[{"data":1,"prerenderedAt":1174},["ShallowReactive",2],{"layer:four-operations:extend":3},{"layer":4,"contentHash":1146,"dependencyHashes":1147,"approval":1168,"releaseId":1173},{"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":1141,"reviewStatus":1142,"authoring":1143},1,"four-operations","en","extend","Other ways to calculate, and harder puzzles","Lattices, Vedic-style shortcuts, doubling, binary, classic puzzles, olympiad problems and real projects","Try the lattice, Napier's bones, Vedic-style shortcuts and Russian peasant multiplication and see why each works. Crack classic puzzles and olympiad problems, then plan real projects: a trip budget, a kirana bill, a harvest and a run chase.",[13,14,15,16,17],"Multiply with the lattice, near-a-base, × 11 and doubling-and-halving methods, and explain each with place value.","Link doubling-and-halving to binary and to how computers add with carries.","Solve remainder, missing-digit and counting puzzles, checking answers by reasoning and by search.","Plan and cost a real project, making sensible decisions about remainders and estimates.","Tell apart the ancient Indian mathematical tradition and the modern 'Vedic Mathematics' collection.",50,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 50 minutes",{"label":29,"value":30},"Prior knowledge","Column methods and checking",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","× 11 sprint, trick sort, project sprint, methods match",{"label":38,"value":39},"Try at home","Magic square, rainfall diary, kirana bill",[41,45,51,57,60,80,98,103,117,154,157,162,165,175,186,225,229,232,237,242,260,329,341,346,349,376,385,389,404,416,421,424,435,439,460,465,468,478,484,496,501,511,522,527,532,542,545,556,568,573,587,592,595,608,658,661,670,680,730,734,746,751,754,785,789,794,797,802,868,873,905,961,965,1096,1100,1118,1122],{"id":42,"type":43,"markdown":44},"intro","prose","You can already add, subtract, multiply and divide numbers as big as crores, and you know how to check your answers. So is there anything left to explore? A great deal.\n\nThe column methods you learned at school are only **one way** to calculate. For centuries people in India, the Arab world, China, Egypt, Russia and Europe used other methods: filling in a grid of diagonals, laying out carved rods, doubling and halving, sliding beads on an abacus. Each method works for the same deep reason (place value and the way multiplication spreads over addition), but each one looks completely different.\n\nThis layer is a playground. You will try **other ways to calculate**, crack **puzzles** that have entertained people for hundreds of years, tackle **olympiad-style problems**, plan **real projects** with real budgets, and meet people whose jobs depend on the four operations. There is no single right order. Dip in wherever looks interesting.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how","callout","observation","How to use this layer","Keep a pencil and a rough page nearby. For every new method, **try it on your own numbers first**, then check with the column method you already trust. For every puzzle, have a real go (at least five minutes) before you read the answer. Getting stuck is part of the fun: most puzzles in this layer were designed to make clever people stuck.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","Lattice multiplication: the grid of diagonals","Chapter 01","1 Lattice method",{"id":58,"type":43,"markdown":59},"lattice-intro","Here is a method that feels like drawing, not calculating. It is called **lattice multiplication**, or the **gelosia method** (a *gelosia* is an Italian window screen with a diagonal criss-cross pattern, which is what the grid looks like).\n\nTo multiply **473 × 58**, draw a grid with one column for each digit of 473 and one row for each digit of 58. Write 473 along the top and 58 down the right-hand side. Draw a diagonal line through every box, from its top-right corner to its bottom-left corner.\n\nIn each box, write the product of its column digit and row digit, with the **tens above the diagonal** and the **ones below it**. So 4 × 5 = 20 is written as 2 \u002F 0. Every box holds a two-digit answer (a single digit gets a 0 in front: 2 × 3 = 06).",{"id":61,"type":62,"caption":63,"columns":64,"rows":69},"lattice-grid","table","The filled lattice for 473 × 58: each box shows tens \u002F ones of one digit product",[65,66,67,68],"Row digit","Column 4","Column 7","Column 3",[70,75],[71,72,73,74],"× 5","2 \u002F 0","3 \u002F 5","1 \u002F 5",[76,77,78,79],"× 8","3 \u002F 2","5 \u002F 6","2 \u002F 4",{"id":81,"type":82,"title":83,"problem":84,"steps":85,"help":93},"we-lattice","worked_example","Lattice multiplication of 473 × 58","Use the lattice to find 473 × 58. Add along the diagonals, starting from the bottom-right corner.",[86,87,88,89,90,91,92],"Fill every box: 4 × 5 = 20; 7 × 5 = 35; 3 × 5 = 15; 4 × 8 = 32; 7 × 8 = 56; 3 × 8 = 24.","**Ones diagonal:** 4 = 4. Write **4**.","**Tens diagonal:** 5 + 6 + 2 = 13. Write **3**, carry 1 into the next diagonal.","**Hundreds diagonal:** 5 + 1 + 2 + 5 + 1 (carried) = 14. Write **4**, carry 1 into the next diagonal.","**Thousands diagonal:** 0 + 3 + 3 + 1 (carried) = 7. Write **7**.","**Ten-thousands diagonal:** 2 = 2. Write **2**.","Read the digits from the top-left down and round to the bottom-right: **27,434**. Check with long multiplication: 473 × 8 = 3,784, 473 × 50 = 23,650, and 3,784 + 23,650 = 27,434. ✓",{"simplerExplanation":94,"hints":95},"Every box is a small times-table fact. Each diagonal collects the pieces that belong to the same place value (ones, tens, hundreds…), so adding along a diagonal is just adding a column, tilted.",[96,97],"Tens go above the diagonal, ones below.","Start adding at the bottom-right: that diagonal holds only the ones.",{"id":99,"type":47,"variant":100,"title":101,"markdown":102},"aha-lattice","aha","Why the diagonals work","Every diagonal strip collects numbers with the **same place value**. The bottom-right triangle holds the ones of 3 × 8. The next strip holds the tens of 3 × 8, the ones of 7 × 8 (which is really 70 × 8, so its ones are tens) and the ones of 3 × 5 (really 3 × 50). All of those are **tens**. The lattice is ordinary long multiplication with the partial products chopped into pieces and slid into place-value lanes. The big advantage: you do all the small multiplications first and **all the carrying at the very end**, so you never have to hold a carry in your head while multiplying.",{"id":104,"type":105,"itemId":106,"prompt":107,"check":108,"hints":112,"feedback":114},"prac-lattice","practice","four-operations.x-lattice-diagonal","In a lattice for **36 × 24**, the boxes hold 3 × 2 = 06, 6 × 2 = 12, 3 × 4 = 12 and 6 × 4 = 24. The ones diagonal is 4. What number do you get when you add the **tens diagonal** (before carrying)?",{"kind":109,"answer":110,"tolerance":111},"number",6,0,[113],"The tens diagonal holds the tens digit of 6 × 4 and the ones digits of 6 × 2 and 3 × 4.",{"correct":115,"incorrect":116},"Yes: 2 + 2 + 2 = 6. The full answer is 36 × 24 = 864.","The tens diagonal holds 2 (tens of 24), 2 (ones of 12 from 6 × 2) and 2 (ones of 12 from 3 × 4): 2 + 2 + 2 = 6. The answer is 36 × 24 = 864.",{"id":118,"type":119,"title":120,"items":121},"tl-tools","timeline","Tools and methods for calculating, through history",[122,126,130,134,138,142,146,150],{"time":123,"title":124,"text":125},"Ancient","Counting boards and abacuses","Many civilisations moved pebbles or beads in columns to add and subtract. The Latin word *calculus* means a small pebble.",{"time":127,"title":128,"text":129},"500s-800s","Place value with zero in India","Indian mathematicians write numbers with nine digits and a place-holder zero, making written column methods possible. The earliest undisputed dated example is the Gwalior inscription of 876 AD; earlier claims are argued over.",{"time":131,"title":132,"text":133},"1200s-1400s","Lattice method spreads","Grid multiplication is recorded by Ibn al-Banna' in the Arab world in the late 1200s, in an anonymous Latin treatise in England around 1300, and by Wu Jing in China in 1450. Its first origin is uncertain, and a similar method is described in a 12th-century Indian commentary.",{"time":135,"title":136,"text":137},"1202","Liber Abaci","Fibonacci's book teaches Europeans the Hindu–Arabic numerals and written methods, including grid-style multiplication.",{"time":139,"title":140,"text":141},"1617","Napier's bones","John Napier of Scotland describes numbered rods that turn multiplication into reading and adding diagonals, like a portable lattice.",{"time":143,"title":144,"text":145},"1642","Pascal's calculator","Blaise Pascal builds a gear machine that adds and subtracts, with an automatic carry from one wheel to the next.",{"time":147,"title":148,"text":149},"1940s","Electronic computers","Machines like ENIAC do thousands of additions a second using electronic switches.",{"time":151,"title":152,"text":153},"1970s","Pocket calculators","Cheap electronic calculators arrive; today a phone does billions of operations every second.",{"id":155,"type":43,"markdown":156},"napier","**Napier's bones** are a set of rods, one for each digit 0 to 9. Each rod shows that digit's times table, written in little diagonal-split squares exactly like a lattice. To multiply 473 × 8, you lay the 4, 7 and 3 rods side by side, look along the row for 8, and add along the diagonals. Scottish mathematician John Napier published the idea in 1617. The rods were popular across Europe, and sets were even made in China and Japan. They are really a **lattice you can carry in your pocket**.",{"id":158,"type":53,"title":159,"eyebrow":160,"navLabel":161},"ch2","Vedic-style shortcuts, and why they work","Chapter 02","2 Clever shortcuts",{"id":163,"type":43,"markdown":164},"vedic-intro","In India you may have seen books and classes on **Vedic Maths**, full of fast tricks with Sanskrit names. The tricks are real, clever and fun, and every one of them works because of place value. Here are four of the best, each with the reason it works. Treat them as extra tools for special numbers, not as a replacement for the general methods.",{"id":166,"type":82,"title":167,"problem":168,"steps":169},"we-crosswise","Vertically and crosswise: 47 × 36","Multiply 47 × 36 in one line, using the pattern *vertically, crosswise, vertically*.",[170,171,172,173,174],"**Ones (vertically, right):** 7 × 6 = 42. Write **2**, carry 4.","**Tens (crosswise):** 4 × 6 + 7 × 3 = 24 + 21 = 45, plus the carried 4 = 49. Write **9**, carry 4.","**Hundreds (vertically, left):** 4 × 3 = 12, plus the carried 4 = **16**.","Answer: **1,692**.","**Why it works:** 47 × 36 = (40 + 7) × (30 + 6). Multiplying out gives 40 × 30 (hundreds), 40 × 6 and 7 × 30 (both tens), and 7 × 6 (ones). The crosswise step simply adds the two 'tens' pieces together: 24 tens + 21 tens = 45 tens.",{"id":176,"type":82,"title":177,"problem":178,"steps":179},"we-nikhilam","Near a base: 97 × 96","Multiply 97 × 96 using how far each number is from 100 (the *nikhilam* idea, 'all from nine and the last from ten').",[180,181,182,183,184,185],"97 is **3** short of 100; 96 is **4** short of 100.","**Left part:** take one number minus the other's shortfall: 97 − 4 = **93** (or 96 − 3 = 93, the same).","**Right part:** multiply the shortfalls: 3 × 4 = **12**. It must fill two places, because the base 100 has two zeros.","Put them together: **9,312**. Check: 97 × 96 = 9,312. ✓","**Above the base works too:** 104 × 107: 104 + 7 = 111, and 4 × 7 = 28, so the answer is **11,128**.","**Why it works:** 97 × 96 = 97 × (100 − 4) = 9700 − 388. Rearranging, that is (97 − 4) × 100 + 3 × 4 = 9300 + 12. The trick just does this bookkeeping for you.",{"id":187,"type":62,"caption":188,"columns":189,"rows":195},"tab-eleven","Multiplying a two-digit number by 11: split the digits, put their sum in the middle",[190,191,192,193,194],"Number","Digits","Sum of digits","Answer","Check",[196,202,208,213,219],[197,198,199,200,201],"23","2 _ 3","5","253","23 × 11 = 253",[203,204,205,206,207],"54","5 _ 4","9","594","54 × 11 = 594",[209,210,205,211,212],"63","6 _ 3","693","63 × 11 = 693",[214,215,216,217,218],"78","7 _ 8","15","858 (carry 1)","78 × 11 = 858",[220,221,222,223,224],"95","9 _ 5","14","1,045 (carry 1)","95 × 11 = 1,045",{"id":226,"type":47,"variant":100,"title":227,"markdown":228},"aha-eleven","Why the 11 trick works","11 = 10 + 1, so 63 × 11 = 63 × 10 + 63 × 1 = 630 + 63. Line them up: the 6 of 630 sits in the hundreds, the 3 of 630 and the 6 of 63 land together in the tens (3 + 6 = 9), and the 3 of 63 sits in the ones. So 63 × 11 = 693. When the middle sum is 10 or more, carry the 1 into the hundreds: 78 × 11 = 7 | 15 | 8 = 858.",{"id":230,"type":43,"markdown":231},"squares-five","**Squaring numbers that end in 5.** Take the tens part, multiply it by the next number up, and write 25 after it. 35 × 35: 3 × 4 = 12, then 25, giving **1225**. 85 × 85: 8 × 9 = 72, giving **7225**. It even works for 105 × 105: 10 × 11 = 110, giving **11,025**.\n\n*Why?* 35 × 35 = 30 × 35 + 5 × 35. Split the second part: 5 × 35 = 5 × 30 + 5 × 5. Now 30 × 35 + 5 × 30 = 30 × (35 + 5) = 30 × 40 = 1,200, and 5 × 5 = 25. So 35 × 35 = 30 × 40 + 25. The 'tens times next number' is just 30 × 40 in disguise.",{"id":233,"type":47,"variant":234,"title":235,"markdown":236},"ml-vedic","model_limit","Where do these tricks really come from?","The book *Vedic Mathematics* was written by Bharati Krishna Tirtha, a Shankaracharya of Puri, and first published in 1965, five years after his death. It presents sixteen short *sutras* (rules), but historians and Sanskrit scholars have not found these sutras in the Vedas themselves, and the author said they came from his own study and insight. So the tricks are a **modern (twentieth-century) collection**, many of them rediscoveries of shortcuts known in several countries.\n\nThat does not make them less useful. India does have a genuinely ancient and brilliant mathematical tradition, from place-value numerals and zero to Aryabhata, Brahmagupta and Bhāskara II, which you can read about in the Go deeper layer. It is just worth knowing which is which.",{"id":238,"type":47,"variant":239,"title":240,"markdown":241},"nuance-tricks","nuance","Shortcuts are specialists","Each trick is fast only for **special numbers**: the 11 trick for multiplying by 11, nikhilam for numbers near 10, 100 or 1,000, the 5-square trick for numbers ending in 5. For 473 × 58 none of them helps much. Good calculators first **look at the numbers**, then pick a tool. That judgement, knowing when a shortcut fits, is the real skill.",{"id":243,"type":244,"component":245,"componentVersion":5,"config":246,"objective":258,"textAlternative":259},"lab-sprint-eleven","interactive","arith-sprint",{"operations":247,"ranges":249,"rounds":254,"secondsTotal":255,"estimateFirst":256,"wordProblems":257},[248],"×",{"a":250,"b":253},{"min":251,"max":252},11,99,{"min":251,"max":251},12,90,false,[],"Race against the clock multiplying two-digit numbers by 11 with the split-and-add trick.","This sprint gives 12 questions of the form (a two-digit number) × 11, with 90 seconds on the clock. Levels rise as your streak grows.\n\nUse the trick: write the first digit, then the sum of the two digits, then the last digit.\n- 23 × 11: 2 | 5 | 3 = **253**.\n- 54 × 11: 5 | 9 | 4 = **594**.\n- 78 × 11: 7 | 15 | 8, carry the 1 into the hundreds = **858**.\n- 99 × 11: 9 | 18 | 9, carry the 1 = **1,089**.\n\nWhy it works: 11 = 10 + 1, so the number × 11 is the number × 10 plus the number itself. The two copies overlap in the tens place, which is why the middle digit is the sum of the two digits. Play once slowly, then try to beat your best score.",{"id":261,"type":244,"component":262,"componentVersion":5,"config":263,"objective":327,"textAlternative":328},"lab-sort-tricks","sort-game",{"prompt":264,"bins":265,"items":278,"seconds":111},"Which shortcut fits each multiplication best? Look at the numbers first.",[266,269,272,275],{"id":267,"label":268},"eleven","× 11 split-and-add",{"id":270,"label":271},"base","Near a base (nikhilam)",{"id":273,"label":274},"five","Square ending in 5",{"id":276,"label":277},"halve","Halve and double",[279,283,287,291,295,299,303,307,311,315,319,323],{"id":280,"label":281,"bin":267,"why":282},"i1","62 × 11","6 | 6+2 | 2 = 682.",{"id":284,"label":285,"bin":270,"why":286},"i2","98 × 97","Shortfalls 2 and 3: 98 − 3 = 95, 2 × 3 = 06, so 9,506.",{"id":288,"label":289,"bin":273,"why":290},"i3","45 × 45","4 × 5 = 20, then 25: 2,025.",{"id":292,"label":293,"bin":276,"why":294},"i4","16 × 25","Halve 16 and double 25 twice: 4 × 100 = 400.",{"id":296,"label":297,"bin":267,"why":298},"i5","87 × 11","8 | 15 | 7, carry 1: 957.",{"id":300,"label":301,"bin":270,"why":302},"i6","103 × 106","Extras 3 and 6: 103 + 6 = 109, 3 × 6 = 18, so 10,918.",{"id":304,"label":305,"bin":273,"why":306},"i7","75 × 75","7 × 8 = 56, then 25: 5,625.",{"id":308,"label":309,"bin":276,"why":310},"i8","125 × 24","Double 125 and halve 24 until easy: 1,000 × 3 = 3,000.",{"id":312,"label":313,"bin":270,"why":314},"i9","996 × 998","Base 1,000, shortfalls 4 and 2: 996 − 2 = 994, 4 × 2 = 008, so 9,94,008.",{"id":316,"label":317,"bin":273,"why":318},"i10","95 × 95","9 × 10 = 90, then 25: 9,025.",{"id":320,"label":321,"bin":276,"why":322},"i11","35 × 18","Double 35, halve 18: 70 × 9 = 630.",{"id":324,"label":325,"bin":267,"why":326},"i12","44 × 11","4 | 8 | 4 = 484.","Look at the numbers and choose the shortcut that makes each multiplication easy.","This sorting game shows 12 multiplications to drop into four bins, one per shortcut.\n\n- **× 11 split-and-add:** 62 × 11 = 682; 87 × 11 = 957 (8 | 15 | 7 with a carry); 44 × 11 = 484.\n- **Near a base (nikhilam):** 98 × 97 = 9,506 (98 − 3 = 95, then 2 × 3 = 06); 103 × 106 = 10,918 (103 + 6 = 109, then 3 × 6 = 18); 996 × 998 = 9,94,008 (base 1,000, so the right part needs three digits: 008).\n- **Square ending in 5:** 45 × 45 = 2,025; 75 × 75 = 5,625; 95 × 95 = 9,025.\n- **Halve and double:** 16 × 25 = 8 × 50 = 4 × 100 = 400; 125 × 24 = 250 × 12 = 500 × 6 = 1,000 × 3 = 3,000; 35 × 18 = 70 × 9 = 630.\n\nThe skill being practised is noticing: a multiplier of 11, numbers hugging 100 or 1,000, a repeated number ending in 5, or a factor that doubles into 10, 100 or 1,000.",{"id":330,"type":105,"itemId":331,"prompt":332,"check":333,"hints":335,"feedback":338},"prac-nikhilam","four-operations.x-near-base","Use the near-a-base idea to find **93 × 98**.",{"kind":109,"answer":334,"tolerance":111},9114,[336,337],"93 is 7 short of 100; 98 is 2 short.","Left part: 93 − 2. Right part: 7 × 2, written with two digits.",{"correct":339,"incorrect":340},"Yes: 93 − 2 = 91 and 7 × 2 = 14, so 9,114.","Shortfalls are 7 and 2. Left: 93 − 2 = 91. Right: 7 × 2 = 14. Answer 9,114.",{"id":342,"type":53,"title":343,"eyebrow":344,"navLabel":345},"ch3","Doubling and halving: the Egyptian and Russian way","Chapter 03","3 Doubling methods",{"id":347,"type":43,"markdown":348},"peasant-intro","Imagine you only knew how to **double**, **halve** and **add**, with no times tables at all. Could you still multiply? Yes. The ancient Egyptians multiplied this way. The Rhind papyrus, which the scribe Ahmes copied about 1550 BCE from an older document of the Middle Kingdom, shows the method; the same idea is known today as **Russian peasant multiplication**.\n\nTo find **37 × 24**: write the two numbers at the top of two columns. Keep **halving** the left number (throw away any half) and **doubling** the right number, until the left column reaches 1. Then cross out every row where the left number is **even**, and add what remains in the right column.",{"id":350,"type":62,"caption":351,"columns":352,"rows":356},"tab-peasant","Russian peasant multiplication: 37 × 24",[353,354,355],"Halve (drop halves)","Double","Keep?",[357,361,365,367,370,373],[358,359,360],"37","24","odd: keep",[362,363,364],"18","48","even: cross out",[205,366,360],"96",[368,369,364],"4","192",[371,372,364],"2","384",[374,375,360],"1","768",{"id":377,"type":82,"title":378,"problem":379,"steps":380},"we-peasant","Finishing 37 × 24","Add the right-hand numbers in the rows you kept, then explain why this works.",[381,382,383,384],"Kept rows: 24 + 96 + 768 = **888**.","Check: 37 × 24 = 888. ✓","**Why:** the odd rows secretly split 37 into doubles: 37 = 32 + 4 + 1. (Every time we dropped a half, that row was odd, and we kept it.)","So 37 × 24 = 32 × 24 + 4 × 24 + 1 × 24 = 768 + 96 + 24 = 888. The right column held exactly those doubles of 24.",{"id":386,"type":47,"variant":100,"title":387,"markdown":388},"aha-binary","You just used binary","Writing 37 as 32 + 4 + 1 is writing it in **binary**, the base-two place-value system computers use. In binary, 37 is **100101**: one 32, no 16, no 8, one 4, no 2, one 1. The kept rows (odd) are exactly the 1s; the crossed-out rows are the 0s, read from the bottom up. Russian peasant multiplication is how a computer's circuits multiply: shift (double), test the last binary digit, and add.",{"id":390,"type":391,"prompt":392,"options":393,"explanation":403},"pred-peasant-swap","prediction","Would Russian peasant multiplication give the same answer if you put **24** on the left and **37** on the right?",[394,397,400],{"id":395,"label":396},"a","Yes, the same answer with a different set of rows",{"id":398,"label":399},"b","No, the method only works with the bigger number on the left",{"id":401,"label":402},"c","Only if both numbers are even","**Yes.** Halving 24: 24, 12, 6, 3, 1; doubling 37: 37, 74, 148, 296, 592. The odd rows are 3 and 1, so add 296 + 592 = **888**. Multiplication is commutative, and the method is correct for any whole numbers. Putting the smaller number on the left usually means fewer rows.",{"id":405,"type":105,"itemId":406,"prompt":407,"check":408,"hints":410,"feedback":413},"prac-peasant","four-operations.x-peasant-sum","Use doubling and halving for **13 × 45**. The halving column goes 13, 6, 3, 1. Add the doubled numbers in the odd rows. What is the answer?",{"kind":109,"answer":409,"tolerance":111},585,[411,412],"The doubling column goes 45, 90, 180, 360.","Rows 13, 3 and 1 are odd.",{"correct":414,"incorrect":415},"Correct: 45 + 180 + 360 = 585 = 13 × 45.","Odd rows are 13 (45), 3 (180) and 1 (360). 45 + 180 + 360 = 585.",{"id":417,"type":53,"title":418,"eyebrow":419,"navLabel":420},"ch4","Beads, brains and binary: calculating machines","Chapter 04","4 Machines and minds",{"id":422,"type":43,"markdown":423},"abacus","An **abacus** is a frame of rods with sliding beads, one rod per place value. On the Japanese **soroban**, each rod has one 'heaven' bead worth 5 and four 'earth' beads worth 1 each, so a rod can show any digit 0 to 9. Adding means pushing beads towards the bar; when a rod would pass 9, you clear it and push one bead on the next rod to the left. That is **carrying**, done with your fingers.\n\nChildren who train on an abacus for a long time often learn to picture it in their heads and slide imaginary beads, adding long lists of numbers astonishingly fast. Abacus and mental-arithmetic classes are popular in many Indian cities today.\n\nIndia also produced one of the most famous mental calculators of all time: **Shakuntala Devi** (1929–2013), often called the 'human computer'. On 18 June 1980, at Imperial College London, she multiplied two 13-digit numbers in her head and gave the correct 26-digit answer in 28 seconds — a feat recorded in the 1982 *Guinness Book of World Records*.",{"id":425,"type":82,"title":426,"problem":427,"steps":428},"we-binary","How a computer adds 11 + 6","Computers store numbers in binary, using only the digits 0 and 1. Add 1011 (eleven) and 0110 (six) in binary.",[429,430,431,432,433,434],"In binary each place is worth **twice** the one to its right: 1, 2, 4, 8, 16… And 1 + 1 = 10 (that is 'two': write 0, carry 1).","**1s column:** 1 + 0 = 1. Write 1.","**2s column:** 1 + 1 = 10. Write 0, carry 1.","**4s column:** 0 + 1 + 1 (carry) = 10. Write 0, carry 1.","**8s column:** 1 + 0 + 1 (carry) = 10. Write 0, carry 1 into the 16s column.","Answer: **10001**, which is 16 + 1 = **17**. ✓ The same column method you use, with carrying, just with 2 instead of 10.",{"id":436,"type":47,"variant":239,"title":437,"markdown":438},"nuance-calculator","If the calculator is always right, why learn this?","A calculator does exactly what you type, and nothing more. Press 250 ÷ 45 and it says 5.5555556: it will not tell you that you need **6 buses**. Type 4,872 + 3,915 when you meant 3,195 and it confidently gives the wrong total. The human jobs are **choosing the operation**, **estimating** so you notice a wrong key, and **interpreting** the answer in the story. Those are exactly the skills machines cannot do for you.",{"id":440,"type":441,"tone":442,"items":443},"spec-speeds","spec","neutral",[444,448,452,456],{"label":445,"big":446,"value":447},"Human, pencil","≈ 1 min","A careful four-digit × two-digit long multiplication, including checking.",{"label":449,"big":450,"value":451},"Abacus expert","seconds","Trained soroban users can add ten 3-digit numbers faster than most people can type them.",{"label":453,"big":454,"value":455},"Pocket calculator","instant","Any four-operation sum within its 8–12 digit display.",{"label":457,"big":458,"value":459},"A phone chip","billions \u002F s","Modern processors do billions of simple operations every second.",{"id":461,"type":53,"title":462,"eyebrow":463,"navLabel":464},"ch5","Puzzles with remainders, missing digits and a magic number","Chapter 05","5 Classic puzzles",{"id":466,"type":43,"markdown":467},"egg-puzzle","Here is a puzzle that has been told, in different versions, for about a thousand years in India, the Arab world and Europe:\n\n> A woman takes her eggs to market. When she puts them in rows of 2, one egg is left over. The same happens with rows of 3, 4, 5 and 6: always one left. But rows of 7 fit exactly. What is the smallest number of eggs she could have?\n\nTry it before reading on. Hint: first find numbers that leave remainder 1 for all of 2, 3, 4, 5 and 6.",{"id":469,"type":82,"title":470,"problem":471,"steps":472},"we-egg","Cracking the egg puzzle","Find the smallest number of eggs that leaves remainder 1 when divided by 2, 3, 4, 5 and 6, and none when divided by 7.",[473,474,475,476,477],"Remove the leftover egg: the number **minus 1** must divide exactly by 2, 3, 4, 5 and 6.","The smallest number that all of 2, 3, 4, 5 and 6 divide is their **LCM**, 60. So the number minus 1 is one of 60, 120, 180, 240, 300…","So the number is one of 61, 121, 181, 241, **301**, …","Test each for rows of 7: 61 ÷ 7 = 8 r 5; 121 ÷ 7 = 17 r 2; 181 ÷ 7 = 25 r 6; 241 ÷ 7 = 34 r 3; 301 ÷ 7 = 43 r 0. ✓","The smallest answer is **301 eggs**. (Next one: 301 + 420 = 721, because 420 is the LCM of 2 to 7.)",{"id":479,"type":480,"conceptId":481,"relation":482,"explanation":483},"conn-hcf","connection","hcf-and-lcm","applied_in","The egg puzzle is really an LCM question in disguise: the smallest number every divisor fits into, plus the leftover.",{"id":485,"type":105,"itemId":486,"prompt":487,"check":488,"hints":490,"feedback":493},"prac-missing","four-operations.x-missing-digits","A multiplication has two digits smudged: **□7 × □3 = 2,491**. What is the first number (the one ending in 7)?",{"kind":109,"answer":489,"tolerance":111},47,[491,492],"Estimate: 2,491 is close to 2,500 = 50 × 50, so both numbers are near 50.","Try 47 × 53.",{"correct":494,"incorrect":495},"Yes: 47 × 53 = 2491. Estimating first (≈ 50 × 50) narrows the search to one or two tries.","2,491 ≈ 2,500 = 50 × 50, so try numbers near 50 that end in 7 and 3: 47 × 53 = 2491. The first number is 47.",{"id":497,"type":47,"variant":498,"title":499,"markdown":500},"q-reverse","question","A number that flips when you multiply it","There is exactly **one** four-digit number which, when multiplied by 4, gives the same digits in reverse order. It is **2,178**: 2,178 × 4 = 8,712. A computer can find it by checking all 9,000 four-digit numbers, but can you find it by reasoning? Start with the first digit: if the number × 4 is still four digits, the first digit must be 1 or 2, and the last digit of the answer must be even… Try it, then look for a five-digit number that does the same (hint: slip a 9 into the middle).",{"id":502,"type":82,"title":503,"problem":504,"steps":505},"we-kaprekar","Kaprekar's routine: every road leads to 6174","Take a four-digit number whose digits are not all the same, such as 3524. Arrange its digits from largest to smallest, and from smallest to largest, and subtract. Repeat with the answer.",[506,507,508,509,510],"5,432 − 2345 = **3,087**","8,730 − 0378 = **8,352**","8,532 − 2358 = **6,174**","From 6174: 7641 − 1467 = 6174. It stays at **6174** forever.","The Indian mathematician **D. R. Kaprekar** (born in Dahanu in 1905, died in Devlali in 1986), a school teacher in Devlali near Nashik, found this in 1946, presented it at the Madras Mathematical Conference in 1949 and published it in 1953. Every four-digit number (not all digits the same) reaches 6174 in **at most 7 steps**. 6174 is called **Kaprekar's constant**.",{"id":512,"type":391,"prompt":513,"options":514,"explanation":521},"pred-kaprekar","Start Kaprekar's routine from **2111** (remember to keep leading zeros, so 0999 counts as a four-digit number). What do you think happens?",[515,517,519],{"id":395,"label":516},"It reaches 6174",{"id":398,"label":518},"It gets stuck at 0",{"id":401,"label":520},"It loops forever without reaching 6174","**It reaches 6174.** 2111 − 1112 = 0999; 9990 − 0999 = 8991; 9981 − 1899 = 8082; 8820 − 0288 = 8532; 8532 − 2358 = 6174. That took 5 steps. A computer check of all 8,991 allowed starting numbers shows none takes more than 7.",{"id":523,"type":480,"conceptId":524,"relation":525,"explanation":526},"conn-patterns","patterns","related_to","Kaprekar's routine, the ×11 digit pattern and the reversing number 2178 are all number patterns that come out of the four operations.",{"id":528,"type":53,"title":529,"eyebrow":530,"navLabel":531},"ch6","Olympiad-style problems","Chapter 06","6 Olympiad problems",{"id":533,"type":82,"title":534,"problem":535,"steps":536},"we-gauss","Adding 1 to 100 in seconds","Find 1 + 2 + 3 + … + 100 without adding them one by one.",[537,538,539,540,541],"Write the sum forwards and backwards, one above the other: 1 + 2 + … + 100 and 100 + 99 + … + 1.","Add the two rows column by column: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101… Every column makes **101**.","There are 100 columns, so the two rows together make 100 × 101 = 10,100.","That is **two** copies of the sum, so one copy is 10,100 ÷ 2 = **5,050**.","A famous story says the young Carl Friedrich Gauss did this in Germany in the 1780s when his teacher set the class a long addition. The story may have grown in the telling, but the method is real and works for any list of numbers going up in equal steps.",{"id":543,"type":43,"markdown":544},"consecutive","**Sums of consecutive numbers.** 15 can be written as a sum of two or more consecutive whole numbers in three ways: 7 + 8, 4 + 5 + 6 and 1 + 2 + 3 + 4 + 5. Try some other numbers: 9 = 4 + 5 = 2 + 3 + 4. 10 = 1 + 2 + 3 + 4.\n\nNow try 8. Or 16. Or 32. You will not manage it. A computer search of 1 to 64 shows the only numbers that **cannot** be written this way are 1, 2, 4, 8, 16, 32, 64: exactly the **powers of 2**. Proving why is a lovely olympiad challenge. (A clue: a run with an odd number of terms is that many copies of its middle number, so its total has an odd factor.)",{"id":546,"type":82,"title":547,"problem":548,"steps":549},"we-pages","How many pages?","A printer used **792 digits** to number all the pages of a book, starting from page 1. How many pages does the book have?",[550,551,552,553,554,555],"Pages 1 to 9 use **1 digit** each: 9 pages × 1 = 9 digits.","Pages 10 to 99 use **2 digits** each: 90 pages × 2 = 180 digits. Running total: 9 + 180 = 189.","Digits left for three-digit pages: 792 − 189 = **603**.","Each page from 100 onwards uses 3 digits: 603 ÷ 3 = **201** three-digit pages.","Those pages are 100 to 100 + 201 − 1 = 300. So the book has **300 pages**.","Check by counting: 9 + 180 + 201 × 3 = 9 + 180 + 603 = 792. ✓",{"id":557,"type":105,"itemId":558,"prompt":559,"check":560,"hints":562,"feedback":565},"prac-ones","four-operations.x-count-ones","When you write all the numbers from 1 to 100, how many times do you write the digit **1**?",{"kind":109,"answer":561,"tolerance":111},21,[563,564],"Count 1s in the ones place (1, 11, 21, … 91) and in the tens place (10 to 19) separately.","Don't forget 100.",{"correct":566,"incorrect":567},"Right: 10 in the ones place, 10 in the tens place (10–19) and 1 in 100 makes 21.","Ones place: 1, 11, 21, …, 91 gives 10. Tens place: 10 to 19 gives 10. Plus the 1 in 100. Total 21.",{"id":569,"type":47,"variant":570,"title":571,"markdown":572},"tryit-magic","try_it","Build a magic square","Place the numbers 1 to 9 in a 3 × 3 grid so that every row, column and diagonal adds to the same total.\n\n1. First work out what that total must be. All nine numbers add to 45, and the three rows share them equally, so each row must add to 45 ÷ 3 = **15**.\n2. Which number must go in the centre? (It belongs to four lines at once: the middle row, middle column and both diagonals.)\n3. Try: **2 7 6 \u002F 9 5 1 \u002F 4 3 8**. Check every line.\n\nThen try with 2 to 10, or with 11 to 19. How does the magic total change?",{"id":574,"type":391,"prompt":575,"options":576,"explanation":586},"pred-trick","Think of any number. Multiply it by 3, add 12, divide by 3, then subtract the number you first thought of. What do you get?",[577,579,581,583],{"id":395,"label":578},"It depends on the number you chose",{"id":398,"label":580},"Always 4",{"id":401,"label":582},"Always 12",{"id":584,"label":585},"d","Always 0","**Always 4.** Picture your number as a bag of marbles. × 3 gives three bags; + 12 adds 12 loose marbles. ÷ 3 shares everything into three equal heaps: each heap is one bag plus 4 marbles. Take away the bag (your number) and **4** remain, whatever was in the bag. Every step was undone except the 12 ÷ 3.",{"id":588,"type":53,"title":589,"eyebrow":590,"navLabel":591},"ch7","Projects: real budgets, bills, harvests and run chases","Chapter 07","7 Real projects",{"id":593,"type":43,"markdown":594},"proj-intro","The best way to master the four operations is to **use** them for something that matters. Each project below is a real task: you gather numbers, choose operations, estimate, calculate and present a decision. Work alone, in a pair or as a class.",{"id":596,"type":82,"title":597,"problem":598,"steps":599},"we-trip","Project 1: plan a school trip budget","Class 7 (48 students and 4 teachers) is visiting a science centre. Buses have 40 seats and cost ₹6,800 each for the day. Entry is ₹80 per student (teachers free). Lunch is ₹95 per person. How much should each student pay?",[600,601,602,603,604,605,606,607],"**People:** 48 + 4 = 52.","**Buses:** 52 ÷ 40 = 1 r 12. The 12 extra people still need seats, so **round up** to 2 buses. Cost: 2 × ₹6,800 = ₹13,600.","**Entry:** 48 × ₹80 = ₹3,840.","**Lunch:** 52 × ₹95 = ₹4,940.","**Estimate the total:** about 14,000 + 4,000 + 5,000 = ₹23,000.","**Exact total:** ₹13,600 + ₹3,840 + ₹4,940 = **₹22,380**. Close to the estimate. ✓","**Per student** (students pay for teachers' lunch too): ₹22,380 ÷ 48 = 466 r 12.","Charging ₹466 would leave the class ₹12 short, so **round up to ₹467**. That collects 48 × ₹467 = ₹22,416, leaving ₹36 spare for emergencies.",{"id":609,"type":62,"caption":610,"columns":611,"rows":616},"tab-kirana","Project 2: a kirana bill to check. The rates are illustrative, not current shop prices — make up your own from a real bill.",[612,613,614,615],"Item","Quantity","Rate","Amount",[617,622,627,632,637,642,646,651,655],[618,619,620,621],"Rice","5 kg","₹62","₹310",[623,624,625,626],"Toor dal","2 kg","₹148","₹296",[628,629,630,631],"Sugar","3 kg","₹44","₹132",[633,634,635,636],"Groundnut oil","2 L","₹165","₹330",[638,639,640,641],"Biscuit packets","6 packets","₹30","₹180",[643,644,645,645],"Tea powder","1 packet (250 g)","₹135",[647,648,649,650],"**Total**","6 items","—","**₹1,383**",[652,653,649,654],"Paid with","one note","₹2,000",[656,649,649,657],"Change","₹617",{"id":659,"type":43,"markdown":660},"kirana-task","For the kirana project, visit a local shop with an adult (or use a real bill from home). Record quantities and rates, then calculate each amount and the total **before** you look at the printed bill. In the sample above, the total is ₹1,383, and paying with ₹2,000 gives ₹617 change. Then extend it: what if every price rose by ₹5? What if you bought the same things every week for a year (52 weeks)? The yearly bill would be 52 × ₹1,383 = ₹71,916.",{"id":662,"type":82,"title":663,"problem":664,"steps":665},"we-harvest","Project 3: a farmer's harvest","A farmer grows wheat on 3 hectares. Using made-up but realistic round figures: the field gives about 42 quintals per hectare, she sells at ₹2,300 per quintal, and seeds, fertiliser, water and labour cost ₹95,000 in all. What is her profit?",[666,667,668,669],"**Harvest:** 3 × 42 = 126 quintals (1 quintal = 100 kg, so 12,600 kg).","**Income:** 126 × ₹2,300 = **₹2,89,800**.","**Profit:** ₹2,89,800 − ₹95,000 = **₹1,94,800**.","**Per month:** a year's profit spread over 12 months is ₹1,94,800 ÷ 12 = ₹16,233 r 4, so about **₹16,200** a month. The yield and the rates here are made-up round figures, not real market prices: real farm incomes depend heavily on the monsoon, on prices and on costs, so try the calculation again with a smaller yield, say 30 quintals per hectare.",{"id":671,"type":82,"title":672,"problem":673,"steps":674},"we-cricket","Project 4: the run chase","A T20 team is chasing 172. After 12 overs it has 94 runs. What is its current run rate, and what rate does it now need?",[675,676,677,678,679],"**Current run rate** = runs ÷ overs = 94 ÷ 12 = 7 r 10, which is about **7.83** runs per over.","**Runs still needed:** 172 − 94 = **78**.","**Overs left:** 20 − 12 = 8, which is 8 × 6 = **48 balls**.","**Required run rate** = 78 ÷ 8 = **9.75** runs per over (78 ÷ 8 = 9 r 6, and 6 ÷ 8 = 0.75).","The team must speed up by about 2 runs per over. Commentators work this out ball by ball. Track a real match and make a table of required rate after every over.",{"id":681,"type":244,"component":245,"componentVersion":5,"config":682,"objective":728,"textAlternative":729},"lab-sprint-projects",{"operations":683,"ranges":687,"rounds":693,"secondsTotal":111,"estimateFirst":694,"wordProblems":695},[684,685,248,686],"+","-","÷",{"a":688,"b":691},{"min":689,"max":690},100,999,{"min":692,"max":254},2,20,true,[696,699,703,706,709,712,715,718,721,725],{"prompt":697,"answer":692,"unit":698,"operation":686},"52 people go on a trip. Each bus has 40 seats, and every person needs a seat. How many buses must be hired?","buses",{"prompt":700,"answer":701,"unit":702,"operation":248},"Entry costs ₹80 for each of 48 students. What is the total entry cost?",3840,"₹",{"prompt":704,"answer":705,"unit":702,"operation":248},"Lunch costs ₹95 per person for 52 people. What is the lunch bill?",4940,{"prompt":707,"answer":708,"unit":702,"operation":684},"Bus ₹13,600 + entry ₹3,840 + lunch ₹4,940. What is the total trip cost?",22380,{"prompt":710,"answer":711,"unit":702,"operation":685},"The class collects ₹22,416 and spends ₹22,380. How much is left over?",36,{"prompt":713,"answer":714,"unit":702,"operation":685},"A kirana bill comes to ₹1,383. You pay with ₹2,000. How much change?",617,{"prompt":716,"answer":717,"unit":702,"operation":248},"A family spends ₹1,383 on groceries every week. How much in 52 weeks?",71916,{"prompt":719,"answer":720,"unit":702,"operation":248},"A farmer harvests 126 quintals and sells at ₹2,300 per quintal. What is the income?",289800,{"prompt":722,"answer":723,"unit":724,"operation":686},"A T20 team needs 78 runs from the last 8 overs. What run rate (runs per over) does it need? Decimals are allowed.",9.75,"runs per over",{"prompt":726,"answer":727,"unit":702,"operation":686},"₹22,380 is shared by 60 people in a bigger group. How much does each pay?",373,"Work through the numbers of real projects (trip budget, kirana bill, harvest, run chase): estimate, then calculate exactly.","This untimed 20-round sprint mixes generated calculations (estimate first, then exact) with word problems from the projects, picked at random, so you will meet some of these each time you play:\n\n1. 52 people, 40-seat buses: 52 ÷ 40 = 1 r 12, so hire **2 buses** (round up).\n2. Entry for 48 students at ₹80: **₹3,840**.\n3. Lunch for 52 at ₹95: **₹4,940**.\n4. Trip total: ₹13,600 + ₹3,840 + ₹4,940 = **₹22,380**.\n5. Collected ₹22,416, spent ₹22,380: **₹36** left.\n6. Change from ₹2,000 on a ₹1,383 bill: **₹617**.\n7. ₹1,383 a week for 52 weeks: **₹71,916**.\n8. 126 quintals at ₹2,300: **₹2,89,800**.\n9. 78 runs in 8 overs: 78 ÷ 8 = 9 r 6, and 6 ÷ 8 = 0.75, so the required rate is **9.75** runs per over (9 an over is not quite enough).\n10. ₹22,380 shared by 60 people: **₹373** each.\n\nSeveral of these need a decision about the remainder, not just a calculation.",{"id":731,"type":47,"variant":570,"title":732,"markdown":733},"tryit-rain","Project 5: a monsoon rainfall diary","Put a straight-sided container (a steel tumbler works) in an open place away from roofs and trees. Each morning, measure the depth of water in millimetres with a ruler, write it down and empty the container.\n\nAt the end of a month: **add** to get the monthly total, **subtract** to compare the wettest and driest days, **divide** the total by the number of days to get the average daily rainfall, and **multiply** the average by 30 or 31 to check your total. Compare with the rainfall figures the India Meteorological Department publishes for your district. Is your home wetter or drier than the official station?",{"id":735,"type":105,"itemId":736,"prompt":737,"check":738,"hints":740,"feedback":743},"prac-trip","four-operations.x-trip-share","A class of 36 students spends ₹15,000 on a trip, shared equally. To cover the cost, each student pays a whole number of rupees. What is the smallest amount each should pay?",{"kind":109,"answer":739,"tolerance":111,"unit":702},417,[741,742],"15,000 ÷ 36 = 416 remainder 24.","Would ₹416 each be enough?",{"correct":744,"incorrect":745},"Yes: ₹417. 36 × 416 = ₹14,976, which is ₹24 short, so round up. 36 × 417 = ₹15,012.","15,000 ÷ 36 = 416 r 24. At ₹416 each the class collects ₹14,976, ₹24 short. So each pays ₹417, collecting ₹15,012.",{"id":747,"type":53,"title":748,"eyebrow":749,"navLabel":750},"ch8","Big numbers from the real world","Chapter 08","8 Big real data",{"id":752,"type":43,"markdown":753},"bigdata","India's 2011 Census counted **1,21,08,54,977** people (1,210,854,977 in the international system). Numbers like this come with their own questions, and every one needs the four operations:\n\n- **How many per square kilometre?** Divide population by area. This is *population density*.\n- **How much did it grow?** Subtract one census count from the next, then compare with the earlier count.\n- **How many per polling booth, per school, per doctor?** Divide, then decide whether to round up (you cannot build 0.3 of a school).\n- **Is a headline sensible?** If a news report says a state produced '5 crore tonnes of rice for each person', estimate: that cannot be right. Rounding and estimating protect you from wrong numbers.\n\nThe Go deeper layer works through census and election arithmetic in detail. Here is a project idea: pick one number from a newspaper each day for a week (crop output, railway passengers, cricket attendance, a state budget) and write one sensible calculation with it, with the operation and the answer's unit.",{"id":755,"type":62,"caption":756,"columns":757,"rows":761},"tab-data-ideas","Big-number project ideas and the operation each needs",[758,759,760],"Question","Data you need","Main operation",[762,766,770,774,778,782],[763,764,765],"How many people live in each sq km of your state?","State population and area (Census, state website)","Division, then rounding",[767,768,769],"How many more people live in your district than in 2001?","Two census counts","Subtraction",[771,772,773],"How many litres of water does your school use in a year?","Daily use, school days","Multiplication",[775,776,777],"How many trains would carry a crowd of 1 lakh fans?","Seats per train (e.g. about 1,500 to 2,000)","Division, round up",[779,780,781],"What is the average score of your favourite batter?","Runs and number of times out","Division",[783,784,773],"How much does your family spend on milk in a year?","Daily litres, price per litre",{"id":786,"type":480,"conceptId":787,"relation":482,"explanation":788},"conn-data","data-handling","Averages, totals and differences from real data sets are the four operations put to work; data handling organises and interprets them.",{"id":790,"type":53,"title":791,"eyebrow":792,"navLabel":793},"ch9","Place value, Roman numerals and people who calculate for a living","Chapter 09","9 Wider world",{"id":795,"type":43,"markdown":796},"roman","Try adding **CXXIII + XLVIII** in Roman numerals without converting to our digits. It is awkward: there are no columns, the symbols do not line up, and XL means 'ten before fifty', so the order of symbols changes their value. The answer is **CLXXI** (123 + 48 = 171), but getting there needs a lot of regrouping of symbols. Now try multiplying MCMLXXXVII by XLVII!\n\nThis is why the **Indian place-value system with zero** was such a revolution. With it, the same ten digits line up in columns, and the same short methods work for any size of number: carrying, borrowing, long multiplication and long division all depend on it. For centuries, Roman-numeral users did their actual calculating on counting boards or an abacus and only wrote the answers in Roman numerals.",{"id":798,"type":480,"conceptId":799,"relation":800,"explanation":801},"conn-number","number-system","helps_understand","Every method in this layer, from the lattice to binary, works because of place value: each position is worth a fixed multiple of the one to its right.",{"id":803,"type":804,"title":805,"prompt":806,"options":807},"explorer-careers","explorer","Who uses the four operations at work?","Pick a job to see how its daily work depends on calculating.",[808,821,833,845,856],{"id":809,"label":810,"chain":811,"badge":817,"note":820},"shop","Shopkeeper",[812,813,814,815,816],"Buy stock in bulk","Divide into unit prices","Add up bills","Give change","Total the day's sales",{"text":818,"tone":819},"All four, every hour","yes","A kirana owner buys a 25 kg sack of rice and must work out the cost per kg, add a margin, total customers' bills, subtract to give change, and at night add up the day's sales and compare with yesterday. Estimation keeps mistakes from slipping past at a busy counter.",{"id":822,"label":823,"chain":824,"badge":830,"note":832},"accountant","Accountant",[825,826,827,828,829],"Record income","Record expenses","Subtract for profit","Work out tax","Check totals match",{"text":831,"tone":819},"Checking is the job","Accountants add long columns of income and expenses, subtract to find profit, multiply by tax rates and divide costs between departments. Much of the work is checking: making sure two independently added totals agree, exactly like checking with inverse operations.",{"id":834,"label":835,"chain":836,"badge":842,"note":844},"engineer","Civil engineer",[837,838,839,840,841],"Measure the site","Calculate loads","Multiply by safety factor","Estimate materials","Divide into lorry loads",{"text":843,"tone":819},"Estimate, then calculate","An engineer building a bridge or a school multiplies loads, adds safety margins, and works out how many bags of cement or lorries of sand are needed, always rounding **up** so the work never runs short. A rough estimate first catches errors that could be dangerous.",{"id":846,"label":847,"chain":848,"badge":853,"note":855},"programmer","Programmer",[849,850,851,852],"Break task into steps","Write operations as code","Test with known answers","Fix edge cases",{"text":854,"tone":819},"Teaches machines to calculate","Programmers write the instructions that make computers add, divide and round. They must think about edge cases a calculator user never sees: what happens when you divide by zero, when a remainder is left, or when a number is too big to store.",{"id":857,"label":858,"chain":859,"badge":865,"note":867},"analyst","Data analyst",[860,861,862,863,864],"Collect data","Add and count","Divide for averages","Compare with subtraction","Report the story",{"text":866,"tone":819},"Numbers into decisions","Data analysts in cricket teams, hospitals, banks or government offices turn huge tables of numbers into totals, averages and differences, and then decide what those numbers mean. Choosing the right operation, and knowing when an answer is not sensible, matters more than speed.",{"id":869,"type":53,"title":870,"eyebrow":871,"navLabel":872},"ch10","Open questions and what you found","Chapter 10","10 Open questions",{"id":874,"type":244,"component":875,"componentVersion":5,"config":876,"objective":903,"textAlternative":904},"lab-match-methods","match-pairs",{"prompt":877,"mode":878,"pairs":879},"Match each calculating method or idea to what it does.","connect",[880,883,885,888,891,894,897,900],{"a":881,"b":882},"Lattice (gelosia)","Grid of diagonals; carry only at the end",{"a":140,"b":884},"Rods showing each digit's times table",{"a":886,"b":887},"Russian peasant","Halve one side, double the other, add odd rows",{"a":889,"b":890},"Nikhilam (near a base)","Use how far numbers are from 100",{"a":892,"b":893},"× 11 trick","Put the sum of the two digits in the middle",{"a":895,"b":896},"Kaprekar's routine","Big-first minus small-first until 6174",{"a":898,"b":899},"Gauss's pairing","Pair first and last to add 1 to 100",{"a":901,"b":902},"Abacus","Beads on rods; carrying by moving a bead left","Connect each method from this layer with a short description of how it works.","This matching game has eight pairs. Draw a line from each method to its description.\n\n- **Lattice (gelosia):** a grid with diagonals; all the small multiplications first, all the carrying at the end.\n- **Napier's bones:** rods that each show one digit's times table in split squares (a portable lattice, 1617).\n- **Russian peasant:** halve the left number, double the right, then add the right-hand numbers in rows where the left is odd.\n- **Nikhilam (near a base):** use each number's distance from 100 (or 10 or 1,000): 97 × 96 = 93 | 12 = 9,312.\n- **× 11 trick:** split the two digits and put their sum in the middle: 63 × 11 = 693.\n- **Kaprekar's routine:** arrange digits largest-first and smallest-first and subtract, repeatedly, until 6174.\n- **Gauss's pairing:** 1 + 100, 2 + 99, … each make 101, so 1 to 100 adds to 5,050.\n- **Abacus:** beads on rods, one rod per place value; carrying means moving a bead onto the next rod.",{"id":906,"type":907,"title":908,"terms":909},"glossary-extend","glossary","Words from this layer",[910,914,917,919,923,927,931,935,937,941,945,949,953,957],{"term":911,"meaning":912,"example":913},"lattice multiplication","A written method that fills a grid with digit products, split by diagonals into tens and ones, then adds along the diagonals.","473 × 58 = 27,434 in a 3 × 2 lattice",{"term":915,"meaning":916},"gelosia method","Another name for lattice multiplication, from the Italian word for a criss-cross window screen.",{"term":140,"meaning":918},"A set of rods, one per digit, each carrying that digit's times table in diagonal-split squares; published by John Napier in 1617.",{"term":920,"meaning":921,"example":922},"vertically and crosswise","A one-line method for multiplying two-digit numbers: ones × ones, then the cross products for the tens, then tens × tens.","47 × 36 = 1,692",{"term":924,"meaning":925,"example":926},"nikhilam","A shortcut for numbers near a base such as 100: subtract a shortfall for the left part and multiply the shortfalls for the right part.","97 × 96 = 93 | 12 = 9,312",{"term":928,"meaning":929,"example":930},"doubling and halving","Multiplying by repeatedly halving one number and doubling the other; the basis of Egyptian and Russian peasant multiplication.","37 × 24 = 24 + 96 + 768 = 888",{"term":932,"meaning":933,"example":934},"binary","The base-two place-value system with only the digits 0 and 1; each place is worth twice the place to its right. Computers use it.","37 = 100101 in binary",{"term":422,"meaning":936},"A frame of rods with sliding beads, one rod per place value, used for calculating; the Japanese version is the soroban.",{"term":938,"meaning":939,"example":940},"cryptarithm","A puzzle in which digits in a calculation are hidden or replaced by letters or boxes, and you must work out what they are.","□7 × □3 = 2,491 → 47 × 53",{"term":942,"meaning":943,"example":944},"Kaprekar's constant","The number 6174, which every four-digit number (not all digits the same) reaches by repeatedly subtracting its digits arranged smallest-first from largest-first.","3524 → 3087 → 8352 → 6174",{"term":946,"meaning":947,"example":948},"magic square","A square grid of numbers in which every row, column and diagonal adds to the same total, the magic sum.","1 to 9 in a 3 × 3 grid: magic sum 15",{"term":950,"meaning":951,"example":952},"run rate","In cricket, runs scored divided by overs bowled.","94 runs in 12 overs ≈ 7.83",{"term":954,"meaning":955,"example":956},"required run rate","Runs still needed divided by overs remaining.","78 runs in 8 overs = 9.75",{"term":958,"meaning":959,"example":960},"Roman numerals","An ancient number system using letters (I, V, X, L, C, D, M) without place value or zero, which makes written calculation hard.","CXXIII + XLVIII = CLXXI",{"id":962,"type":47,"variant":498,"title":963,"markdown":964},"open-questions","Open questions for curious learners","Some of these have answers you can find; some are still being argued about. Pick one and dig in.\n\n- Why do all four-digit numbers fall into 6174? What happens with three-digit numbers (hint: try 495)? With five digits?\n- Can you prove that powers of 2 are the only numbers that are not sums of consecutive whole numbers?\n- Is there a multiplication method that needs fewer single-digit multiplications than long multiplication? (Mathematicians found some, starting with Anatoly Karatsuba in 1960, and computers use them for huge numbers.)\n- Should Indian schools teach the abacus, Vedic-style tricks, or neither? What would you gain and lose?\n- If calculators do arithmetic perfectly, which calculating skills will people still need in 50 years?\n- How many different 3 × 3 magic squares can you make with 1 to 9, if rotations and reflections count as the same?",{"id":966,"type":966,"title":967,"questions":968},"quiz","Check yourself",[969,982,995,1008,1021,1034,1046,1059,1072,1085],{"itemId":970,"prompt":971,"options":972,"correct":395,"why":981},"four-operations.x-quiz-lattice","In lattice multiplication, what does each diagonal strip collect?",[973,975,977,979],{"id":395,"label":974},"Digits with the same place value",{"id":398,"label":976},"The digits of the first number",{"id":401,"label":978},"Only the carries",{"id":584,"label":980},"The answer's digits in reverse","Each diagonal gathers pieces of the same place value (ones, tens, hundreds…), so adding along it is like adding one column of long multiplication.",{"itemId":983,"prompt":984,"options":985,"correct":398,"why":994},"four-operations.x-quiz-eleven","What is 86 × 11?",[986,988,990,992],{"id":395,"label":987},"8,146",{"id":398,"label":989},"946",{"id":401,"label":991},"866",{"id":584,"label":993},"1,046","8 | 8 + 6 | 6 = 8 | 14 | 6. Carry the 1 into the hundreds: 946. Check: 86 × 11 = 946.",{"itemId":996,"prompt":997,"options":998,"correct":395,"why":1007},"four-operations.x-quiz-nikhilam","Using the near-100 shortcut, 99 × 95 is:",[999,1001,1003,1005],{"id":395,"label":1000},"9,405",{"id":398,"label":1002},"9,450",{"id":401,"label":1004},"9,045",{"id":584,"label":1006},"9,505","Shortfalls 1 and 5: 99 − 5 = 94, and 1 × 5 = 05. So 9,405. Check: 99 × 95 = 9,405.",{"itemId":1009,"prompt":1010,"options":1011,"correct":398,"why":1020},"four-operations.x-quiz-five","What is 65 × 65?",[1012,1014,1016,1018],{"id":395,"label":1013},"3,625",{"id":398,"label":1015},"4,225",{"id":401,"label":1017},"4,025",{"id":584,"label":1019},"3,925","6 × 7 = 42, then 25: 4,225 = 4,225.",{"itemId":1022,"prompt":1023,"options":1024,"correct":395,"why":1033},"four-operations.x-quiz-peasant","In Russian peasant multiplication, which rows do you add?",[1025,1027,1029,1031],{"id":395,"label":1026},"Rows where the halving number is odd",{"id":398,"label":1028},"Rows where the doubling number is even",{"id":401,"label":1030},"Every row",{"id":584,"label":1032},"Only the first and last rows","The odd rows mark the powers of 2 that make up the halved number (its binary 1s), so their doubled partners add up to the product.",{"itemId":1035,"prompt":1036,"options":1037,"correct":398,"why":1045},"four-operations.x-quiz-binary","What is 1101 in binary, written in our usual numbers?",[1038,1040,1042,1044],{"id":395,"label":1039},"1,101",{"id":398,"label":1041},"13",{"id":401,"label":1043},"11",{"id":584,"label":216},"Places are 8, 4, 2, 1: 8 + 4 + 0 + 1 = 13.",{"itemId":1047,"prompt":1048,"options":1049,"correct":398,"why":1058},"four-operations.x-quiz-eggs","What is the smallest number bigger than 1 that leaves remainder 1 when divided by 2, 3, 4, 5 and 6?",[1050,1052,1054,1056],{"id":395,"label":1051},"31",{"id":398,"label":1053},"61",{"id":401,"label":1055},"121",{"id":584,"label":1057},"301","The number minus 1 must be a multiple of the LCM of 2 to 6, which is 60. So 60 + 1 = 61.",{"itemId":1060,"prompt":1061,"options":1062,"correct":395,"why":1071},"four-operations.x-quiz-pages","How many digits are needed to number the pages of a 120-page book?",[1063,1065,1067,1069],{"id":395,"label":1064},"252",{"id":398,"label":1066},"360",{"id":401,"label":1068},"243",{"id":584,"label":1070},"189","1–9: 9 digits; 10–99: 180 digits; 100–120: 21 pages × 3 = 63. Total 9 + 180 + 63 = 252.",{"itemId":1073,"prompt":1074,"options":1075,"correct":401,"why":1084},"four-operations.x-quiz-kaprekar","Which number does Kaprekar's routine reach for four-digit numbers?",[1076,1078,1080,1082],{"id":395,"label":1077},"1089",{"id":398,"label":1079},"495",{"id":401,"label":1081},"6174",{"id":584,"label":1083},"9999","Every four-digit number whose digits are not all the same reaches 6174 within 7 steps. (For three digits the number is 495.)",{"itemId":1086,"prompt":1087,"options":1088,"correct":398,"why":1095},"four-operations.x-quiz-rate","A team needs 90 runs from the last 10 overs. What is the required run rate?",[1089,1090,1091,1093],{"id":395,"label":33},{"id":398,"label":205},{"id":401,"label":1092},"900",{"id":584,"label":1094},"80","Required run rate = runs needed ÷ overs left = 90 ÷ 10 = 9 runs per over.",{"id":1097,"type":1098,"prompt":1099},"reflect","reflection","Which method from this layer did you enjoy most, and which one would you actually use to multiply 473 × 58 in an exam? Explain why the method you enjoy and the method you trust might be different, and what it would take for you to trust a new method.",{"id":1101,"type":1102,"title":1103,"points":1104},"cheat-sheet","summary","Cheat sheet",[1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117],"**Lattice (gelosia):** write digit products in a grid with tens above and ones below each diagonal; add along diagonals from the bottom-right. 473 × 58 = 27,434.","**Napier's bones (1617):** rods carrying each digit's times table; a portable lattice.","**Vertically and crosswise:** 47 × 36 → ones 7 × 6, tens 4 × 6 + 7 × 3, hundreds 4 × 3, with carries: 1,692.","**Near a base:** 97 × 96 → 97 − 4 = 93 and 3 × 4 = 12 → 9,312. For base 100 the right part has two digits.","**× 11:** split the digits, put their sum in the middle (carry if 10 or more): 78 × 11 = 858.","**Squares ending in 5:** tens × next number, then 25: 85² = 7,225.","**'Vedic Mathematics'** is a 1965 book by Bharati Krishna Tirtha; its sutras are not found in the Vedas. The tricks work because of place value.","**Russian peasant:** halve left (drop halves), double right, add rows where left is odd. 37 × 24 = 24 + 96 + 768 = 888. It is binary in disguise.","**Egg puzzle:** remainder 1 for 2 to 6 → one more than a multiple of 60; also divisible by 7 → 301.","**Kaprekar (6174):** largest-first minus smallest-first, repeated; at most 7 steps for any four-digit number.","**Gauss:** 1 + … + 100 = 100 × 101 ÷ 2 = 5,050. Only powers of 2 are not sums of consecutive numbers.","**Projects:** round up for buses and fees, round down for items you can buy; always estimate before you calculate.","**Place value matters:** Roman numerals have no columns, which is why the Indian place-value system with zero made written methods possible.",{"id":1119,"type":480,"conceptId":1120,"relation":525,"explanation":1121},"conn-prime","prime-and-composite","Remainder puzzles and 'does it divide exactly?' questions lead straight into factors, primes and composite numbers.",{"id":1123,"type":1123,"sourceIds":1124},"sources",[1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140],"four-operations-wiki-lattice-multiplication","four-operations-britannica-arithmetic","four-operations-khan-arithmetic","four-operations-mathsisfun-long-multiplication","four-operations-ncert-math-6","four-operations-wiki-brahmagupta","four-operations-wiki-egyptian-multiplication","four-operations-wiki-rhind-papyrus","four-operations-wiki-karatsuba","four-operations-wiki-vedic-mathematics","four-operations-mactutor-kaprekar","four-operations-wiki-shakuntala-devi","four-operations-wiki-galley-division","four-operations-mactutor-indian-numerals","four-operations-wiki-arithmetic","four-operations-wiki-census-2011",[1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140],"needs_review",{"generatedBy":1144,"notes":1145},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","1b806080a8ee4349e23d1e230ecadb3ce7b7362250e296c5e4b059fc746b4918",{"logic:practice":1148,"component:arith-sprint@1":1149,"component:sort-game@1":1150,"component:match-pairs@1":1151,"source:four-operations-britannica-arithmetic":1152,"source:four-operations-khan-arithmetic":1153,"source:four-operations-mactutor-indian-numerals":1154,"source:four-operations-mactutor-kaprekar":1155,"source:four-operations-mathsisfun-long-multiplication":1156,"source:four-operations-ncert-math-6":1157,"source:four-operations-wiki-arithmetic":1158,"source:four-operations-wiki-brahmagupta":1159,"source:four-operations-wiki-census-2011":1160,"source:four-operations-wiki-egyptian-multiplication":1161,"source:four-operations-wiki-galley-division":1162,"source:four-operations-wiki-karatsuba":1163,"source:four-operations-wiki-lattice-multiplication":1164,"source:four-operations-wiki-rhind-papyrus":1165,"source:four-operations-wiki-shakuntala-devi":1166,"source:four-operations-wiki-vedic-mathematics":1167},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","66e407fe40575b09f96c71af650fbd183b777a5b7047ed8fcc6b693b637804f6","ef6b590392c4299a9ce1be700f25412c2a52dea7c431f6dd28b6d7c1ce15374d","e5cf10b627ae5e1c1d0cb78cbdabf4a54d8dba727ef143b6e7088d5ca2bb4555","144d76bf0a4a3e8b80869a7997eba057680b81b8919f0e4492f8265e18722051","f673458eca6d084a854c7e5fd91ec78c1ed561c54faf5e27538711d81efda7a5","c66e4c408e97e6c5758c1b1160148cfabaf1f5e9ca4caba87383f8ca4e389c95","3fd5b7fcbfec31c612f8356e67126230dae21df1a69f1c1a06216d510015a231","1db8b2f0f76ee0a79d803c4141d87b8198945baeebe2130a015dd376b05aa9b3","8ba18aecc6b48bd05854e49bc36f8655f5b8327b0952c4bfdae46051fb03ed50","dfe770ecef268921a17b49e86a600c5876686e4c7749ba6e50ab03e52577bb33","72e6ef79e0dc90768278cc22338cc136fdc6d8ba028bf31b4966d3910e29ca80","de678a4a6fcb6c73193d9ed8fe9e1046de28d42d399a5eca2bb2a21f582f6771","bb730789f912f54add0c4a57bf935e8be654e892910665839db150297af29dcd","a81515acfb928a7a55299065c455e3b107ec39e2f3e919628634f6bfdd963afd","cca35a4844e81890e1fdb7e42384e4d76c804acd9e1b7eed7fed207834a75002","1890c48aee5faf321185e74b544d758d5e255e4438d374c2764504f469af2fa0",{"state":1169,"reviewer":1170,"selfReview":694,"reviewedAt":1171,"method":1172},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899596872]