Class 7 Mathematics Chapter 6: Number Play

A number can be an answer about a group

Five children of different heights stand in a line. Each one says how many of the others are taller than they are.

Nobody is measuring anything. Each number is an answer to a question about the group, and that makes it a piece of information.

What the numbers come out as

The tallest child can see nobody taller, so says 0. The next tallest says 1. Then 2, then 3, and the shortest says 4.

So the five numbers are 0, 1, 2, 3 and 4, in some order.

Rearrange them and try again

Now let the children shuffle and say their numbers again. The numbers come out in a different order.

But they are still 0, 1, 2, 3 and 4. Not one of them has changed value.

Why that has to happen

Each child has a fixed number of taller children, because heights have not changed. Standing somewhere else does not make anyone taller.

So each child always says the same number, whatever the order. Only the order of the answers moves.

And the total never moves either

Adding them gives 0 and 1 and 2 and 3 and 4, which is 10. Every arrangement gives 10.

A quantity that stays the same however things are rearranged is worth noticing. It is a fact about the group, not about the line.

A rule that does depend on the order

Change the question. Each child now says how many of the children to their LEFT are taller than they are.

Now the arrangement matters. The child at the left end always says 0, whoever it is, because nobody is to their left.

Reading the line back out of the numbers

These numbers carry more information. Given the list of answers you can often rebuild the order the children stood in.

A number can therefore describe an arrangement, not merely count things in it.

The same numbers, two different meanings

So 0, 1, 2, 3, 4 under the first rule say nothing about the order. Under the second rule they say the children stood from shortest to tallest.

The numbers alone are not the information. The rule that produced them is part of it.

Grids: a different kind of play

Take a three-by-three grid and place the numbers 1 to 9 in it, one to a square, so that every row and every column adds to the same total.

Before trying, work out what that total must be.

Working out the row total

The nine numbers add to 45, because 1 and 2 up to 9 make 45.

The three rows between them use every number exactly once, so the three row totals add to 45. Three equal totals adding to 45 are 15 each.

That was found, not guessed

Notice what happened. The total was not discovered by trying arrangements. It followed from the numbers available.

Working out what a total must be, before filling anything in, is usually faster than searching.

Ask for the diagonals too

Now ask that both diagonals add to 15 as well. Four lines then pass through the centre square: the middle row, the middle column and the two diagonals.

That extra demand fixes the centre completely.

Why the centre has to be 5

Those four lines together add to four lots of 15, which is 60.

Between them they cover the centre square four times and each of the other eight squares once. So 60 is the total of all nine squares plus three extra centres: 45 plus three centres. Three centres make 15, so the centre is 5.

What that buys you

One square is now settled before any searching begins. The eight remaining numbers go round it.

Each pair opposite the centre has to add to 10, since a line through the centre totals 15 and the centre takes 5.

Checking a finished grid

Add every row, then every column, then both diagonals. Eight totals, all 15.

If one total is wrong, exactly one line needs changing, and the square where two wrong lines cross is the one to suspect.

A demand that cannot be met

Ask instead that the nine numbers fill a grid where every row adds to 16. Three rows of 16 would total 48, but the numbers only add to 45.

So no arrangement exists at all. Deciding that took one subtraction and no searching.

What carries forward

A number can answer a question about an arrangement, and the rule behind it is part of the information. Some answers do not change when things are rearranged; those are facts about the group.

In a grid, work out what the totals must be before filling anything in. The same arithmetic that finds the answer also proves when there is none.

Scenario:

A teacher lines up five children of different heights for a photograph. She asks each to call out how many of the others are taller. She writes the five answers on a noticeboard. The next day she lines the same five up in a completely different order and asks again. A pupil predicts that the answers will all be different this time. Another pupil says the total at least will change. The teacher then sets a second puzzle on the same noticeboard: fit the numbers 1 to 9 into a three-by-three grid so that every row adds to 16. Several children spend the afternoon trying arrangements.

Analysis:

Both pupils are expecting rearrangement to change an answer that depends only on heights, and the grid puzzle is impossible for a reason that one subtraction reveals.

Activity:

Line up five objects of different heights, apply both rules in turn, and record which set of answers changes when you rearrange them.

Title:

Find the total before you fill the grid

Safety:

No safety concerns. Paper, pencil and cards only.

What You Notice:

The centre comes out as 5 every time, because four lines cross there and that forces it. Rows of 16 would need 48 against the 45 available, so that version has no solution at all, and the subtraction shows it faster than any search.

The five taller-than-me answers are always 0, 1, 2, 3 and 4, so the list of answers never tells you who stood where.

The centre of a three-by-three magic square can only ever be 5, and the proof is a counting argument rather than a search.

Proving a grid impossible can take one subtraction, while searching for the arrangement could take an afternoon.

Work out the total before you fill a square.

If rearranging cannot change it, it is a fact about the group.

Two kinds of whole number

Every whole number is either even or odd. An even number splits into two equal whole halves, and an odd number leaves one over.

That single word for which kind a number is, even or odd, is called its parity.

Parity is a very small piece of information

Knowing a number is 24 tells you a great deal. Knowing only that it is even tells you almost nothing.

Yet that almost nothing is often exactly enough, and this topic is about when.

Adding two evens

Take 6 and 8. Each splits into equal halves, so putting them together leaves nothing over.

6 and 8 make 14, which is even. Two evens always add to an even.

Adding two odds

Take 7 and 9. Each leaves one over. Put them together and the two left-over ones pair up with each other.

7 and 9 make 16, which is even. Two odds always add to an even as well, and that surprises people.

Adding an odd and an even

Take 6 and 9. The 6 leaves nothing over and the 9 leaves one, and that one has nothing to pair with.

6 and 9 make 15, which is odd. An odd and an even always add to an odd.

The three rules together

Even and even make even. Odd and odd make even. Odd and even make odd.

Read it the other way: a sum is odd exactly when one of the two numbers is odd. Pairing up is what the whole thing rests on.

Adding a long list

Now add many numbers. Each odd one contributes a left-over one, and those left-overs pair off with each other.

So the total is odd exactly when the NUMBER OF ODD NUMBERS in the list is odd. The even numbers never matter at all.

That rule in use

Add 1 to 10. Among those there are five odd numbers, and five is odd, so the total must be odd.

Checking, the total is 55, which is odd. We knew the answer was odd before adding anything.

And again

Add 1 to 7. The odd numbers there are 1, 3, 5 and 7, and four of them is an even count, so the total must be even.

The total is 28, which is even. Parity gave the answer; the addition only confirmed it.

Why that is worth having

Parity is cheap to work out. You can tell whether a total will be odd without doing the sum.

So a wrong answer can sometimes be spotted without redoing any arithmetic at all.

Proving something cannot be done

Here is the real power. Can 100 be written as a sum of an odd number of odd numbers?

Every odd count of odds gives an odd total, and 100 is even. So no. Not one case had to be tried.

What that argument did

It did not search. It found a property that every attempt must have and that the target does not.

That is the shape of most impossibility arguments, and parity is the simplest property to use.

A game decided by parity

Fifteen coins lie heads up. A move turns over exactly two of them, any two. Can you ever reach all fifteen tails?

Count the heads. Each move changes that count by two, or leaves it alone when one coin goes each way.

Finishing the argument

So the parity of the head count never changes. It starts at 15, which is odd, and it would have to end at 0, which is even.

No sequence of moves can do it, however long. The parity was the whole proof.

What to look for

When a puzzle asks whether something is reachable, look for a quantity that every move leaves unchanged.

If the start and the target disagree on it, the answer is no. Parity is the first such quantity to try.

What parity cannot tell you

Parity never tells you how large a number is. It cannot distinguish 4 from 400, since both are even.

So it settles impossibility cleanly and almost never settles possibility. Knowing a total is even does not tell you it is reachable.

What carries forward

Parity is which of the two kinds a number is. Two evens or two odds make an even; one of each makes an odd.

A long total is odd exactly when an odd count of its numbers are odd. And a quantity that no move can change is the fastest way to prove something impossible.

Scenario:

A shopkeeper keeps fifteen identical coins in a tray, all lying heads up. A customer offers a wager. He may turn over exactly two coins as often as he likes, in any combination, and if he can leave every coin showing tails the shopkeeper pays him. The shopkeeper accepts at once. Later the same shopkeeper is totalling a raffle sheet. The entries are the numbers 1 to 10 and he writes the total as 54. His assistant says the total cannot possibly be right, without adding anything up. The customer, still turning coins, claims he just needs more time.

Analysis:

Both the wager and the raffle total turn on parity: one quantity that no move can change, and one that can be predicted before adding.

Activity:

Lay out seven coins heads up and try to reach all tails by turning two at a time, then repeat with eight coins and explain the difference.

Title:

Predict the parity, then add

Safety:

No safety concerns. Notebook, pencil and coins only.

What You Notice:

The predictions come out right every time, because only the count of odd numbers matters. Fifteen coins can never reach all tails, since the head parity starts odd and would have to end even; fourteen coins can, because both ends are even.

Two odd numbers always add to an even one, which is the rule people guess wrongly most often.

The parity of a total of 1 to 10 can be known before a single addition, because only the five odd numbers matter.

Fourteen coins heads up CAN be turned to all tails two at a time, while fifteen never can, and the only difference is parity.

Count the odd ones; the evens never matter.

To prove no, find what no move can change.

A sequence that looks back two steps

Most sequences you have met add a fixed amount each time. This one does something else.

Each term is the sum of the two terms before it. That one instruction generates the whole sequence.

Writing it out

Start with 1 and 1. Then 1 and 1 make 2. Then 1 and 2 make 3. Then 2 and 3 make 5.

Carrying on gives 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Every term is built from its two predecessors and nothing else.

Who first wrote them down

These are often called the Virahanka numbers, after the Indian scholar who described them while studying the rhythms of verse.

They were studied in India centuries before they were written about in Europe, and the problem they came from was a counting problem.

The problem they came from

Verse rhythms are built from two kinds of syllable. A short one lasts one beat and a long one lasts two beats.

Ask how many different rhythms last exactly a given number of beats.

Counting the short cases

One beat has one rhythm: a single short syllable. Two beats have two: two shorts, or one long.

Three beats have three: short short short, short long, and long short. Four beats have five.

Why those counts follow the rule

Look at a rhythm of four beats and ask how it begins. It either starts with a short syllable, leaving three beats to fill, or with a long one, leaving two.

So the four-beat count is the three-beat count plus the two-beat count. That is exactly the rule of the sequence, and it is a reason rather than a pattern.

A pattern against a reason

Spotting that 1, 1, 2, 3, 5 continues by adding is noticing a pattern. Seeing that a rhythm must start short or long is a reason.

The reason tells you the rule holds for every case, including ones nobody has counted.

Where the sequence turns up

The same numbers appear in the spiral arrangements of some seed heads and in the branching of some plants.

They do not appear in everything, and a sequence turning up is not an explanation on its own. It is a question worth asking why.

Digits in disguise

Now a different kind of play. A puzzle hides digits behind letters, and the arithmetic has to be reversed to find them.

The same letter always stands for the same digit, and different letters for different digits unless you are told otherwise.

The simplest kind

Take 1A + A1 = 77, where A is a single digit. Reading by place value, 1A is ten and A, and A1 is ten lots of A and one.

Adding gives eleven lots of A, plus eleven. So eleven lots of A plus eleven make 77.

Finishing it

Taking the eleven away leaves eleven lots of A making 66, so A is 6.

Checking, 16 and 61 do make 77. The puzzle was solved by place value, not by trying every digit.

Two letters at once

Now take AB + BA = 132. By the same reading, AB is ten A and B, while BA is ten B and A.

Adding gives eleven lots of A plus B. So eleven times the total of A and B makes 132, which means A and B add to 12.

What that does and does not tell you

It fixes the total of A and B at 12. It does not fix either digit on its own.

A could be 3 with B as 9, or 4 with 8, or 5 with 7, and several more. The puzzle has seven answers, not one.

Knowing how many answers there are

That is worth saying out loud. A puzzle can pin down a total without pinning down the parts.

Asking how many answers a puzzle has is as much a part of solving it as finding one.

A puzzle with no answer at all

Try AB + BA = 100. The same reading gives eleven times the total of A and B making 100.

But 100 is not a multiple of eleven: eleven nines are 99 and eleven tens are 110. So no digits work, and no searching was needed.

The useful habit

In every one of these, the first move was the same. Rewrite the hidden numbers by place value and see what the arithmetic forces.

That turns a guessing game into a calculation, and it is what tells you whether there is one answer, several, or none.

What carries forward

A sequence can be defined by looking back more than one step, and the Virahanka numbers come from counting verse rhythms. A reason for a rule is worth more than a pattern, because it covers cases nobody counted.

In a digit puzzle, rewrite by place value first. The arithmetic then tells you how many answers exist, which may be one, several or none.

Scenario:

A school poster about patterns shows the numbers 1, 1, 2, 3, 5, 8 and asks what comes next. One pupil writes 13 and explains that the differences are growing. Another writes 13 and explains that a verse rhythm must begin with either a short or a long syllable. The teacher says only one of them has given a reason. The same teacher then shows a faded till receipt where two two-digit numbers were added. The digits are hidden but the second number is the first with its digits swapped, and the total reads 132. A pupil says the two numbers must have been 39 and 93.

Analysis:

Both pupils reach 13, but only one has a reason that covers uncounted cases; and the receipt fixes a total without fixing the digits.

Activity:

Write out the first twelve terms of the sequence, then make up a digit puzzle of your own and count how many answers it has.

Title:

Count the rhythms, then crack the digits

Safety:

No safety concerns. Notebook, pencil and ruler only.

What You Notice:

Three beats give three rhythms and four give five, and the four-beat list splits neatly into those starting S and those starting L. The first puzzle has one answer and the second has seven, which is the point: the place-value step tells you how many there are.

The Virahanka numbers were described in India centuries before they were written about in Europe, and they came from a counting problem about verse.

The puzzle AB + BA = 132 has seven answers, so a faded receipt showing that total cannot be read back uniquely.

A sum of the form AB + BA is always eleven times the total of the two digits, which is why only multiples of eleven are reachable.

Place value first, then count the answers.

A pattern records; a reason explains.