Class 6 Mathematics Chapter 3: Number Play

Meera lines up her friends

Meera plays a game in the school ground. Five friends stand in a row, shoulder to shoulder. Each one looks left and right, then calls out a single number. Meera writes down what she hears: 1, 1, 0, 1, 1. Her cousin sees the list and cannot make head or tail of it. The list looks like nonsense. It is not. Those five numbers describe the whole row.

The rule they are using

Every child counts the neighbours who are taller than they are. A child with a taller person on one side only calls out 1. A child with taller people on both sides calls out 2. A child who is taller than everyone beside them calls out 0. So the number is not the height. It is a count of taller neighbours.

Why the two ends are different

A child in the middle has a neighbour on the left and a neighbour on the right. A child at the end of the row has only one neighbour. That single fact decides a lot. An end child can call 0 or 1 and nothing else. Calling 2 would mean two taller neighbours, and there is nobody there to be the second one.

The tallest child always says nothing but zero

Somebody in the row is taller than everybody else. Nobody beside that child is taller. So the tallest child always calls out 0. This works however the children shuffle themselves. It also means a row can never be all ones. Wherever the tallest child stands, a 0 has to appear in the list.

Can every child call zero?

Try it and you will fail. Calling 0 means being taller than both your neighbours. Now think about the shortest child in the row. Somebody standing beside that child is taller, because everybody is. So the shortest child has to call 1 or 2, never 0. A row of all zeros is impossible.

Neighbours can call the same number

Meera's list begins 1, 1. That surprises people. It is easy to arrange. Put a short child, then a medium child, then a tall child. The first child sees a taller neighbour and says 1. The second child sees one shorter and one taller, so says 1 as well. Two neighbours saying the same thing is perfectly ordinary.

Reading the list backwards

The real fun starts when you are given the numbers and asked about the row. A 2 in the list marks a dip: that child is shorter than both neighbours. A 0 marks a peak. Peaks and dips cannot sit side by side, because a child cannot be taller and shorter than the same neighbour. So the list carries the shape of the row.

A table of numbers that compete

Now leave the ground and open your notebook. Write seven numbers in a row of boxes: 412, 806, 375, 940, 268, 719, 153. Colour a box when its number beats every number touching it. A coloured box is called a supercell. Three of these seven qualify. Find them before reading on.

Checking each box in turn

806 sits between 412 and 375, and beats both, so it is a supercell. 940 sits between 375 and 268, and beats both. 719 sits between 268 and 153, and beats both. The other four all lose to a neighbour. So the supercells here are 806, 940 and 719.

The end boxes count their own way

A box at the end of a row touches only one other box. So it has to beat one number, not two. That makes it easier for an end box to win, not harder. In our row 412 still loses, because 806 sits next to it. Swap 412 for 999 and the end box would win.

Two supercells never touch

Suppose two boxes sit side by side and both are coloured. The left one must beat the right one. The right one must beat the left one. Two numbers cannot each be bigger than the other. So that can never happen, and the coloured boxes always have a gap between them.

So how many can a row hold?

Colour one box, skip one, colour the next, and so on. That is the most you can manage. A row of seven boxes holds at most four supercells. A row of eight also holds four. A row of nine holds five. Work it out by filling a row with big and small numbers turn by turn.

Numbers have places, not just sizes

A number line does the same job in a different way. It turns size into position. The further right a number sits, the bigger it is. Two numbers you cannot tell apart in a list are obvious on a line, because one of them stands further along than the other.

One step is worth whatever the line says

Read the labels before you read the marks. On one line each step may be worth 200, so the marks run 3000, 3200, 3400 and onwards. On another each step may be worth 2, so the marks run 45670, 45672, 45674. The picture looks the same. Only the labels tell you the difference.

Finding a mark nobody has labelled

Very often a line labels two marks and leaves the rest blank. Count the gaps between the two labels. Share the difference out equally among them. That gives the worth of one step, and from there every blank mark can be filled in, going left as easily as right.

Scenario:

A class is given two figures for a town: a population of 48,000 and a bus fare of 15 rupees. Meena is asked which number carries more information. She says the bigger one. Their teacher asks what each number would tell a visitor. The population says how many people live there, and that shapes how many schools and buses the town needs. The fare says what one journey costs. The class decides that neither is more informative, because they answer different questions. Meena rewrites her answer. A number means nothing on its own, and what it is counting is as much a part of it as its size.

Analysis:

Meena ranked the numbers by size. What a number counts, and the question it answers, decide what it can tell you.

Activity:

Find two numbers about your town. Write what question each one answers and who would need it.

Title:

Line up and call out

What You Notice:

Every list holds at least one zero, and the ends never say 2. In the boxes, no two coloured cells ever end up touching, and four is the most you can squeeze out of seven boxes.

A single row of numbers can hide a picture. The numbers 0, 2, 0, 2, 0 describe a row that goes tall, short, tall, short, tall.

Computers sort huge lists by asking the same small question again and again: is this item bigger than the one beside it?

Mapmakers mark hilltops and valley floors in much the same way, by comparing a place with the land touching it.

What a child at the end of a row can say — One neighbour, one chance — An end child compares with a single person. So the answer is 0 or 1. There is no second neighbour to make it 2.

Spotting supercells quickly — Beat both sides, or the one side you have — Count the boxes touching yours first. Then check your number against each of them. Nothing else matters.

Arjun counts the numbers

Arjun writes numbers in his book, starting at 1. He wonders how many numbers there are of each size. One-digit numbers run from 1 to 9, so there are nine of them. Two-digit numbers run from 10 to 99, and there are ninety. Three-digit numbers run from 100 to 999, and there are nine hundred. Each step up multiplies the count by ten. Five-digit numbers come to ninety thousand.

Adding the digits instead of using them

Here is a different thing to do with a number. Ignore what it is worth and just add its digits. The digits of 89 add to 17. The digits of 764 also add to 17. So do the digits of 98,000. These three numbers are miles apart in size, yet they share one digit sum.

Hunting for the smallest and the largest

Suppose you want a digit sum of 17 and the smallest number that gives it. Use as few digits as you can, and make the last digit as large as possible. That gives 89. Now ask for the largest five-digit number with the same digit sum. Load the front digits and leave the rest at zero, which gives 98,000.

A pattern in consecutive digits

Try three-digit numbers whose digits run on, like 234 or 567. Add the digits of 234 and you get 9. Add the digits of 567 and you get 18. Each answer is three times the middle digit. That is no accident. One digit is one less than the middle and one is one more, so the two changes cancel out.

Counting a digit rather than a number

Rekha writes out all the numbers from 1 to 100 and asks how many times she has written the digit 4. In the units place a 4 appears ten times, at 4, 14, 24 and so on. In the tens place it appears ten more times, all through the forties. So she has written it twenty times.

Numbers that read the same either way

Some numbers are unchanged when you read them backwards. 66 is one. So are 848 and 1111. These are called palindromes. Checking one is easy: pair the first digit with the last, the second with the second last, and so on. Every pair must match. In a number with an odd count of digits, the middle digit has no partner and can be anything.

How many palindromes are there?

Count the three-digit ones. The first digit may be any of 1 to 9, and the last is forced to copy it. The middle digit is free and may be any of 0 to 9. So there are ninety. Five-digit palindromes work the same way, with two free middle digits, and there are nine hundred of them.

Reverse and add

Now a game. Take a number, reverse it, and add the two together. If the answer is a palindrome, stop. If not, do the same thing again. Start with 23 and one round is enough: 23 plus 32 gives 55. Start with 57 and it takes two rounds, reaching 132 and then 363.

Some numbers keep you waiting

Start at 68 and you need three rounds, passing through 154 and 605 before landing on 1111. Start at 89 and the patience required is astonishing. It takes twenty-four rounds, and the palindrome at the end has thirteen digits. Nobody could guess that from looking at 89.

What nobody has settled

Every two-digit starting number does reach a palindrome in the end. Three-digit numbers are a different story, and this one is not finished. Mathematicians do not know whether every three-digit start reaches a palindrome. The number 196 is the famous doubtful case. Computers have run it for a very long time without a palindrome appearing, and people suspect it never will. Suspecting is not the same as knowing.

A teacher from Devlali

D.R. Kaprekar earned his living teaching mathematics at a state school in the town of Devlali, in Maharashtra. He spent his spare hours playing with numbers and found patterns nobody had noticed before. In 1949 he came across something very strange indeed while fiddling with four-digit numbers.

His routine, step by step

Pick a four-digit number with at least two different digits. Rearrange its digits to make the largest number you can, and call that A. Rearrange them again to make the smallest, and call that B. Work out A minus B. Then start over with the answer, and keep going.

Watching 2059 settle down

The digits of 2059 give a largest of 9520 and a smallest of 0259, which is just 259. Taking 259 from 9520 leaves 9261. Round two turns 9261 into 8352. Round three turns 8352 into 6174. Try a fourth round and nothing moves: the largest is 7641, the smallest is 1467, and the difference is 6174 all over again.

Why the routine has to stop somewhere

That number 6174 is called the Kaprekar constant. Every four-digit start with two different digits reaches it, and never in more than seven rounds. The warning about two different digits matters. A number like 5555 gives the same largest and smallest, so the difference is zero and there is nothing left to work with.

The same idea with three digits

Three-digit numbers have a constant of their own. Begin at 325 and the routine gives 297, then 693, then 594, then 495. From there it repeats 495 for ever. So the three-digit routine settles as well, just at a different resting place.

Scenario:

A class notices a number plate reading 2059 and is asked to make the largest and smallest numbers from those digits. The largest comes quickly: 9520. The smallest causes an argument. Arjun writes 0259 and Nita says that is not a four-digit number at all. The class tests it by reading it aloud, and it says two hundred and fifty-nine. So the zero cannot lead. The smallest four-digit arrangement is 2059, the plate itself. Arjun sees why. A leading zero does not change the value, so it quietly turns a four-digit number into a three-digit one.

Analysis:

The argument is about what a digit in front actually does. A leading zero adds a symbol and no value, so the number shrinks a place.

Activity:

Take a four-digit number plate with a zero in it. Write the largest and smallest four-digit numbers you can make.

Title:

A digit workshop at home

What You Notice:

Your birth year always reaches 6174, and so does everybody else's, though the number of rounds differs from person to person. The reverse-and-add game can finish at once or drag on for a surprising number of rounds.

Kaprekar did all of his exploring while holding down a full timetable at a state-run school in the town of Devlali.

No four-digit number with two different digits needs more than seven rounds of the routine before it reaches 6174.

Palindromes turn up outside mathematics too. Malayalam is a language whose name reads the same both ways.

Testing a palindrome — Outside in, pair by pair — Compare the outermost digits, then step inwards. One mismatch anywhere ends the test. A lone middle digit never needs checking.

The order of Kaprekar's routine — Biggest minus smallest, then go again — Arrange the digits high to make A, arrange them low to make B, and subtract. Feed the answer straight back in.

A clock with something to say

Nisha glances at the wall clock and sees 5:55. A minute later the pattern is gone. She starts watching for readings that look special. Some repeat a digit, like 11:11. Others read the same backwards, like 3:03 or 12:21. A clock face is only a pair of numbers, and yet it throws up patterns all day long.

Testing a time for a palindrome

Ignore the colon and look at the digits alone. The reading 3:03 gives 303, which reads the same either way. The reading 12:21 gives 1221, which also passes. The reading 4:15 gives 415, and backwards that is 514, so it fails. The test is exactly the one you use on a plain number.

How many such times are there?

Take an hour with a single digit, say 3. The reading is 3 then two minute digits, so the last digit has to be a 3 as well. The first minute digit may be 0, 1, 2, 3, 4 or 5. That gives six times in every such hour, and there are nine of those hours. The hours 10, 11 and 12 contribute one each. The total is fifty-seven.

Dates can do it too

A date written as day, month and year makes an eight-digit string. The date 21/02/2012 gives 21022012, and that reads the same from either end. So does 10/02/2001. Such dates are rare, so people notice them and remember them.

Will a calendar ever come round again?

Jaspreet wonders why a new calendar is printed every year. The answer is that the days slide. A year of 365 days is fifty-two weeks and one day over, so every date shifts on by one weekday. A leap year shifts it by two. A calendar can be used again only by a year that starts on the same weekday and is of the same kind, leap or common.

Building the biggest and the smallest

Give yourself four digits, say 8, 1, 5 and 3. Sort them downwards for the largest number, 8531. Sort them upwards for the smallest, 1358. Their difference is 7173 and their sum is 9889. Change the digits you start from and both answers change, which makes a good puzzle for a friend.

Sums you can do without writing

A large addition is often easier in the head than on paper, provided you break it up sensibly. Suppose you have the pieces 30,000, 9,000, 1,200 and 200 to build with. You may take any piece as often as you like. To reach 40,600, take 30,000 and 9,000 and 1,200, then take 200 twice.

Why repeating a piece is allowed

Children often assume each piece is used once and then give up. Nothing in the rules says that. Reaching 40,600 needs the 200 twice. Once you see that a piece can come back, many more totals fall into place, and a short set of pieces goes a long way.

Taking away as well as adding

Now allow subtraction too. Some totals are much easier that way round. To reach 29,700 you could hunt for pieces that add up to it, or you could start from 30,000, take away 500 and put 200 back. Working down from above is often the quicker route.

How many digits does the answer have?

A five-digit number added to another five-digit number sometimes gives five digits and sometimes six. Try 10,000 and 10,000, which gives 20,000. Now try 99,999 twice, which gives 199,998. So the answer is sometimes five digits and sometimes six, and the only way to know is to look.

Some answers are simply out of reach

Other cases can be ruled out at once. A four-digit number plus a two-digit number can never have six digits. The largest you can manage is 9,999 plus 99, which is 10,098, and that has five. In the same way, a five-digit number take away a two-digit number never drops to three digits.

Examples worth keeping

It helps to have one clean example of each kind in your notebook. Adding 54,321 and 67,890 gives 122,211, which is two five-digit numbers making six digits. Taking 40,905 from 41,208 leaves 303, which is two five-digit numbers making three digits. Writing your own examples is better practice than reading mine.

Adding a figure the quick way

A pattern of numbers can be added one at a time, but that is slow and mistakes creep in. Look for equal numbers instead and count them. Four rows of six twenty-fives is twenty-four twenty-fives, and 24 x 25 = 600. One multiplication has replaced twenty-three additions.

Splitting a figure into groups

Some patterns hold two different numbers. Deal with each group on its own and add at the end. Twelve fifteens come to 180, because 12 x 15 = 180. Eight thirty-fives come to 280. Adding those two totals gives 460. The order in which you group them does not change the answer, so choose whichever grouping you find easiest to see.

Scenario:

Fatima helps her teacher set the bell times for the school day in Bhopal. The first period begins at 9:00 in the morning. Each period runs for forty minutes and there is no gap between them. Fatima writes out the starting time of every period in a list. Partway down she notices that one of the times reads the same backwards. Her teacher then asks her a second question. The school wants to reuse the same printed timetable card next year. Fatima says that the times will be fine but the dates and weekdays printed beside them will not, because a year does not divide evenly into whole weeks. She works out both answers in her notebook.

Analysis:

Number patterns and calendar arithmetic both turn up in an ordinary school day, and both can be settled by careful counting rather than by guesswork.

Activity:

Write out the start times of six lessons in your own school and mark any that read the same backwards.

A common year runs one day past fifty-two whole weeks, which is why your birthday lands on a new weekday each year.

Palindromic dates cluster in certain years and are missing altogether from others, depending on how the digits of the year fall.

Testing a clock reading — Drop the colon, read it back — The colon is punctuation and carries no value. What is left is a plain number, and the usual palindrome test applies.

Building from pieces — Any piece, any number of times — Nothing limits you to one go with each piece. Remembering this is what unlocks totals that look impossible at first.

A rule anyone can follow

Here is a rule for making a run of numbers. Start anywhere you like. An even number gets cut in half. An odd number gets trebled, with one more added on. Then do the same to the answer. Nothing in that rule is hard. A child can follow it, and so can a computer.

Where the rule takes you

Start at 15. It is odd, so trebling and adding one gives 46. That is even, so halving gives 23. Carry on and the run climbs to 106, falls away, climbs again, and after seventeen steps arrives at 1. Start at 7 instead and it takes sixteen steps. Start at 32 and the whole run is a slide: 32, 16, 8, 4, 2, 1.

Powers of two fall straight down

A power of two is the easiest start of all. Halve it and you get another power of two. There is never an odd number to send you climbing again. So the run drops step by step and lands on 1 without a single detour.

Some runs wander a very long way

Not every start behaves so nicely. Begin at 27 and the run rises as high as 9,232 before it comes down. It takes a hundred and eleven steps to reach 1. Nothing about the number 27 warns you of that. Two starts that look alike can behave completely differently.

The part nobody has settled

In 1937 the German mathematician Lothar Collatz suggested that every whole number, followed by this rule, ends at 1. Mathematicians have worked on it ever since. Computers have checked an enormous stretch of numbers and every single one has reached 1. Even so, nobody has proved it for all numbers. It is still open, and among the best known questions of its kind anywhere.

Checking is not proving

The gap here is worth understanding. A computer can test the first many billions of starting numbers. That leaves infinitely many untested, because the whole numbers never run out. So a search can support a belief and can never finish the job. The honest sentence is that the conjecture is unproved.

When an exact number is not needed

Now put the puzzles aside for something practical. Your headteacher knows exactly how many children are on the roll. You do not, and you do not need to. Is it about two hundred, or about a thousand? That is an estimate, and for most purposes it is all anyone wants.

How an estimate is built

Maryam counts her own section: 38 children. The other two sections have 41 and 36. That comes to 115, so she calls her whole class about 120. Her school has five classes of a similar size. Five lots of 120 gives about 600. She has not counted a single child beyond her own section.

Rounding on purpose

Notice that Maryam rounded 115 up to 120 deliberately. An estimate is meant to be easy to carry in the head and easy to multiply. Carrying 115 through the next step would have given a false air of accuracy. An estimate that looks exact is worse than one that is plainly rough.

Estimating things you cannot count

Some quantities cannot be counted directly at all. How many breaths do you take in a day? Count them for one minute and suppose you get twenty. An hour has sixty minutes, and 20 x 60 = 1200. A day has twenty-four hours, and 1200 x 24 = 28800. The answer is rough, and it is far better than a shrug.

A game with a hidden rule

Now a game for two players. Each of you in turn puts 1, 2 or 3 on to a running total that begins at nothing. Whoever says 21 wins. Play it a few times and you will begin to feel that the first player has an advantage. There is a reason, and it is worth working out.

Working backwards from the win

Whoever says 21 wins, so 21 is a safe number. What about 17? From 17 your opponent can only reach 18, 19 or 20, and from any of those you can say 21. So 17 is safe too. Keep going down in fours: 13, 9, 5 and 1 are all safe. The first player says 1 and never lets go.

Why the gap is four

The largest jump allowed is 3. So whatever your opponent adds, you can add something that brings the pair of moves to exactly 4. Your rival adds 1 and you add 3. Your rival adds 3 and you add 1. Either way you move four along and stay on the safe list.

Changing the game changes the winner

Try a version where players add anything from 1 to 10 and the winner says 99. Now the gap is eleven, and the safe numbers are 11, 22, 33 and onwards to 99. The first player cannot reach 11 in one move, so the second player takes control. Here the second player wins with correct play.

Making the rule your own

The pattern is easy to state. Add one to the largest jump allowed, and that is the gap between safe numbers. If the target sits exactly on that gap, the second player wins. If it does not, the first player wins by opening on the remainder. Invent your own target and jump size and test it on a friend.

Scenario:

The tank on the roof of a school in Nagpur runs dry by the middle of the afternoon, and the headmaster asks the Class 6 children to work out why. Nobody can measure the tank, so the children estimate instead. Ritu counts the pupils at one water point during a single break and finds that about forty fill a bottle. There are three such points and three breaks in the day. A bottle holds about one litre. The children also allow roughly two hundred litres for the washrooms and the kitchen together. They write their working on the board in stages, rounding at each stage, and mark every figure with the word about so that nobody mistakes it for a measurement.

Analysis:

An estimate built in clear stages can answer a practical question that no available instrument can settle, provided each rounding is deliberate and openly recorded.

Activity:

Estimate how many litres your own household uses in a day by counting one bucket, then scaling up.

Title:

Invent a game and crack it

What You Notice:

The first five rounds look like luck and the last five do not. Whether the first or the second player wins is settled before anyone moves, by whether the target lands exactly on the gap.

Lothar Collatz put his question forward in 1937, and it has resisted every attempt since.

The run that begins at 27 climbs to 9,232 before it turns, and needs 111 steps to reach 1.

The rule is often called the 3n plus 1 problem, after the step that every odd number has to take.

Applying the Collatz rule — Even halve, odd climb — An even number is cut in two. An odd number is trebled and given one more. Those are the only two moves there are.

Finding a winning strategy — Start at the finish and walk back — Mark the target as safe, then step back by one more than the largest jump, again and again, until you reach the beginning.