[{"data":1,"prerenderedAt":898},["ShallowReactive",2],{"layer:patterns:investigate":3},{"layer":4,"contentHash":880,"dependencyHashes":881,"approval":891,"releaseId":897},{"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":875,"reviewStatus":876,"authoring":877},1,"patterns","en","investigate","Pattern detectives: predict, test, explain","Matchstick challenges, Gauss’s trick, calendar magic, growth races and patterns that fool you","Investigate growing patterns like a detective: predict first, collect small cases, find the rule, test it and explain why it works. Includes far predictions, working backwards, odd sums, Gauss’s pairing, grid tricks and always-sometimes-never reasoning.",[13,14,15,16,17],"Use the predict, collect, spot, test and explain method on a pattern investigation.","Predict the 100th picture of a growing shape pattern and work backwards from a total to a position.","Explain with pictures why odd numbers add to square numbers and why two staircases make a rectangle.","Use pairing to add arithmetic sequences, and explain grid and calendar tricks.","Decide whether statements are always, sometimes or never true, using examples and counterexamples.",50,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 50 minutes",{"label":29,"value":30},"Prior knowledge","Understand: rules, arithmetic and geometric",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Pattern machine ×2, sort game, match pairs",{"label":38,"value":39},"Motto","Examples suggest; reasons prove",[41,45,78,84,90,108,144,157,160,176,181,184,197,202,207,220,222,234,239,242,254,265,278,283,286,289,300,305,343,348,351,386,397,402,407,410,423,428,484,489,492,573,583,614,619,622,627,649,654,699,832,836,842,847,851,866],{"id":42,"type":43,"markdown":44},"intro","prose","In the first two layers you learned to spot rules and use them. Now you become a **pattern detective**. Detectives do not just guess; they **predict**, **test**, **check** and **explain**.\n\nThis layer is full of investigations. Each one starts with a question like *“how many sticks for 100 squares?”*, *“does this trick always work?”* or *“which grows faster?”*. You will make a prediction first (and write it down, so you cannot pretend afterwards!), then test it with the labs, tables and a pencil, and finally try to explain **why** the answer comes out the way it does.\n\nSome of your predictions will be wrong. That is the point. A wrong prediction you can explain teaches you more than a lucky right one.",{"id":46,"type":47,"title":48,"items":49},"steps-detective","steps","The pattern detective’s method",[50,54,58,62,66,70,74],{"title":51,"tag":52,"text":53},"Ask","Question","Choose something to find out: “How many sticks for 100 squares?”",{"title":55,"tag":56,"text":57},"Predict","Write it down","Make a guess before you calculate. Say why you think so.",{"title":59,"tag":60,"text":61},"Collect","Small cases","Build or draw the 1st, 2nd, 3rd, 4th and 5th cases and count. Put the results in a table.",{"title":63,"tag":64,"text":65},"Spot","Rule","Look at differences and ratios. Write a rule in words, then as a formula in n.",{"title":67,"tag":68,"text":69},"Test","New case","Use your rule to predict a case you have not counted (say the 6th), then count it to check.",{"title":71,"tag":72,"text":73},"Explain","Why?","Use the picture to explain why the rule works. An explained rule is far stronger than a guessed one.",{"title":75,"tag":76,"text":77},"Use","Predict far","Now answer the question: the 100th case, or which case gives a certain number.",{"id":79,"type":80,"variant":81,"title":82,"markdown":83},"def-conjecture","callout","definition","Conjecture and counterexample","A **conjecture** is a statement you think is true because it has worked in every case you have tried, but you have not yet shown it must always be true. Example: *“the sum of two odd numbers is always even.”*\n\nA **counterexample** is a single case that shows a conjecture is false. One counterexample is enough to break a conjecture, however many examples agreed with it. Example: *“all odd numbers are prime”* is broken by 9 = 3 × 3.",{"id":85,"type":86,"title":87,"eyebrow":88,"navLabel":89},"ch01","chapter","The matchstick lab","Chapter 01","1 Matchstick lab",{"id":91,"type":92,"prompt":93,"options":94,"explanation":107},"predict-100-triangles","prediction","Matchstick triangles are joined in a row, each sharing a side with the next: 3, 5, 7, … sticks. **Predict:** how many sticks for **100 triangles**?",[95,98,101,104],{"id":96,"label":97},"a","300",{"id":99,"label":100},"b","201",{"id":102,"label":103},"c","203",{"id":105,"label":106},"d","200","**201.** Each new triangle shares one side, so it adds 2 sticks. Think of the first stick on its own, then 2 sticks for each triangle: 2 × n + 1. For n = 100: 2 × 100 + 1 = **201**. 300 (3 × 100) forgets that 99 sides are shared: 300 − 99 = 201 ✓.",{"id":109,"type":110,"component":111,"componentVersion":5,"config":112,"objective":138,"textAlternative":139,"help":140},"lab-far-shapes","interactive","pattern-machine",{"puzzles":113},[114,119,121,125,127,130,134],{"kind":115,"shape":116,"show":117,"askTerm":118},"shape","matchstick-squares",4,100,{"kind":115,"shape":120,"show":117,"askTerm":118},"matchstick-triangles",{"kind":115,"shape":122,"show":123,"askTerm":118,"hint":124},"hexagon-chain",3,"Count the sticks each new hexagon adds.",{"kind":115,"shape":126,"show":117,"askTerm":18},"l-shapes",{"kind":115,"shape":128,"show":117,"askTerm":129},"dot-squares",25,{"kind":115,"shape":131,"show":117,"askTerm":132,"hint":133},"staircase",20,"Two staircases fit together into a rectangle.",{"kind":115,"shape":135,"show":136,"askTerm":118,"hint":137},"dot-triangles",5,"n × (n + 1) ÷ 2","Predict far-away pictures (the 20th to the 100th) of growing shape patterns without drawing them, then choose the rule.","Seven far-prediction puzzles. The machine draws the first few pictures; you type the count for a far picture and then pick the rule.\n\n1. Matchstick squares (4, 7, 10, 13): 100th picture **301** sticks (3 × n + 1).\n2. Matchstick triangles (3, 5, 7, 9): 100th **201** sticks (2 × n + 1).\n3. Matchstick hexagons (6, 11, 16): 100th **501** sticks (5 × n + 1).\n4. L-shapes (1, 3, 5, 7 tiles): 50th **99** tiles (2 × n − 1).\n5. Dot squares (1, 4, 9, 16): 25th **625** dots (n × n).\n6. Staircases (1, 3, 6, 10 blocks): 20th **210** blocks (n × (n + 1) ÷ 2).\n7. Dot triangles (1, 3, 6, 10, 15): 100th **5,050** dots (n × (n + 1) ÷ 2).\n\nNobody could draw the 100th triangle of 5,050 dots by hand. The rule does it in one line.",{"hints":141},[142,143],"For stick patterns, ask: how many new sticks does each picture add? That number multiplies n.","For dot squares and triangles, think about rows and columns.",{"id":145,"type":146,"title":147,"problem":148,"steps":149,"help":155},"we-reverse","worked_example","Working backwards: the 2026-stick challenge","A school wants to lay out matchstick squares in a row for its **2026** Annual Day, using exactly **2,026** sticks. Is that possible, and how many squares would there be?",[150,151,152,153,154],"Matchstick squares use 3 × n + 1 sticks.","We need 3 × n + 1 = 2,026, so 3 × n = 2,025.","2,025 ÷ 3 = **675** exactly, so it is possible: **675 squares**.","Check: 3 × 675 + 1 = 2,025 + 1 = 2,026 ✓.","What about triangles? 2 × n + 1 = 2,026 gives 2 × n = 2,025, which is odd, so it cannot be done: a triangle row always uses an **odd** number of sticks.",{"simplerExplanation":156},"Take away the 1 extra stick, then divide the rest into groups of 3. Each group is one square.",{"id":158,"type":43,"markdown":159},"two-rows","**Investigation: two rows of squares.** Build a rectangle of matchstick squares **2 rows high** and n squares long. Count the sticks for n = 1, 2, 3, 4, 5.\n\n| Squares long (n) | 1 | 2 | 3 | 4 | 5 |\n| --- | --- | --- | --- | --- | --- |\n| Sticks | 7 | 12 | 17 | 22 | 27 |\n\nThe differences are all **5**, so the rule is 5 × n + something. For n = 1 we need 7, so the rule is **5 × n + 2**.\n\n**Why 5?** Each new column adds 3 horizontal sticks (top, middle, bottom) and 2 vertical sticks: 5. **Why + 2?** The 2 vertical sticks on the far left edge.\n\nTry **3 rows high** yourself before reading on. (You should get 10, 17, 24, 31, …, which is **7 × n + 3**: 4 horizontal and 3 vertical sticks per column, plus 3 on the left edge.)",{"id":161,"type":162,"itemId":163,"prompt":164,"check":165,"hints":170,"feedback":173},"pr-grid","practice","patterns.investigate-grid-sticks","How many matchsticks make a rectangle of squares **2 rows high and 10 squares long**?",{"kind":166,"answer":167,"tolerance":168,"unit":169},"number",52,0,"matchsticks",[171,172],"Use the rule 5 × n + 2.","Or count: 3 rows of 10 horizontal sticks, and 11 columns of 2 vertical sticks.",{"correct":174,"incorrect":175},"Yes: 5 × 10 + 2 = 52. Counted another way: 30 horizontal + 22 vertical = 52.","Horizontal sticks: 3 lines of 10 = 30. Vertical sticks: 11 lines of 2 = 22. Total 52 (and 5 × 10 + 2 = 52).",{"id":177,"type":86,"title":178,"eyebrow":179,"navLabel":180},"ch02","Dot patterns: squares, triangles and rectangles","Chapter 02","2 Dot patterns",{"id":182,"type":43,"markdown":183},"dots-rect","**Investigation: how many dots in the 100th triangle?** Adding 1 + 2 + 3 + … + 100 by hand is slow. Here is a trick you can test with counters.\n\nMake a triangle of dots with rows 1, 2, 3, 4 (10 dots). Make a second, identical triangle and turn it upside down. Push the two together: they make a **rectangle** 4 dots wide and 5 dots tall, which has 4 × 5 = 20 dots. One triangle is half of that: **10**. ✓\n\n| Triangle n | Rectangle | Rectangle dots | Triangle dots (half) |\n| --- | --- | --- | --- |\n| 3 | 3 × 4 | 12 | 6 |\n| 4 | 4 × 5 | 20 | 10 |\n| 5 | 5 × 6 | 30 | 15 |\n| 10 | 10 × 11 | 110 | 55 |\n| 100 | 100 × 101 | 10,100 | 5,050 |\n\nSo the nth triangular number is **n × (n + 1) ÷ 2**.",{"id":185,"type":92,"prompt":186,"options":187,"explanation":196},"predict-tri-sum","Add two **neighbouring** triangular numbers: 1 + 3, 3 + 6, 6 + 10, 10 + 15. **Predict** what kind of numbers you will get.",[188,190,192,194],{"id":96,"label":189},"Triangular numbers again",{"id":99,"label":191},"Square numbers",{"id":102,"label":193},"Always even numbers",{"id":105,"label":195},"Cube numbers","**Square numbers:** 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25. With counters you can see why: a staircase of 1, 2, 3, 4 and a staircase of 1, 2, 3 (turned over) fit together into a 4 by 4 square. They are not always even (9 and 25 are odd).",{"id":198,"type":80,"variant":199,"title":200,"markdown":201},"try-counters","try_it","Build it with counters","Use buttons, coins or bottle tops.\n\n1. Make dot triangles of 1, 3, 6 and 10.\n2. Make two triangles of 10 and fit them into a 4 by 5 rectangle.\n3. Fit a triangle of 6 and a triangle of 10 into a 4 by 4 square.\n4. Now try: can **eight** copies of a triangle, plus one extra counter, make a square? (Try with the triangle of 3: 8 × 3 + 1 = 25 = 5 × 5. With 6: 8 × 6 + 1 = 49 = 7 × 7. It always works! Mathematicians have known this for over 1,800 years.)",{"id":203,"type":86,"title":204,"eyebrow":205,"navLabel":206},"ch03","Odd sums and L-shapes","Chapter 03","3 Odd sums",{"id":208,"type":92,"prompt":209,"options":210,"explanation":219},"predict-odd-sum","**Predict** the total: **1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19** (the first ten odd numbers).",[211,213,215,217],{"id":96,"label":212},"90",{"id":99,"label":214},"100",{"id":102,"label":216},"110",{"id":105,"label":218},"95","**100.** Add the first few: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. The totals are the square numbers! The first **n** odd numbers add to **n × n**, so the first ten add to 10 × 10 = **100**.",{"id":126,"type":43,"markdown":221},"**Why do odd numbers build squares?** Picture a 1 by 1 square of one tile. To make it into a 2 by 2 square, wrap an **L-shape** of 3 tiles around two sides. To grow that into a 3 by 3 square, wrap an L of 5 tiles. Then an L of 7 tiles makes a 4 by 4 square.\n\n| Step | L-shape added | Square made | Total tiles |\n| --- | --- | --- | --- |\n| 1 | 1 (just the corner) | 1 × 1 | 1 |\n| 2 | 3 | 2 × 2 | 1 + 3 = 4 |\n| 3 | 5 | 3 × 3 | 4 + 5 = 9 |\n| 4 | 7 | 4 × 4 | 9 + 7 = 16 |\n| 5 | 9 | 5 × 5 | 16 + 9 = 25 |\n\nEach L has one tile on the corner and two equal arms. To go from an (n − 1) square to an n square, each arm has n − 1 tiles, so the L has (n − 1) + (n − 1) + 1 = **2 × n − 1** tiles: exactly the nth odd number. These L-shapes are the growing L-shapes in the pattern machine. The Greeks called them **gnomons**, the name for the L-shaped pointer on a sundial.",{"id":223,"type":162,"itemId":224,"prompt":225,"check":226,"hints":228,"feedback":231},"pr-odd-range","patterns.investigate-odd-range","Use the square-number pattern to find **21 + 23 + 25 + … + 39** (all the odd numbers from 21 to 39).",{"kind":166,"answer":227,"tolerance":168},300,[229,230],"1 + 3 + … + 39 is the sum of the first 20 odd numbers.","Subtract 1 + 3 + … + 19, the first 10 odd numbers.",{"correct":232,"incorrect":233},"Yes: 20 × 20 − 10 × 10 = 400 − 100 = 300.","The odd numbers up to 39 are the first 20 odd numbers, adding to 400. Those up to 19 are the first 10, adding to 100. So 21 + … + 39 = 400 − 100 = 300.",{"id":235,"type":86,"title":236,"eyebrow":237,"navLabel":238},"ch04","Staircases and the story of young Gauss","Chapter 04","4 Gauss's trick",{"id":240,"type":43,"markdown":241},"gauss","A famous story is told about Carl Friedrich Gauss, who became one of the greatest mathematicians in history. When he was about nine, his teacher asked the class to add up all the numbers from 1 to 100, hoping for some peace and quiet. Gauss wrote the answer almost immediately: **5,050**.\n\nTreat that as a **story, not history**. It is an anecdote — historians class it as apocryphal — that has grown in the retelling, and the pairing trick itself was written down centuries before Gauss was born. What is real is the idea, so here it is.\n\n**Pair** the numbers from the two ends:\n\n- 1 + 100 = 101\n- 2 + 99 = 101\n- 3 + 98 = 101\n- …\n- 50 + 51 = 101\n\nThere are 50 pairs, each adding to 101, so the total is 50 × 101 = **5,050**.\n\nThis is the same as the two-staircase rectangle from the dot chapter: 100 × 101 ÷ 2 = 5,050.",{"id":243,"type":146,"title":244,"problem":245,"steps":246,"help":252},"we-gauss-evens","Pairing for other sums","Use pairing to find **2 + 4 + 6 + … + 100** and **5 + 10 + 15 + … + 200**.",[247,248,249,250,251],"2 + 4 + … + 100 has 50 terms. Pair the ends: 2 + 100 = 102, 4 + 98 = 102, … There are 25 pairs.","Total: 25 × 102 = **2,550**. (Or: it is 2 × (1 + 2 + … + 50) = 2 × 1,275 = 2,550.)","5 + 10 + … + 200 has 200 ÷ 5 = 40 terms. Pair the ends: 5 + 200 = 205. There are 20 pairs.","Total: 20 × 205 = **4,100**.","General trick for any arithmetic sequence: **(first + last) × number of terms ÷ 2**.",{"simplerExplanation":253},"Write the list forwards and backwards, one under the other. Every column adds to the same number. Multiply, then halve.",{"id":255,"type":92,"prompt":256,"options":257,"explanation":264},"predict-odd-count","Gauss's pairing works neatly with 100 numbers (an even number of terms). What happens with **1 + 2 + 3 + … + 9**, where there are 9 terms?",[258,260,262],{"id":96,"label":259},"The trick fails because the middle number has no partner",{"id":99,"label":261},"The trick still works: (first + last) × terms ÷ 2 = 45",{"id":102,"label":263},"You get 50, because you round up","**It still works.** 1 + 9 = 10, 2 + 8 = 10, 3 + 7 = 10, 4 + 6 = 10, and 5 is left in the middle, which is half of 10. So 4 × 10 + 5 = 45, and (1 + 9) × 9 ÷ 2 = 45 too. Writing the sum forwards and backwards (instead of folding it) avoids the problem entirely: 9 columns of 10 is 90, half is 45.",{"id":266,"type":162,"itemId":267,"prompt":268,"check":269,"hints":272,"feedback":275},"pr-gauss-20","patterns.investigate-sum-to-20","The school hall has **20 rows** of chairs. The front row has 1 chair, the next has 2, the next 3, and so on up to 20 chairs in the back row. How many chairs are there altogether?",{"kind":166,"answer":270,"tolerance":168,"unit":271},210,"chairs",[273,274],"Pair the rows: 1 + 20, 2 + 19, …","Or use n × (n + 1) ÷ 2 with n = 20.",{"correct":276,"incorrect":277},"Yes: 20 × 21 ÷ 2 = 210 chairs (10 pairs of 21).","Pair the front row with the back row: 1 + 20 = 21. There are 10 such pairs, so 10 × 21 = 210 chairs.",{"id":279,"type":86,"title":280,"eyebrow":281,"navLabel":282},"ch05","Investigations on the hundred square and calendar","Chapter 05","5 Grid detective",{"id":284,"type":43,"markdown":285},"grid-2x2","**Investigation: the 2 × 2 box.** On a hundred square, draw a 2 by 2 box around four numbers, say 23, 24, 33, 34. Multiply the numbers on each **diagonal**:\n\n- 23 × 34 = 782\n- 24 × 33 = 792\n\nThe difference is **10**. Try another box: 56, 57, 66, 67 gives 56 × 67 = 3752 and 57 × 66 = 3762: again a difference of **10**.\n\nNow try it on a **calendar**, where rows have 7 days. The box 8, 9, 15, 16 gives 8 × 16 = 128 and 9 × 15 = 135: difference **7**.\n\n**Conjecture:** the difference is always the **row length**: 10 on the hundred square, 7 on a calendar. In the Deepen layer you can prove it with a little algebra: if the top-left number is a and the rows have length r, the box is a, a + 1, a + r, a + r + 1, and (a + 1)(a + r) − a(a + r + 1) = r.",{"id":287,"type":43,"markdown":288},"grid-3x3","**Investigation: the 3 × 3 box.** In Discover you met the calendar trick: the nine dates in any 3 by 3 box add up to **9 × the middle date**. Why?\n\nLook at the box around a middle date m on a calendar:\n\n| | | |\n| --- | --- | --- |\n| m − 8 | m − 7 | m − 6 |\n| m − 1 | **m** | m + 1 |\n| m + 6 | m + 7 | m + 8 |\n\nEvery number above or to the left of the middle has a partner below or to the right that is the same amount **bigger**: m − 8 pairs with m + 8, m − 7 with m + 7, m − 6 with m + 6, m − 1 with m + 1. The pluses and minuses cancel, leaving nine m's: **9 × m**.\n\nThe same works on the hundred square (with 11, 10, 9 and 1 instead of 8, 7, 6 and 1), and for any 3 × 3 box on any grid.",{"id":290,"type":162,"itemId":291,"prompt":292,"check":293,"hints":295,"feedback":297},"pr-3x3","patterns.investigate-3x3-box","On a hundred square, a 3 × 3 box has **47** in the middle. What is the total of its nine numbers?",{"kind":166,"answer":294,"tolerance":168},423,[296],"The total is 9 times the middle number.",{"correct":298,"incorrect":299},"Yes: 9 × 47 = 423.","Pair each number with the one opposite it through the middle: they add to 2 × 47. Four pairs plus the middle give 9 × 47 = 423.",{"id":301,"type":80,"variant":302,"title":303,"markdown":304},"aha-three-consecutive","aha","The same idea in one line","The sum of **three consecutive numbers** is 3 × the middle one: 19 + 20 + 21 = 60 = 3 × 20. The number below the middle is one less and the number above is one more, so they balance. That is why the sum of any three consecutive whole numbers is always a **multiple of 3**.",{"id":306,"type":110,"component":111,"componentVersion":5,"config":307,"objective":338,"textAlternative":339,"help":340},"lab-grid-paths",{"puzzles":308},[309,315,320,324,328,334],{"kind":166,"rule":310,"show":117,"ask":313,"hint":314},{"type":311,"start":5,"step":312},"add",11,2,"A diagonal on the hundred square: down one row and right one column.",{"kind":166,"rule":316,"show":117,"ask":313,"hint":319},{"type":311,"start":317,"step":318},10,9,"The other diagonal: down one row and left one column.",{"kind":166,"rule":321,"show":117,"ask":313,"hint":323},{"type":311,"start":123,"step":322},7,"Down a calendar column: the same weekday each week.",{"kind":166,"rule":325,"show":117,"ask":5,"hint":327},{"type":311,"start":5,"step":326},8,"A calendar diagonal going down to the right.",{"kind":166,"rule":329,"show":332,"ask":313,"hint":333},{"type":330,"start":5,"steps":331},"alternate",[5,317],6,"A zigzag path on the hundred square: right, down, right, down.",{"kind":166,"rule":335,"show":332,"ask":313,"hint":337},{"type":330,"start":136,"steps":336},[322,5],"A staircase path on a calendar: down a week, then one day right.","Follow paths across the hundred square and a calendar, predicting where each path lands next.","Six paths across number grids.\n\n1. Hundred-square diagonal ↘ from 1: 1, 12, 23, 34, … next **45, 56** (add 11).\n2. Hundred-square diagonal ↙ from 10: 10, 19, 28, 37, … next **46, 55** (add 9).\n3. Calendar column from the 3rd: 3, 10, 17, 24, … next **31, 38** (add 7). A month has at most 31 days, so the calendar path stops at 31.\n4. Calendar diagonal ↘ from the 1st: 1, 9, 17, 25, … next **33** (add 8).\n5. Hundred-square zigzag: 1, 2, 12, 13, 23, 24, … next **34, 35** (right +1, down +10).\n6. Calendar staircase: 5, 12, 13, 20, 21, 28, … next **29, 36** (down +7, right +1). As a real calendar path it would stop at 31.\n\nEvery move on a grid is an addition, so every straight path is an arithmetic sequence, and every zigzag is an alternating one.",{"hints":341},[342],"Right is +1. Down is + the row length.",{"id":344,"type":86,"title":345,"eyebrow":346,"navLabel":347},"ch06","Digit detective","Chapter 06","6 Digit detective",{"id":349,"type":43,"markdown":350},"eleven","**Investigation: multiplying a two-digit number by 11.** Try a few:\n\n| Number | × 11 | Notice |\n| --- | --- | --- |\n| 23 | 253 | 2 _ 3 with 2 + 3 = 5 in the middle |\n| 45 | 495 | 4 _ 5 with 4 + 5 = 9 in the middle |\n| 61 | 671 | 6 _ 1 with 6 + 1 = 7 in the middle |\n| 78 | 858 | 7 + 8 = 15: write 5, carry 1 into the 7 |\n| 99 | 1,089 | 9 + 9 = 18: write 8, carry 1 into the 9 |\n\n**Conjecture:** to multiply ab by 11, write a, then a + b, then b. **Test:** it works whenever a + b is 9 or less. When a + b is 10 or more, you must carry. So the neat version is only **sometimes** true, but the version with carrying is **always** true, because ab × 11 = ab × 10 + ab.",{"id":352,"type":353,"caption":354,"columns":355,"rows":358},"table-8s","table","The ×8 staircase: test every line (computed)",[356,357],"Calculation","Result",[359,362,365,368,371,374,377,380,383],[360,361],"1 × 8 + 1","9",[363,364],"12 × 8 + 2","98",[366,367],"123 × 8 + 3","987",[369,370],"1,234 × 8 + 4","9,876",[372,373],"12,345 × 8 + 5","98,765",[375,376],"123,456 × 8 + 6","987,654",[378,379],"1,234,567 × 8 + 7","9,876,543",[381,382],"12,345,678 × 8 + 8","98,765,432",[384,385],"123,456,789 × 8 + 9","987,654,321",{"id":387,"type":92,"prompt":388,"options":389,"explanation":396},"predict-break-8","The ×8 staircase works for nine lines, ending with 123,456,789 × 8 + 9 = 987,654,321. The natural “tenth line” writes the counting numbers 1 to 10 side by side: **12,345,678,910 × 8 + 10**. Predict the answer.",[390,392,394],{"id":96,"label":391},"9,876,543,210: the digits keep counting down",{"id":99,"label":393},"A number that does not continue the pattern",{"id":102,"label":395},"It cannot be calculated","**The pattern breaks.** 12,345,678,910 × 8 + 10 = **98,765,431,290**, not a neat count-down. The staircase relies on every line being built from the single digits 1 to 9. There is no single “digit 10”, so writing 10 on the end pushes the other digits along and the carries spoil everything. Many digit patterns are like this: beautiful for a while, then they run out of digits. Always test before you trust!",{"id":398,"type":80,"variant":399,"title":400,"markdown":401},"nuance-next-line","nuance","What counts as “the next line”?","Deciding what the next line of a digit pattern *should* be is itself a choice. Someone could cleverly pick 1,234,567,900 × 8 + 10 and get 9,876,543,210, which looks like the pattern carrying on with a 0. That is a trick, not the pattern. A good detective states the rule for building each line first, and then tests the line that rule really gives.",{"id":403,"type":86,"title":404,"eyebrow":405,"navLabel":406},"ch07","Growth races: adding against multiplying","Chapter 07","7 Growth races",{"id":408,"type":43,"markdown":409},"race","**Investigation: who saves more?** Arjun saves **₹500 every week**. Bela saves **₹1 in week 1**, then doubles what she saves each week: ₹1, ₹2, ₹4, ₹8, …\n\n| Week | Arjun that week | Bela that week | Arjun total | Bela total |\n| --- | --- | --- | --- | --- |\n| 1 | ₹500 | ₹1 | ₹500 | ₹1 |\n| 5 | ₹500 | ₹16 | ₹2,500 | ₹31 |\n| 9 | ₹500 | ₹256 | ₹4,500 | ₹511 |\n| 10 | ₹500 | ₹512 | ₹5,000 | ₹1,023 |\n| 12 | ₹500 | ₹2,048 | ₹6,000 | ₹4,095 |\n| 13 | ₹500 | ₹4,096 | ₹6,500 | ₹8,191 |\n| 15 | ₹500 | ₹16,384 | ₹7,500 | ₹32,767 |\n\n- Bela's **weekly** amount first beats Arjun's in **week 10** (₹512).\n- Bela's **total** first beats Arjun's in **week 13** (₹8,191 against ₹6,500).\n- By week 20 Bela would be saving ₹524,288 in one week. (Her parents might have something to say about that!)\n\nAdding the same amount each week (arithmetic) grows **steadily**. Doubling (geometric) starts slowly and then **explodes**. This is why scientists worry about anything that doubles, such as an infection spreading or a debt with high interest.",{"id":411,"type":92,"prompt":412,"options":413,"explanation":422},"predict-fold","A sheet of paper is about **0.1 mm** thick. Each fold in half doubles the thickness. If you could keep folding (you can't, but imagine!), about how many folds would make it thicker than the distance to the **Moon** (about 3,84,400 km)?",[414,416,418,420],{"id":96,"label":415},"About 42 folds",{"id":99,"label":417},"About 1,000 folds",{"id":102,"label":419},"About a million folds",{"id":105,"label":421},"It would never reach","**Only about 42.** After 42 folds the thickness is 0.1 mm × 2⁴² ≈ **439,805 km** (4,39,805 km in the Indian system), past the Moon. After 41 folds it is only about 219,902 km. Doubling is astonishingly powerful. In real life, a sheet of paper is very hard to fold more than 7 or 8 times: after 7 folds it is 128 layers thick (12.8 mm).",{"id":424,"type":80,"variant":425,"title":426,"markdown":427},"limit-doubling","model_limit","Nothing doubles for ever","Bela's savings, the folded paper and the forwarded message are **models**. Real doubling always hits a limit: Bela's family runs out of money, paper cannot be folded more than a few times, and a message runs out of new people (India has about 140 crore people, so a message forwarded to 3 new people each round would run out of people during round 19: after 18 rounds it has reached about 58 crore people, and the 19th round alone would need about 116 crore more). Bacteria run out of food and space. Geometric growth is a wonderful description of the **start** of many processes, not of their whole story.",{"id":429,"type":110,"component":111,"componentVersion":5,"config":430,"objective":479,"textAlternative":480,"help":481},"lab-growth",{"puzzles":431},[432,435,438,451,453,458,470],{"kind":166,"rule":433,"show":332,"ask":313},{"type":434,"start":5,"factor":313},"multiply",{"kind":166,"rule":436,"show":117,"ask":313},{"type":311,"start":437,"step":437},500,{"kind":166,"rule":439,"show":136,"ask":313,"hint":450},{"type":440,"terms":441,"ruleText":449},"list",[5,123,322,442,443,444,445,446,447,448],15,31,63,127,255,511,1023,"Bela’s running totals: double the last total and add 1 (one less than a power of 2)","Compare each term with 2, 4, 8, 16, 32.",{"kind":166,"rule":452,"show":117,"ask":313},{"type":434,"start":123,"factor":123},{"kind":166,"rule":454,"show":117,"ask":313,"hint":457},{"type":434,"start":455,"factor":456},1000,0.5,"Halving: the terms get smaller but never reach zero.",{"kind":166,"rule":459,"show":136,"ask":313,"hint":469},{"type":440,"terms":460,"ruleText":468},[313,332,461,132,462,463,464,465,466,467],12,30,42,56,72,90,110,"n × (n + 1): two staircases make a rectangle","Differences 4, 6, 8, 10, …",{"kind":166,"rule":471,"show":117,"ask":313},{"type":440,"terms":472,"ruleText":478},[5,117,318,473,129,474,475,476,477,118],16,36,49,64,81,"Running totals of the odd numbers 1 + 3 + 5 + …: the square numbers","Compare sequences that add and sequences that multiply, and extend running totals.","Seven growth puzzles.\n\n1. Doubling from 1: 1, 2, 4, 8, 16, 32, … next **64, 128**.\n2. Arjun's totals, adding ₹500: 500, 1,000, 1,500, 2,000, … next **2,500, 3,000**.\n3. Bela's totals, 1, 3, 7, 15, 31, … next **63, 127** (double and add 1; each is one less than a power of 2).\n4. Multiply by 3 from 3: 3, 9, 27, 81, … next **243, 729**.\n5. Halving from 1,000: 1,000, 500, 250, 125, … next **62.5, 31.25**.\n6. Rectangle numbers 2, 6, 12, 20, 30, … next **42, 56** (n × (n + 1)).\n7. Running totals of odd numbers 1, 4, 9, 16, … next **25, 36** (the squares).\n\nThe doubling and tripling puzzles outrun the adding ones very quickly; the halving puzzle shrinks towards zero without ever reaching it.",{"hints":482},[483],"If the differences keep getting bigger in proportion, divide instead of subtracting.",{"id":485,"type":86,"title":486,"eyebrow":487,"navLabel":488},"ch08","Always, sometimes or never true?","Chapter 08","8 Always true?",{"id":490,"type":43,"markdown":491},"asn","Mathematicians love the question **“Is it always true?”**. For any statement about numbers there are three possibilities:\n\n- **Always true**: it works for every number you could try. You need a *reason* (an argument or proof) to be sure, because you can never try every number.\n- **Sometimes true**: it works for some numbers and not others. Find one example where it works and one **counterexample** where it doesn't.\n- **Never true**: it fails for every number. Again, you need a reason.\n\nTesting examples is the way to start. Explaining is the way to finish.",{"id":493,"type":110,"component":494,"componentVersion":5,"config":495,"objective":567,"textAlternative":568,"help":569},"lab-asn","sort-game",{"prompt":496,"bins":497,"items":507,"seconds":168},"Is each statement always true, sometimes true or never true (for whole numbers)?",[498,501,504],{"id":499,"label":500},"always","Always true",{"id":502,"label":503},"sometimes","Sometimes true",{"id":505,"label":506},"never","Never true",[508,512,516,520,524,528,532,536,540,544,548,552,556,560,563],{"id":509,"label":510,"bin":499,"why":511},"oddodd","The sum of two odd numbers is even.","The two leftover counters pair up with each other.",{"id":513,"label":514,"bin":499,"why":515},"three","The sum of three consecutive numbers is a multiple of 3.","It equals 3 × the middle number.",{"id":517,"label":518,"bin":502,"why":519},"sqeven","A square number is even.","4 and 16 are even; 9 and 25 are odd.",{"id":521,"label":522,"bin":499,"why":523},"sqdiff","The difference between neighbouring square numbers is odd.","The L-shape that turns one square into the next always has 2 × n − 1 tiles.",{"id":525,"label":526,"bin":502,"why":527},"triodd","A triangular number is odd.","1, 3, 15, 21 are odd; 6, 10, 28 are even.",{"id":529,"label":530,"bin":505,"why":531},"mult4odd","A multiple of 4 is odd.","Every multiple of 4 is also a multiple of 2, so it is even.",{"id":533,"label":534,"bin":499,"why":535},"consecprod","The product of two consecutive numbers is even.","One of any two neighbouring numbers is even, so the product is even.",{"id":537,"label":538,"bin":505,"why":539},"double","Doubling a whole number gives an odd number.","Doubling makes pairs, so the answer is always even.",{"id":541,"label":542,"bin":502,"why":543},"cubeend","A number and its cube end in the same digit.","4 × 4 × 4 = 64 ends in 4, but 2 × 2 × 2 = 8 does not end in 2.",{"id":545,"label":546,"bin":499,"why":547},"oddsum","Adding the first few odd numbers (1 + 3 + 5 + …) gives a square number.","The L-shapes wrap a square into the next square every time.",{"id":549,"label":550,"bin":505,"why":551},"sqend","A square number ends in 2, 3, 7 or 8.","Squares can only end in 0, 1, 4, 5, 6 or 9: check the last digits of 0² to 9².",{"id":553,"label":554,"bin":499,"why":555},"tri2","Two neighbouring triangular numbers add up to a square number.","Two neighbouring staircases fit together into a square.",{"id":557,"label":558,"bin":502,"why":559},"fibeven","A Fibonacci number is even.","Every third Fibonacci number is even (2, 8, 34, …); the rest are odd.",{"id":349,"label":561,"bin":502,"why":562},"To multiply a two-digit number by 11, put the sum of its digits in the middle.","It works for 23 × 11 = 253 but not for 78 × 11 = 858, where you must carry.",{"id":564,"label":565,"bin":505,"why":566},"grow","Adding the same amount each time eventually beats doubling from 1.","Doubling can lose for a while, but it always overtakes any steady adding in the end.","Decide whether statements about number patterns are always, sometimes or never true, using examples and counterexamples.","Fifteen statements to sort into three bins.\n\n**Always true:** odd + odd is even; the sum of three consecutive numbers is a multiple of 3; neighbouring squares differ by an odd number; the product of two consecutive numbers is even; 1 + 3 + 5 + … is a square; two neighbouring triangular numbers make a square.\n\n**Sometimes true:** a square number is even (4 yes, 9 no); a triangular number is odd (3 yes, 6 no); a number and its cube end in the same digit (4 and 64 yes, 2 and 8 no); a Fibonacci number is even (every third one); the “digit-sum in the middle” trick for × 11 (works for 23, fails for 78 without carrying).\n\n**Never true:** a multiple of 4 is odd; doubling a whole number gives an odd number; a square number ends in 2, 3, 7 or 8; adding a fixed amount beats doubling for ever.\n\nFor “sometimes”, one example and one counterexample settle it. For “always” and “never”, you need a reason that covers every number.",{"hints":570},[571,572],"Try at least five numbers, including small, large, odd and even ones.","Look for a picture or pairing argument for “always”.",{"id":574,"type":146,"title":575,"problem":576,"steps":577},"we-prove-sqdiff","Explaining an “always” statement","Show that the difference between two **neighbouring square numbers** is always odd, and that it equals the two numbers being squared added together.",[578,579,580,581,582],"Examples: 4² − 3² = 16 − 9 = 7 = 4 + 3. 10² − 9² = 100 − 81 = 19 = 10 + 9.","Picture a 9 by 9 square of tiles. To make it 10 by 10, add a row of 9 along the top, a column of 9 down the side, and 1 tile in the corner.","That is 9 + 9 + 1 = 19 tiles, which is 9 + 10.","In general, going from (n − 1) × (n − 1) to n × n adds (n − 1) + (n − 1) + 1 = 2 × n − 1 tiles, which is (n − 1) + n.","2 × n − 1 is always odd (an even number minus 1). ✓",{"id":584,"type":110,"component":585,"componentVersion":5,"config":586,"objective":612,"textAlternative":613},"lab-match-sums","match-pairs",{"prompt":587,"mode":588,"pairs":589},"Match each sum to its total. Use a pattern, not a long addition!","memory",[590,593,596,599,602,604,607,609],{"a":591,"b":592},"1 + 3 + 5 + 7 + 9","25",{"a":594,"b":595},"1 + 2 + 3 + … + 10","55",{"a":597,"b":598},"1 + 2 + 4 + 8 + 16","31",{"a":600,"b":601},"1³ + 2³ + 3³","36",{"a":603,"b":214},"1 + 3 + 5 + … + 19",{"a":605,"b":606},"1 + 2 + 3 + … + 100","5,050",{"a":608,"b":216},"2 + 4 + 6 + … + 20",{"a":610,"b":611},"19 + 20 + 21","60","Use patterns (square numbers, pairing, doubling) to match sums to their totals quickly.","A memory game of eight sums and eight totals. Flip two cards; keep them if they match.\n\n- 1 + 3 + 5 + 7 + 9 = **25** (the first five odd numbers make 5 × 5).\n- 1 + 2 + … + 10 = **55** (10 × 11 ÷ 2).\n- 1 + 2 + 4 + 8 + 16 = **31** (one less than the next power of 2, 32).\n- 1³ + 2³ + 3³ = 1 + 8 + 27 = **36** (which is 6², and 6 is 1 + 2 + 3).\n- 1 + 3 + … + 19 = **100** (the first ten odd numbers).\n- 1 + 2 + … + 100 = **5,050** (Gauss's pairing: 50 × 101).\n- 2 + 4 + … + 20 = **110** (twice 1 + … + 10).\n- 19 + 20 + 21 = **60** (3 × the middle number).",{"id":615,"type":86,"title":616,"eyebrow":617,"navLabel":618},"ch09","When a pattern fools you","Chapter 09","9 Patterns that fool",{"id":620,"type":43,"markdown":621},"fool","Here is a warning every pattern detective needs. Look at **1, 2, 4, …** and you will probably say the next term is 8 (doubling). But these rules all start 1, 2, 4:\n\n- **Double each time:** 1, 2, 4, 8, 16, 32, …\n- **Add 1, then 2, then 3, …:** 1, 2, 4, 7, 11, 16, …\n- **Add the two terms before, plus 1:** 1, 2, 4, 7, 12, 20, …\n\nThree different rules, three different futures, all from the same three numbers. And in the Deepen layer you will meet a real geometry problem, cutting a circle into pieces, whose answers go **1, 2, 4, 8, 16** and then, astonishingly, **31**.\n\nSo when you find a rule, ask yourself two questions: **does it fit every term I have?** and **do I know why it should keep working?** The matchstick rules pass both tests, because you can see why each square adds 3 sticks. A rule guessed from a few numbers only passes the first.",{"id":623,"type":80,"variant":624,"title":625,"markdown":626},"mis-examples-prove","misconception","“It worked for ten examples, so it is proved”","Ten examples, or even a million, do not prove that something is **always** true. The expression n × n − n + 41 gives a prime number for n = 1, 2, 3, … all the way to 40. Anyone testing it would be convinced. But for n = 41 it gives 41 × 41 − 41 + 41 = 41 × 41 = 1,681, which is not prime. Examples suggest; reasons prove.",{"id":628,"type":162,"itemId":629,"prompt":630,"check":631,"hints":643,"feedback":646},"pr-rule-fit","patterns.investigate-rule-fit","Which rule fits **all** of these terms: **2, 5, 10, 17, 26**?",{"kind":632,"options":633,"correct":642},"choice",[634,636,638,640],{"id":96,"label":635},"Add 3 each time",{"id":99,"label":637},"n × n + 1",{"id":102,"label":639},"Multiply by 2, then add 1",{"id":105,"label":641},"3 × n − 1",[99],[644,645],"Check each rule on every term, not just the first two.","Compare the terms with 1, 4, 9, 16, 25.",{"correct":647,"incorrect":648},"Yes: 1 + 1 = 2, 4 + 1 = 5, 9 + 1 = 10, 16 + 1 = 17, 25 + 1 = 26. Each is a square number plus 1.","Add 3, 2 × term + 1 and 3 × n − 1 all fit the first two terms (2 → 5) but fail at the third. n × n + 1 fits all five.",{"id":650,"type":86,"title":651,"eyebrow":652,"navLabel":653},"ch10","Check your investigations","Chapter 10","10 Check yourself",{"id":655,"type":656,"title":657,"terms":658},"glossary-investigate","glossary","Investigation words",[659,663,667,671,675,679,683,687,691,695],{"term":660,"meaning":661,"example":662},"investigation","A careful exploration of a question: collect cases, spot a rule, test it and explain it.","How many sticks for 100 squares?",{"term":664,"meaning":665,"example":666},"conjecture","A statement you believe is true from examples but have not yet proved.","“Two neighbouring triangular numbers always make a square.”",{"term":668,"meaning":669,"example":670},"counterexample","One example that shows a statement is false.","9 is a counterexample to “all odd numbers are prime”.",{"term":672,"meaning":673,"example":674},"generalise","To state a rule that works for every case, often using n.","“n squares need 3 × n + 1 sticks.”",{"term":676,"meaning":677,"example":678},"prove","To give a reason that shows a statement must be true in every case.","The L-shape picture proves 1 + 3 + … + (2n − 1) = n².",{"term":680,"meaning":681,"example":682},"gnomon","An L-shaped piece that turns one square (or rectangle) into the next larger one.","Adding 5 tiles turns a 2 × 2 square into 3 × 3.",{"term":684,"meaning":685,"example":686},"consecutive","Following one after another without gaps.","17, 18, 19 are consecutive numbers.",{"term":688,"meaning":689,"example":690},"running total","The total so far as you add terms one by one.","Running totals of 1, 3, 5, 7 are 1, 4, 9, 16.",{"term":692,"meaning":693,"example":694},"power of 2","A number made by multiplying 2 by itself some number of times.","2, 4, 8, 16, 32, 64",{"term":696,"meaning":697,"example":698},"rectangular number","A number of dots that makes an n by (n + 1) rectangle; twice a triangular number.","2, 6, 12, 20, 30",{"id":700,"type":701,"title":702,"questions":703},"quiz-investigate","quiz","Detective check",[704,717,730,743,756,768,781,793,806,819],{"itemId":705,"prompt":706,"options":707,"correct":99,"why":716},"patterns.investigate-q-hex100","Matchstick hexagons in a row need 5 × n + 1 sticks. How many for 100 hexagons?",[708,710,712,714],{"id":96,"label":709},"600",{"id":99,"label":711},"501",{"id":102,"label":713},"505",{"id":105,"label":715},"500","5 × 100 + 1 = 501. 600 would be 100 separate hexagons with no shared sides.",{"itemId":718,"prompt":719,"options":720,"correct":99,"why":729},"patterns.investigate-q-back","How many matchstick squares in a row use exactly 61 sticks (3 × n + 1)?",[721,723,725,727],{"id":96,"label":722},"19",{"id":99,"label":724},"20",{"id":102,"label":726},"21",{"id":105,"label":728},"15","3 × n + 1 = 61, so 3 × n = 60 and n = 20.",{"itemId":731,"prompt":732,"options":733,"correct":96,"why":742},"patterns.investigate-q-odd","What is 1 + 3 + 5 + … + 29 (the first 15 odd numbers)?",[734,736,738,740],{"id":96,"label":735},"225",{"id":99,"label":737},"210",{"id":102,"label":739},"240",{"id":105,"label":741},"450","The first 15 odd numbers add to 15 × 15 = 225.",{"itemId":744,"prompt":745,"options":746,"correct":99,"why":755},"patterns.investigate-q-gauss","What is 1 + 2 + 3 + … + 50?",[747,749,751,753],{"id":96,"label":748},"1,250",{"id":99,"label":750},"1,275",{"id":102,"label":752},"2,550",{"id":105,"label":754},"1,300","50 × 51 ÷ 2 = 1,275 (25 pairs of 51).",{"itemId":757,"prompt":758,"options":759,"correct":102,"why":767},"patterns.investigate-q-cal","On a calendar, a 2 × 2 box has diagonal products whose difference is:",[760,762,764,766],{"id":96,"label":761},"1",{"id":99,"label":763},"6",{"id":102,"label":765},"7",{"id":105,"label":33},"The difference is the row length. Calendar rows have 7 days.",{"itemId":769,"prompt":770,"options":771,"correct":99,"why":780},"patterns.investigate-q-3x3","A 3 × 3 box on a calendar has 18 in the middle. What is the total of the nine dates?",[772,774,776,778],{"id":96,"label":773},"144",{"id":99,"label":775},"162",{"id":102,"label":777},"180",{"id":105,"label":779},"153","9 × 18 = 162.",{"itemId":782,"prompt":783,"options":784,"correct":102,"why":792},"patterns.investigate-q-counter","Which is a counterexample to “every square number is even”?",[785,787,788,790],{"id":96,"label":786},"16",{"id":99,"label":601},{"id":102,"label":789},"49",{"id":105,"label":791},"64","49 = 7 × 7 is a square number and it is odd, so the statement is false.",{"itemId":794,"prompt":795,"options":796,"correct":99,"why":805},"patterns.investigate-q-race","Plan A adds ₹100 a day. Plan B starts at ₹1 and doubles daily. Which is true?",[797,799,801,803],{"id":96,"label":798},"Plan A is always ahead",{"id":99,"label":800},"Plan B overtakes eventually",{"id":102,"label":802},"They are equal on day 10",{"id":105,"label":804},"Plan B is always ahead","Doubling starts slowly but always overtakes steady adding in the end: on day 12 Plan B pays ₹2,048 against Plan A’s ₹1,200.",{"itemId":807,"prompt":808,"options":809,"correct":99,"why":818},"patterns.investigate-q-11","What is 87 × 11?",[810,812,814,816],{"id":96,"label":811},"8,157",{"id":99,"label":813},"957",{"id":102,"label":815},"967",{"id":105,"label":817},"1,057","8 + 7 = 15: write 5 in the middle and carry 1 into the 8, giving 957. Check: 87 × 10 + 87 = 870 + 87 = 957.",{"itemId":820,"prompt":821,"options":822,"correct":99,"why":831},"patterns.investigate-q-grid","Squares 2 rows high and n long use 5 × n + 2 sticks. How many for n = 7?",[823,825,827,829],{"id":96,"label":824},"35",{"id":99,"label":826},"37",{"id":102,"label":828},"42",{"id":105,"label":830},"39","5 × 7 + 2 = 37. Count check: 3 rows of 7 horizontal (21) plus 8 columns of 2 vertical (16) = 37.",{"id":833,"type":834,"prompt":835},"reflect-investigate","reflection","Choose one investigation from this layer. Write a short report as a detective would: the question, your first prediction (right or wrong), the table of small cases, the rule you found, how you tested it, and your explanation of **why** it works. What would you investigate next?",{"id":837,"type":838,"conceptId":839,"relation":840,"explanation":841},"conn-properties","connection","properties-of-numbers","helps_understand","Always\u002Fsometimes\u002Fnever questions about odd, even and square numbers are properties of numbers; investigating patterns is how you discover them.",{"id":843,"type":838,"conceptId":844,"relation":845,"explanation":846},"conn-shape","shape-and-space","related_to","Dot squares, dot triangles, rectangles and L-shaped gnomons are shapes; counting them turns geometry into number patterns.",{"id":848,"type":838,"conceptId":849,"relation":845,"explanation":850},"conn-data","data-handling","Collecting cases in a table and looking for a trend is the same habit used in data handling, where patterns in data suggest (but do not prove) a rule.",{"id":852,"type":853,"title":854,"points":855},"cheat-investigate","summary","Cheat sheet",[856,857,858,859,860,861,862,863,864,865],"**Detective method:** ask → predict → collect small cases → spot a rule → test a new case → explain why → predict far.","**Conjecture:** believed from examples. **Counterexample:** one case that breaks it. Examples suggest; reasons prove.","**Far terms:** matchstick squares 3n + 1 (100th: 301), triangles 2n + 1 (201), hexagons 5n + 1 (501). Work backwards by undoing: 3n + 1 = 2,026 → n = 675.","**Rectangles of squares:** 2 rows high need 5n + 2 sticks; 3 rows high need 7n + 3.","**Triangular numbers:** two copies make an n by (n + 1) rectangle, so Tₙ = n(n + 1) ÷ 2. The 100th is 5,050.","**Odd sums:** 1 + 3 + 5 + … (n terms) = n × n, because each odd number is an L-shape around a square.","**Gauss’s pairing:** sum of an arithmetic sequence = (first + last) × number of terms ÷ 2.","**Grids:** in a 2 × 2 box the diagonal products differ by the row length (10 on a hundred square, 7 on a calendar). A 3 × 3 box adds to 9 × the middle.","**Growth races:** doubling always overtakes steady adding in the end; 42 paper folds would pass the Moon.","**Beware:** 1, 2, 4 can continue in many ways. Always ask: does my rule fit every term, and do I know why?",{"id":867,"type":868,"sourceIds":869},"sources-investigate","sources",[870,871,872,873,874],"patterns-ncert-ganita-prakash-6","patterns-ncert-class6-algebra","patterns-mathsisfun-sequences","patterns-wiki-circle-regions","patterns-wiki-gauss",[870,871,872,873,874],"needs_review",{"generatedBy":878,"notes":879},"claude-code","Draft generated locally from a Python script with every number computed and asserted; pending owner review.","66c5fa6b8253f3d768bbd1eb797962c0283d5c1cbd9ff171658e8871bd024954",{"component:pattern-machine@1":882,"logic:practice":883,"component:sort-game@1":884,"component:match-pairs@1":885,"source:patterns-mathsisfun-sequences":886,"source:patterns-ncert-class6-algebra":887,"source:patterns-ncert-ganita-prakash-6":888,"source:patterns-wiki-circle-regions":889,"source:patterns-wiki-gauss":890},"b3d384b49b4d138a00726767a816928da91f44fad70d7d5ccfca6db297c45839","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","1c634ad2570a389d1bc96431d0a220aa359ad82b4d33f5b55eb0b12dab8bab4c","b31773e8224c076a8a34aa4e6df704febaac5ecd4d6e1d3ba5041a31aad871e1","36726d3ff16d73627216db999b62b46580dfaf514f313774e79669de57d7c595","3036d0a86299354fe4f0f1df748a233c1fe9fdf208b4e75f1169be57492e7095","81c9809a1cbaa777fc7846784b96a2e10997501e6d2142d0b1458eea22e052b0",{"state":892,"reviewer":893,"selfReview":894,"reviewedAt":895,"method":896},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598664]