12 × 12, the only Fibonacci number bigger than 1 that is a square.)",[593],"Fibonacci",{"id":595,"section":31,"level":115,"prompt":596,"check":597,"hints":599,"solution":601,"skills":602},"patterns.q045","What is the **13th** term of the Fibonacci sequence 1, 1, 2, 3, 5, …?",{"kind":90,"answer":598,"tolerance":92},233,[600],"Write the terms out carefully, counting positions.","Continue: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, **233**. The 13th term is 89 + 144 = 233.",[593],{"id":604,"section":31,"level":115,"prompt":605,"check":606,"hints":608,"solution":610,"skills":611},"patterns.q046","How many square numbers are there from **1 to 200** (including 1)?",{"kind":90,"answer":607,"tolerance":92},14,[609],"Find the biggest number whose square is at most 200.","14 × 14 = 196 is at most 200, but 15 × 15 = 225 is more. So the squares are 1², 2², …, 14²: **14** square numbers.",[516],{"id":613,"section":31,"level":87,"prompt":614,"check":615,"hints":617,"solution":619,"skills":620},"patterns.q047","Use the pattern 11 × 11 = 121, 111 × 111 = 12,321, 1,111 × 1,111 = 1,234,321 to find **111,111 × 111,111**.",{"kind":90,"answer":616,"tolerance":92},12345654321,[618],"Count the 1s: the answer counts up to that number and back.","With six 1s, the answer counts up to 6 and back down: **12,345,654,321**. The pattern works because each column of the long multiplication adds up to at most 9, so there is no carrying.",[621],"digit patterns",{"id":623,"section":31,"level":87,"prompt":624,"check":625,"hints":627,"solution":629,"skills":630},"patterns.q048","Continue the pattern 1 × 8 + 1 = 9, 12 × 8 + 2 = 98, 123 × 8 + 3 = 987. What is **12,345 × 8 + 5**?",{"kind":90,"answer":626,"tolerance":92},98765,[628],"Look at how the answers grow by one digit each line.","Each line adds a digit to the count-down: 9, 98, 987, 9,876, **98,765**. Check: 12,345 × 8 = 98,760, and 98,760 + 5 = 98,765 ✓.",[621],{"id":632,"section":35,"level":49,"prompt":633,"check":634,"hints":637,"solution":639,"skills":640},"patterns.q049","Matchstick squares in a row need 4, 7, 10, 13, … sticks. How many sticks for **6 squares**?",{"kind":90,"answer":635,"tolerance":92,"unit":636},19,"matchsticks",[638],"Add 3 each time.","Each new square adds 3 sticks: 4, 7, 10, 13, 16, **19**. (Rule: 3 × 6 + 1 = 19.)",[641],"matchstick patterns",{"id":643,"section":35,"level":49,"prompt":644,"check":645,"hints":647,"solution":649,"skills":650},"patterns.q050","Square dot patterns have 1, 4, 9, 16, … dots. How many dots are in the **9th** pattern?",{"kind":90,"answer":348,"tolerance":92,"unit":646},"dots",[648],"The nth pattern is n dots by n dots.","The 9th pattern is a 9 by 9 square: 9 × 9 = **81** dots.",[651,516],"dot patterns",{"id":653,"section":35,"level":87,"prompt":654,"check":655,"hints":657,"solution":659,"skills":660},"patterns.q051","Matchstick triangles in a row use 3, 5, 7, … sticks (2 × n + 1). How many sticks for **25 triangles**?",{"kind":90,"answer":656,"tolerance":92,"unit":636},51,[658],"Use 2 × n + 1.","2 × 25 + 1 = 50 + 1 = **51** sticks. Each triangle after the first adds 2 sticks because it shares a side.",[641,343],{"id":662,"section":35,"level":87,"prompt":663,"check":664,"hints":666,"solution":668,"skills":669},"patterns.q052","Hexagons made of matchsticks are joined in a row: 6, 11, 16, … sticks. How many sticks for **12 hexagons**?",{"kind":90,"answer":665,"tolerance":92,"unit":636},61,[667],"How many new sticks does each hexagon add?","Each new hexagon shares one side, so it adds 5 sticks. Rule: 5 × n + 1. For n = 12: 60 + 1 = **61**.",[641],{"id":671,"section":35,"level":87,"prompt":672,"check":673,"hints":676,"solution":678,"skills":679},"patterns.q053","L-shapes of tiles grow 1, 3, 5, 7, … tiles (each arm one tile longer). How many tiles in the **20th** L-shape?",{"kind":90,"answer":674,"tolerance":92,"unit":675},39,"tiles",[677],"These are the odd numbers.","The rule is 2 × n − 1 (the odd numbers). 2 × 20 − 1 = **39** tiles: two arms of 19 tiles plus the corner tile.",[680],"shape patterns",{"id":682,"section":35,"level":87,"prompt":683,"check":684,"hints":687,"solution":688,"skills":689},"patterns.q054","A staircase of blocks has 1 block in the first column, 2 in the next, 3 in the next, and so on. How many blocks in a staircase with **15 columns**?",{"kind":90,"answer":685,"tolerance":92,"unit":686},120,"blocks",[552],"1 + 2 + … + 15 = 15 × 16 ÷ 2 = **120** blocks. Two identical staircases fit together into a 15 by 16 rectangle of 240 blocks; one staircase is half.",[535,680],{"id":691,"section":35,"level":87,"prompt":692,"check":693,"hints":704,"solution":706,"skills":707},"patterns.q055","Matchstick **pentagons** are joined in a row, each sharing one side with the next: 5, 9, 13, … sticks. Which rule gives the number of sticks for n pentagons?",{"kind":52,"options":694,"correct":703},[695,697,699,701],{"id":55,"label":696},"5 × n",{"id":58,"label":698},"4 × n + 1",{"id":61,"label":700},"4 × n + 5",{"id":64,"label":702},"5 × n − 1",[58],[705],"How many sticks does each extra pentagon add?","Each new pentagon shares one side, so it adds 4 sticks: the coefficient of n is 4. For n = 1 we need 5, so add 1: **4 × n + 1**. Check: n = 3 gives 13 ✓.",[641,708],"nth term",{"id":710,"section":35,"level":115,"prompt":711,"check":712,"hints":715,"solution":717,"skills":718},"patterns.q056","How many hexagons can you make in a row with exactly **151** matchsticks (rule 5 × n + 1)?",{"kind":90,"answer":713,"tolerance":92,"unit":714},30,"hexagons",[716],"Undo the + 1, then divide by 5.","Solve 5 × n + 1 = 151: 5 × n = 150, so n = **30**. Check: 5 × 30 + 1 = 151 ✓.",[641,374],{"id":720,"section":35,"level":115,"prompt":721,"check":722,"hints":724,"solution":726,"skills":727},"patterns.q057","A rectangle of matchstick squares is **2 rows high** and **20 squares long**. How many matchsticks does it use?",{"kind":90,"answer":723,"tolerance":92,"unit":636},102,[725],"Count horizontal and vertical sticks separately.","Horizontal sticks: 3 lines (top, middle, bottom) of 20 = 60. Vertical sticks: 21 upright lines of 2 = 42. Total 60 + 42 = **102**. (The rule is 5 × n + 2: 5 × 20 + 2 = 102.)",[641,323],{"id":729,"section":35,"level":115,"prompt":730,"check":731,"hints":732,"solution":734,"skills":735},"patterns.q058","Priya builds the **first five** pictures of the matchstick-squares pattern (1 square, 2 squares, … 5 squares), each separately. How many matchsticks does she need altogether?",{"kind":90,"answer":453,"tolerance":92,"unit":636},[733],"Find the sticks for each picture first.","The pictures need 4, 7, 10, 13 and 16 sticks. Pair them: 4 + 16 = 20, 7 + 13 = 20, and 10 in the middle. Total 20 + 20 + 10 = **50**.",[641,506],{"id":737,"section":39,"level":87,"prompt":738,"check":739,"hints":750,"solution":752,"skills":753},"patterns.q059","What is the nth-term rule for **6, 10, 14, 18, …**?",{"kind":52,"options":740,"correct":749},[741,743,745,747],{"id":55,"label":742},"4 × n + 6",{"id":58,"label":744},"4 × n + 2",{"id":61,"label":746},"6 × n + 4",{"id":64,"label":748},"n + 4",[58],[751],"The step becomes the number multiplying n.","The step is 4, so the rule starts 4 × n. For n = 1, 4 × 1 = 4, but we need 6, so add 2: **4 × n + 2**. Check: n = 3 gives 14 ✓.",[708],{"id":755,"section":39,"level":87,"prompt":756,"check":757,"hints":767,"solution":769,"skills":770},"patterns.q060","What is the nth-term rule for the odd numbers **1, 3, 5, 7, …**?",{"kind":52,"options":758,"correct":766},[759,761,763,765],{"id":55,"label":760},"2 × n + 1",{"id":58,"label":762},"2 × n − 1",{"id":61,"label":764},"n + 2",{"id":64,"label":513},[58],[768],"Compare with the even numbers 2, 4, 6, 8.","Step 2 gives 2 × n: 2, 4, 6, 8. Each odd number is one less, so the rule is **2 × n − 1**. Check: n = 4 gives 7 ✓.",[708,565],{"id":772,"section":39,"level":87,"prompt":773,"check":774,"hints":785,"solution":787,"skills":788},"patterns.q061","Which expression is **equivalent** to **4 + 3 × (n − 1)**?",{"kind":52,"options":775,"correct":784},[776,778,780,782],{"id":55,"label":777},"3 × n + 4",{"id":58,"label":779},"3 × n + 1",{"id":61,"label":781},"4 × n − 1",{"id":64,"label":783},"7 × n − 7",[58],[786],"Multiply out the bracket: 3 × (n − 1) = 3 × n − 3.","Expand: 4 + 3 × n − 3 = **3 × n + 1**. Both describe matchstick squares: 4 sticks for the first square, then 3 for each extra square.",[789],"equivalent expressions",{"id":791,"section":39,"level":115,"prompt":792,"check":793,"hints":804,"solution":806,"skills":807},"patterns.q062","What is the nth-term rule for **20, 17, 14, 11, …**?",{"kind":52,"options":794,"correct":803},[795,797,799,801],{"id":55,"label":796},"3 × n + 17",{"id":58,"label":798},"20 − 3 × n",{"id":61,"label":800},"23 − 3 × n",{"id":64,"label":802},"17 − 3 × n",[61],[805],"The first term minus the step gives the constant.","The step is −3, so the rule contains −3 × n. For n = 1, −3 × 1 = −3, but we need 20, so add 23: **23 − 3 × n**. Check: n = 4 gives 23 − 12 = 11 ✓.",[708,383],{"id":809,"section":39,"level":115,"prompt":810,"check":811,"hints":813,"solution":815,"skills":816},"patterns.q063","A sequence has nth term **7 × n − 3**. What is its **50th** term?",{"kind":90,"answer":812,"tolerance":92},347,[814],"Multiply first, then subtract.","Put n = 50: 7 × 50 − 3 = 350 − 3 = **347**.",[708,343],{"id":818,"section":39,"level":115,"prompt":819,"check":820,"hints":831,"solution":833,"skills":834},"patterns.q064","Asha says the nth term of **5, 8, 11, 14, …** is **3 × n + 5**. What is wrong, and what is the correct rule?",{"kind":52,"options":821,"correct":830},[822,824,826,828],{"id":55,"label":823},"Nothing is wrong",{"id":58,"label":825},"The step should be 5, so 5 × n + 3",{"id":61,"label":827},"She used the first term as the constant; it should be 3 × n + 2",{"id":64,"label":829},"It should be n + 3",[61],[832],"Test her rule with n = 1.","Test her rule: n = 1 gives 3 + 5 = 8, not 5. The constant must make n = 1 give the first term: 3 × 1 + ? = 5, so ? = 2. The correct rule is **3 × n + 2**. Using the first term as the constant is a very common slip.",[708,835],"spot the mistake",{"id":837,"section":39,"level":115,"prompt":838,"check":839,"hints":841,"solution":843,"skills":844},"patterns.q065","A sequence has nth term **n × n + 3**. What is its **12th** term?",{"kind":90,"answer":840,"tolerance":92},147,[842],"Square first, then add.","12 × 12 + 3 = 144 + 3 = **147**. (The sequence starts 4, 7, 12, 19, 28, …)",[708,333],{"id":846,"section":39,"level":847,"prompt":848,"check":849,"hints":860,"solution":862,"skills":863},"patterns.q066","challenge","Which rule gives **2, 6, 12, 20, 30, …**?",{"kind":52,"options":850,"correct":859},[851,853,855,857],{"id":55,"label":852},"4 × n − 2",{"id":58,"label":854},"n × n + 1",{"id":61,"label":856},"n × (n + 1)",{"id":64,"label":858},"2 × n × n",[61],[861],"The second differences are all 2, so look for n × n inside the rule.","Test each rule on several terms. 4n − 2 gives 2, 6, 10 ✗. n² + 1 gives 2, 5 ✗. 2n² gives 2, 8 ✗. **n × (n + 1)** gives 1 × 2 = 2, 2 × 3 = 6, 3 × 4 = 12, 4 × 5 = 20, 5 × 6 = 30 ✓. These are twice the triangular numbers.",[708,333],{"id":865,"section":43,"level":87,"prompt":866,"check":867,"hints":869,"solution":871,"skills":872},"patterns.q067","Use Gauss’s pairing trick to find **1 + 2 + 3 + … + 60**.",{"kind":90,"answer":868,"tolerance":92},1830,[870],"How many pairs are there?","Pair the ends: 1 + 60 = 61, 2 + 59 = 61, … There are 30 pairs, so the total is 30 × 61 = **1,830**. (Formula: 60 × 61 ÷ 2 = 1,830.)",[506,873],"Gauss",{"id":875,"section":43,"level":87,"prompt":876,"check":877,"hints":879,"solution":881,"skills":882},"patterns.q068","At a party, **8 friends** each shake hands once with every other friend. How many handshakes are there?",{"kind":90,"answer":328,"tolerance":92,"unit":878},"handshakes",[880],"Count the new handshakes each person makes.","The first friend shakes 7 hands, the next 6 new hands, then 5, 4, 3, 2, 1: 7 + 6 + 5 + 4 + 3 + 2 + 1 = **28**. (Or 8 × 7 ÷ 2: each handshake is counted once for each of the two people.)",[535,883],"counting",{"id":885,"section":43,"level":115,"prompt":886,"check":887,"hints":889,"solution":891,"skills":892},"patterns.q069","Find **3 + 6 + 9 + … + 90** (the multiples of 3 up to 90).",{"kind":90,"answer":888,"tolerance":92},1395,[890],"Count the terms first.","There are 90 ÷ 3 = 30 terms. Sum = (first + last) × number of terms ÷ 2 = (3 + 90) × 30 ÷ 2 = 93 × 15 = **1,395**. (Or 3 × (1 + 2 + … + 30) = 3 × 465 = 1,395.)",[506,23],{"id":894,"section":43,"level":115,"prompt":895,"check":896,"hints":898,"solution":900,"skills":901},"patterns.q070","On a calendar, a 3 × 3 box of nine dates has **17** in the middle. What do the nine dates add up to?",{"kind":90,"answer":897,"tolerance":92},153,[899],"The answer is 9 times the middle date.","The dates are 9, 10, 11, 16, 17, 18, 23, 24, 25. Each date above or left of the middle pairs with one below or right that is the same amount bigger, so the total is 9 × 17 = **153**.",[902,323],"calendar patterns",{"id":904,"section":43,"level":115,"prompt":905,"check":906,"hints":907,"solution":909,"skills":910},"patterns.q071","Four dates in the **same column** of a calendar (one below the other) add up to **70**. What is the first (top) date?",{"kind":90,"answer":474,"tolerance":92},[908],"Dates in a column go up by 7.","If the top date is d, the others are d + 7, d + 14 and d + 21. Total 4 × d + 42 = 70, so 4 × d = 28 and d = **7**. The dates are 7, 14, 21, 28 ✓.",[902,911],"algebra",{"id":913,"section":43,"level":115,"prompt":914,"check":915,"hints":916,"solution":918,"skills":919},"patterns.q072","In a hundred square, a 2 × 2 box contains 45, 46, 55, 56. What is **46 × 55 − 45 × 56**?",{"kind":90,"answer":368,"tolerance":92},[917],"Work out both products.","46 × 55 = 2,530 and 45 × 56 = 2,520. The difference is **10**, the length of a row in the hundred square. It is 10 for every 2 × 2 box, because (a + 1)(a + 10) − a(a + 11) = 10.",[920,323],"hundred square",{"id":922,"section":43,"level":87,"prompt":923,"check":924,"hints":926,"solution":928,"skills":929},"patterns.q073","A 6 by 6 magic square uses the numbers 1 to 36 once each. What is its **magic constant** (the total of each row)?",{"kind":90,"answer":925,"tolerance":92},111,[927],"Add 1 to 36 first.","1 + 2 + … + 36 = 36 × 37 ÷ 2 = 666. The six rows share this equally: 666 ÷ 6 = **111**. (Formula: n(n² + 1) ÷ 2 = 6 × 37 ÷ 2 = 111.)",[930],"magic squares",{"id":932,"section":43,"level":847,"prompt":933,"check":934,"hints":935,"solution":937,"skills":938},"patterns.q074","In the sequence **1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …** each number n is written n times. What is the **50th** term?",{"kind":90,"answer":368,"tolerance":92},[936],"Where does the last 9 appear?","The last copy of each number sits at a triangular position: the last 9 is at position 1 + 2 + … + 9 = 45, and the last 10 at position 55. Positions 46 to 55 are all 10, so the 50th term is **10**.",[535,323],{"id":940,"section":43,"level":847,"prompt":941,"check":942,"hints":945,"solution":947,"skills":948},"patterns.q075","Points on a circle are all joined by straight lines (no three lines meeting inside). With 1 to 6 points the most regions are 1, 2, 4, 8, 16, 31. With **7 points**, how many regions are there? (Regions = 1 + number of lines + number of crossing points; 7 points give 21 lines and 35 crossings.)",{"kind":90,"answer":943,"tolerance":92,"unit":944},57,"regions",[946],"Use the formula in the question.","Regions = 1 + 21 + 35 = **57**, not 64. The pattern 1, 2, 4, 8, 16 looked like doubling but was never doubling: it broke at 31. This is why mathematicians need proofs, not just patterns.",[949],"patterns that break",{"id":951,"section":43,"level":847,"prompt":952,"check":953,"hints":955,"solution":958,"skills":959},"patterns.q076","What is the **last digit** of **3²⁰²⁶**?",{"kind":90,"answer":954,"tolerance":92},9,[956,957],"Write the last digits of 3, 9, 27, 81, 243, …","Divide the power by the cycle length.","Last digits of powers of 3 cycle: 3, 9, 7, 1, 3, 9, 7, 1, … (length 4). 2,026 ÷ 4 = 506 remainder 2, so 3²⁰²⁶ ends like 3², in **9**.",[139,448],{"id":961,"section":43,"level":847,"prompt":962,"check":963,"hints":966,"solution":968,"skills":969},"patterns.q077","How many squares of **all sizes** can you find on a **6 by 6** grid of small squares?",{"kind":90,"answer":964,"tolerance":92,"unit":965},91,"squares",[967],"How many positions are there for a 2 × 2 square?","1 × 1: 36. 2 × 2: 25. 3 × 3: 16. 4 × 4: 9. 5 × 5: 4. 6 × 6: 1. Total 36 + 25 + 16 + 9 + 4 + 1 = **91**, the sum of the first six square numbers.",[516,883],{"id":971,"section":43,"level":847,"prompt":972,"check":973,"hints":976,"solution":978,"skills":979},"patterns.q078","Indian poets counted rhythms made of short syllables (1 beat) and long syllables (2 beats). There are 1, 2, 3, 5, 8, … rhythms of 1, 2, 3, 4, 5 beats. How many rhythms fill exactly **10 beats**?",{"kind":90,"answer":974,"tolerance":92,"unit":975},89,"rhythms",[977],"Keep adding the last two counts.","Each count is the sum of the two before (a rhythm ends in a short or a long syllable): 1, 2, 3, 5, 8, 13, 21, 34, 55, **89**. These are the Virahānka–Fibonacci numbers, known in India centuries before Fibonacci.",[593,980],"history",{"id":982,"section":43,"level":847,"prompt":983,"check":984,"hints":985,"solution":987,"skills":988},"patterns.q079","In the Meru Prastara (Pascal’s triangle), row 5 is 1 5 10 10 5 1. Each number is the sum of the two above it. What is the **middle number of row 6**?",{"kind":90,"answer":397,"tolerance":92},[986],"Add neighbouring numbers of row 5.","Row 6 is 1, 1 + 5 = 6, 5 + 10 = 15, 10 + 10 = **20**, 15, 6, 1. The middle number is 20. Check: the row adds to 64 = 2⁶ ✓.",[989],"Meru Prastara",{"id":991,"section":43,"level":847,"prompt":992,"check":993,"hints":995,"solution":997,"skills":998},"patterns.q080","Using the pattern 1³ = 1², 1³ + 2³ = 3², 1³ + 2³ + 3³ = 6², find **1³ + 2³ + 3³ + … + 20³**.",{"kind":90,"answer":994,"tolerance":92},44100,[996],"The numbers being squared (1, 3, 6, …) are triangular numbers.","The sum of the first n cubes is the square of the nth triangular number. 1 + 2 + … + 20 = 20 × 21 ÷ 2 = 210, and 210² = **44,100**.",[545,535],{"id":1000,"section":43,"level":847,"prompt":1001,"check":1002,"hints":1013,"solution":1015,"skills":1016},"patterns.q081","The rule n × n − n + 41 gives a prime number for every n from 1 to 40. What happens at **n = 41**?",{"kind":52,"options":1003,"correct":1012},[1004,1006,1008,1010],{"id":55,"label":1005},"It gives another prime",{"id":58,"label":1007},"It gives 41 × 41 = 1,681, which is not prime",{"id":61,"label":1009},"It gives a negative number",{"id":64,"label":1011},"It gives 41",[58],[1014],"Simplify: − 41 + 41 cancels.","At n = 41: 41 × 41 − 41 + 41 = 41 × 41 = **1,681**, which is divisible by 41, so it is not prime. Forty examples in a row did not prove the pattern. One counterexample breaks it.",[949,1017],"counterexample",[1019,1020,1021,1022,1023,1024,1025,1026],"patterns-ncert-ganita-prakash-6","patterns-ncert-class6-algebra","patterns-mathsisfun-sequences","patterns-mathsisfun-fibonacci","patterns-wiki-fibonacci","patterns-wiki-pascal","patterns-wiki-magic-square","patterns-wiki-circle-regions","needs_review",{"generatedBy":1029,"notes":1030},"claude-code","Draft generated with Python; every numeric answer was computed and asserted. Pending owner review.","997cd475fc2b0e03d2a512cefcadc1ce00f1ab92fc75ab101b8979e731715500",{"logic:questions":1033,"source:patterns-mathsisfun-fibonacci":1034,"source:patterns-mathsisfun-sequences":1035,"source:patterns-ncert-class6-algebra":1036,"source:patterns-ncert-ganita-prakash-6":1037,"source:patterns-wiki-circle-regions":1038,"source:patterns-wiki-fibonacci":1039,"source:patterns-wiki-magic-square":1040,"source:patterns-wiki-pascal":1041},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","b77a2fee9032c27c6a0b903d14287fc8a01076a000aba32f0ffc0e3ff54db124","1c634ad2570a389d1bc96431d0a220aa359ad82b4d33f5b55eb0b12dab8bab4c","b31773e8224c076a8a34aa4e6df704febaac5ecd4d6e1d3ba5041a31aad871e1","36726d3ff16d73627216db999b62b46580dfaf514f313774e79669de57d7c595","3036d0a86299354fe4f0f1df748a233c1fe9fdf208b4e75f1169be57492e7095","2b7f65c540dd62068d0f516bd71c5de387803d7540246de06abc95456694188a","73b34d05e25e1a384421070a78b4e9263f7cab7309df8ca4978360e2d30306ae","8b958ff4286a612ef0e8589f7261e9e18d2878dcfc122f6277212421fcccfa92","preview-7e1cbbcc4f",1789899599933]