Number Magic 9982120

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NUMBER MAGIC

Eric Lord

for Janet

Twelve articles first published in the magazine hoop-la* in 2008

*http://www.hooplaclub.com/member_schools.htm

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Fig.1

Number Magic 1

When your editor asked if I could do something forhoop-la my first thought was to do some sillydrawings of the kind that made my classmates laughwhen I was at school (Fig.1). Then I decided to

write a maths column instead, because Im amathematician. But it should be interesting for all ofyou. Thats not so easy. Some people dont likemaths they think its hard and boring. If youreone of those, be patient with me and Ill try to showthat maths is really aboutseeing patterns, and abouthow one idea can lead to another (see Fig.1) andthat it can be fun. On the other hand, a lot of Indianchildren are wizards at maths better than I am. Ifyoure one of those, be patient with me. Some ofwhat I write youll already know, but I hope thatnow and then Ill show you something youll findchallenging.

1089: a Special Number

Write 1089 secretly on a piece of paper. Tell a friend to write a three-figure number (thefirst and last digits must be different). Then tell them to reverse the digits turn thenumber back to front and subtract the smaller of the two numbers from the larger. Ifthe result is a two-digit number, a 0 should be inserted on the left to make it three digits.Now reverse this number, and add.

Example: 925 529 = 396, 396 + 693 = 1089.

The answer is always 1089. It doesnt matter which 3-digit number you start with. Yourfriend will be surprised when you reveal the piece of paper with your prediction on it.

Those maths wizards among you might like to use a bit of algebra to try to figure outwhy this works.

Another curious property of 1089 is that the number got by writing it backwards is amultiple of it:

9801 = 1089 9

Theres only one other 4-figure number with this property. Can you find it?

If hoop-la readers like Number Magic this can become a regular column at least untilI run out of ideas! Please write in especially if you know any interesting number factsyoud like to share with other readers. Next time, Ill write about another very interesting

and peculiar number, 142857.

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Number Magic 2


142857 1 = 142857142857 2 = 285714

142857 3 = 428571

et cetera. Notice whats happening. For each multiplication, the answer is thesame stringof six digits, in thesame order only the starting place changes (in mathematiciansjargon, the six digits are cyclically permuted). This goes on if we multiply by 4, 5 and6. A surprise comes when we multiply by 7:

142857 7 = 999999.

The pattern continues when we multiply by numbers larger than 7, so long as we keep thenumber of digits to six by splitting the answer into groups of six, from the right, andadding. For example,

142857 123 = 17571411, 17 + 571411 = 571428,142857 142857 = 20408122449, 20408 + 122449 = 142857.

Another curious property is revealed if we split the number into groups of two or groupsof three:

14 + 28 + 57 = 99, 42 + 85 + 71 = 99,142 + 857 = 428 + 571 = 285 + 714 = 999.

One seventh, expressed as a decimal, is recurring. In fact,

1

7= 0.1428571428571428571428

Again, we have the same string of six digits. All these strange properties of 142857 arerelated to the fact that this number is a factor of 999999. One idea leads to another. Arethere any other factors of 999999 that will do the same tricks? The answer is not quite 142957 is unique. But interesting, more intricate patterns emerge when we investigate.The prime factors of 999999 are 3, 7, 11, 13 and 37:

999999 = 33 7 11 13 37

(aprime number is one that has no factors, so when a number has been split into its primefactors it cant be factorised any further). We have seen what happens when 999999 isdivided by its factor 7, so its worth dividing it by other factors (not necessarily prime) tosee what will happen. The interesting cases are

999999 = 13 76923 and 999999 = 21 47619

We can try the trick that worked for 1452857 with the six-digit numbers 076923 and047619 (the zero has been put in to make the sixth digit). Multiply 076923 by 1, 2, 3, etc.,up to 13. Try it! Again we get cyclic permutations of six digits, but there are now twostrings of six digits at work, instead of only one. We also find that

07 + 69 + 23 = 99 and 078 + 923 = 999, etc.

Multiplying 047619 by 1, 2, 3, etc., up to 21, we again get cyclic permutations, but nowthere are three different strings of six digits. And surprises when you multiply by 7 andby 14. Try it (better use a calculator or youll get fed up...). A further surprise comeswhen we split 047619 into groups of three, and add. We dont get 999.

For some mysterious reason, we get the number of the beast

047 + 619 = 666 (!)

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[In the very strange book Revelation (sometimes called the Apocalypse), the final bookof the Christian Bible, it says Let him that hath understanding count the number of thebeast: for it is the number of a man; and his number is six hundred threescore and six.Nobody knows what this means.]

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Number Magic 3

The Tower of Brahma

The French mathematician Edouard Lucas invented an interesting puzzle-toy. It consistsof three sticks and a set of discs all of different sizes. The picture shows a version with

eight discs.

The problem is to transfer the discs one at a time to a different stick, until the wholeconical tower has been transferred to a different stick. The rule is that no disc must everbe placed on top of a smaller disc.

An interesting bit of mathematics comes from the question: for a given number of discs,how many moves will it need to transfer the whole tower? Consider the 8-disc version inthe picture. Its not difficult to figure out that, if it takes n7 moves to solve the 7- discpuzzle, it will take n8 = 2 n7 + 1 moves to solve the 8-disc puzzle. [Before the largestdisc (the eighth) can be moved we need to have the 7 discs above it transferred to anotherstick, with one stick free to receive the eighth disc. That takes n7moves. One more move

transfers the eighth disc. Then n7 more moves are neededto get the 7-disc tower on topof it.] More generally, if the puzzle with kdiscs needs nkmoves, the puzzle with k+ 1discs will need nk+1 = 2 nk + 1 moves. We know that n1 = 1, so we can calculate:

n2 = 2 1 + 1 = 3, n3 = 2 3 + 1 = 7, n4 = 2 7 + 1 = 15,n5 = 2 15 + 1 = 31, n6 = 2 31 + 1 = 63, and so on...

Now a pattern is emerging. Notice that each answer is a power of 2, minus 1:

3 = 22 1,7 = 23 1, 15 = 24 1,31 = 25 1, 63 = 26 1, etc.

So we can guess that nk= 2k 1. (Notice that we havent proved this, weve onlynoticed its a reasonable guess. Thats how mathematical discoveries are made bymaking a reasonable guess and then trying to prove that it must be so.)

The 8-disc puzzle in the picture needs 28 1 = 255 moves.

The inventor of the puzzle called it the Tower of Brahma and made up a little storyabout it (which isnt true!): in Banaras theres a temple containing a version with threediamond rods and sixty-four golden discs. Brahma himself put it there when the worldwas created. The priests work night day moving the discs. When their task is completedthe world will end!

The number of moves the priests need to complete the task is 264 1. This is anenormous number! Even if the priests work at lightning speed its likely that the universe

itself really will end long before theyve finished!

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Number Magic 4

Number Symbols: part 1

All over the world its now taken forgranted that any (whole) number can

be written down using just tensymbols, and that arithmetic can bedone on a piece of paper by methodseveryone learns at school. Differentcultures use different shapes for thesymbols, but the idea is the same.[Just a few examples: the rows in thetable show the modern ten numerals

we are all familiar with, Arabic numerals, Hindi (Devanagari) numerals, and Kannadanumerals.]

The beauty and simplicity of this way of writing and working with numbers was theinvention of an unknown Indian, probably about 1700 years ago. The methods wereadopted by the Arabs, entered Italy in the thirteenth century, and slowly through thecenturies spread throughout Europe and eventually throughout the whole world. Thebrilliance of the idea can only be fully appreciated if we compare it with other ways ofwriting numbers.

The Romans represented numbers by letters of the alphabet. I = 1, V = 5, X = 10, L = 50,C = 100, D = 500, M = 1000. They also used IV = 4 (one less than five), IX = 9, XL =40, XC = 90, CD = 400, CM = 900. Using this system, any number up to a few thousandcan be written down. For example, MMCDXCVII means 2497 [2000 + 400 + 90 + 5 +2]. Roman numerals continued in use throughout Europe for centuries, side by side withthe modern Indo-Arabic numerals. Even today the Roman system is occasionally seen:at the end of every movie the year the movie was made appears on the screen in Romannumerals! (2008 is MMVIII).

The answers to 1089 + 410 or 222 3 are obvious, but MLXXXIX + CDX and CCXXII III look terrifying! So how did the Romans do arithmetic with numbers written likethis? The answer is that they didnt. They used an abacus, and used written numbersonly to record the answer. The earliest kind of abacus was just lines drawn in thesand, or grooves in a board, to represent the unit, tens, hundreds, thousands (etc.)columns, and arithmetic was done with little stones on these lines The wordcalculate comes from the Latin word calculus, meaning a little stone! Later abacuses(or abaci) had wires with beads on them. The enormous advantage of the modern Indo-Arabic way of writing numbers comes from having a symbol (0 = zero) to represent anempty column on the abacus. Then its no longer necessary to use differentsymbols (eg.I, X, C, M,...) for units, tens, hundreds, etc. Theposition of a symbol in the writtennumber will tell you whether its unit, tens, or hundreds, etc. You can then throw yourabacus away you can calculate with the written numbers in the way its done today.

Abaci are still used in China and Japan by shopkeepers. An expert on the abacus canmove the beads very fast, in intricate ways, to add, subtract, multiply and divide, andeven to find square roots and cube roots! The picture shows a Japanese abacus. Eachbead above the horizontal bar represents five and each bead below represents one.Numbers are represented by sliding the beads into contact with the bar.

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Number Magic 5

Number Symbols: part 2

In the previous issue I wrote about the Roman number system and about how it wasgradually replaced by the Indo-Arabic system used today throughout the world. In both

these systems, the numberten plays a very special role; we use a base ten or decimalsystem for naming and writing numbers and for doing arithmetic. Thats not becausetheres anything mathematically special about the number ten. It came about because wehappen to have ten fingers, so counting in tens comes naturally to us when we givenames to numbers in any language and when we want to write numbers down. When wewrite 365 we mean 3 102 + 6 10 + 5. If human beings had evolved with sixteenfingers (eight on each hand) they might have been writing 365 to mean the number thatwe call eight hundred and sixty-nine! [because 3 162 + 6 16 + 5 = 869. ] This is thebase sixteen or hexadecimal system of writing numbers. Computer specialistssometimes use this system (even though they have only ten fingers like everybodyelse...), with the extra symbols A, B, C, D, E and F to mean ten, eleven, twelve, thirteen,fourteen and fifteen. [Example: CAB in hexadecimal means, in our familiar decimalsystem, 12 162 + 10 16 + 11 = 3243.]

Computers do arithmetic using the base two orbinary system. Their silicon chips are circuits withswitches that are eitherofforon, which we canrepresent by 0 and 1. Any number can be written usingjust these two symbols. For example, 10001000001 inbinary means, in our decimal system, 1089 [because210 + 26 +1 = 1024 + 64 +1 = 1089]. Its surprising torealise that, by using the binary system and agreeingthat the various fingers shall mean powers of two as inthe picture, any number up to 1023 can be indicated byholding up fingers!

The Mayan civilization of Mexico used base

twenty (the vigesimal system). Numbers up to19 were written with dots representing ones andbars representing fives. They also had a zerosymbol, so any number could be writtenunambiguously.

The Babylonian civilization (from about 4300 years ago to 2500 years ago, in what isnow Iraq) used a base sixty number system (the hexagesimal system) . They wrote onclay by making marks with a triangular stick. Numbers were written with two kinds ofmarks , representing 1 and 10, and they wrote all the numbers up to 59 by combiningthem . For bigger numbers they used the same marks again: the symbol for 1 could alsomean 60 or 602 or 603 etc, and the symbol for 10 could also mean 10 60 or 10 602,etc. The system could be ambiguous because they didnt have a zero mark, but theyseemed to manage fairly well. Here, Ive shown how theyd have written the number thatwed write as 142857 expressing it in powers of sixty: 39 60

2 + 40 60 + 57.

The division of an hour into sixty minutes and a minute into sixty seconds, and thedivision of a circle into 360 degrees, comes down to us all the way from this ancientBabylonian way of reckoning in sixties.

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Number Magic 6


Write the number 987654321 backwards and subtract. The answer contains just the samenine digits, in a mixed-up order:

987654321 123456789 = 864197532.

Now try adding instead of subtracting:

987654321 + 123456789 = 1111111110

Multiplying by 9, we get 987654321 9 = 8888888889. This result belongs to a set ofsurprising number patterns:

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More Number Patterns

In Number Magic 1 I mentioned that 1089 reversedis a multiple of 1089 (9801 = 1089 9) and I said there was only one other four-figure number with this property. Its 2178(8712 = 2178 4). Each of these numbers is the first of an infinite sequence of numberswhose reversal is a multiple of the original number:

Next time, Ill show you some more of these strange multiplication patterns.

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Number Magic 7

Strange Number Patterns

Some of the things that numbers do are really weird! Look at this:

and so on, all the way up to

I wont try to explain it! Ill just give you a few more such things for you to think about:

Finally, two oddities:

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Number Magic 8


Notice that this ten-digit number contains all the digits of the decimal system, in a funnyorder. Its the only ten-digit number that does the following clever trick:

3 is exactly divisible by 1 (trivial, but true nonetheless!),38 is exactly divisible by 2,381 is exactly divisible by 3,3816 is exactly divisible by 4,

and so it continues (check it out!) until finally (and this time obviously),

3816547290 is exactly divisible by 10.

Divisibility Tests

You probably already know a few of the tricks for finding out whether a large number isexactly divisible (i.e., without a remainder) by a given small number, without actuallydoingthe division. A well-known trick is to add up all the digits. If the result is divisibleby 3, the number you started with is divisible by 3. Otherwise it is not. For example,

3 + 8 + 1 + 6 + 5 + 4 = 27, 2 + 7 = 9, which is a multiple of 3, so 381654 is divisible by 3.The same test works for divisibility by 9, so we see straight away that 381654 is alsodivisible by 9.

A number is divisible by 2 if its final digit is a multiple of 2;A number is divisible by 4 if the number made by its final two digits is a multiple of 4;A number is divisible by 8 if the number made by its final three digits is a multiple of 8.

For example, 38165472 is divisible by 8 because 472 is.

Obviously, a number is divisible by 6 if its divisible by 2 and also by 3. A number isdivisible by 5 if its final digit is 0 or 5, and by 10 if its final digit is 0. To test whether alarge number is divisible by 11, add alternate digits andsubtract. For example, is987563201 is divisible by 11?

9 + 7 + 6 + 2 + 1 = 25; 8 + 5 + 3 + 0 = 14; 25 14 = 11. So the answer is yes, 987563201is divisible by 11.

Obviously, a number is divisible by 12 if its divisible by 3 and also by 4.

So here we have simple tests for divisibility by any number up to twelve, except seven.Tests for divisibility by seven are a bit trickier, but they do exist. They can sometimestake longer than actually going ahead and dividing! But they are interesting curiositiesanyway. Here is one way: split off the final digit (eg., think of 3816547 as 381654 and 7).Do a subtraction with twice the final digit and the rest of the number:

381654 2 7 = 381640; 3816 2 4 = 3808; 380 2 8 = 364;36 2 4 = 28; and, finally, 2 8 2 = 14; 2 4 1 = 7. Therefore 3816547 is divisibleby 7.

For a large number (more than three digits) the work can be cut short by first reducing itto a three-figure number. Split off the last three digits (think of 3816547 as 3816 and547). Then keep subtracting:

3816 547 = 3269; 269 3 = 266.

Now we need only to test whether 266 is divisible by 7. Applying the first test gives

26 2 6 = 14; 2 4 1 = 7,which tells us that 3816547 is divisible by 7.

So here are tests for divisibility of any large number by any given number up to 12.What about 13?....

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Number Magic 9

Fibonacci Numbers

In the year 1202 a book calledLiber Abaci (the Book of the Abacus) was published inItaly by Leonardo of Pisa, now better known as Fibonacci which is short forfilius

Bonacci the son of Bonaccio. The book introduced into Europe, for the first time, theHindu-Arabic numerals that Fibonacci had learned about from Arab mathematiciansduring his travels. InLiber Abaci Fibonacci demonstrated that this way of writingnumbers and doing calculations was far superior to the clumsy Roman numbers then inuse. The idea caught on and spread, very slowly and gradually over the next fewcenturies, and is now the number system we all use and take for granted.

InLiber Abaci the sequence of numbers now called Fibonacci numbers was used as oneof the examples. (Fibonacci didnt discover it. The sequence is discussed in the ChandrasShastra, a treatise on rhythms in Sanskrit poetry written by Pingala, perhaps as long agoas 400BC.) It is very simple and easy to construct, but turns out to be have many quitefascinating properties. It is just

1 1 2 3 5 8 13 21 34 55 89 144 ...

where each number (after the first two) is just the sum of the two previous numbers. If wecall the number in the nth position in the sequenceFn, this rule is

Fn+1 = Fn1 + Fn

Fibonacci explained the sequence in terms of the increase of a population of rabbits. Startwith a pair of baby rabbits, denoted byB . We suppose that, after a month, they havebecome adults:B A. At the end of the next month we suppose that the adult pair hasproduced a new pair of baby rabbits:AAB. The rabbits are supposed to be immortal, sothat, month by month, it goes like this:

B A AB ABA ABAAB ABAABABA ABAABABAABAAB

and so on. The population grows month by month in accordance with the Fibonaccinumber sequence.

Here are a few of the strange properties of the sequence .

(1) The square of a Fibonacci number is always one less or one more than the product ofthe numbers that come before and after it:

12 = 1 2 122 = 1 3 + 132 = 2 5 152 = 3 8 + 182 = 8 13 1

and so on.... [The general formulaFn2 =Fn1Fn+1 (1)

n ]

(2) The sum of the first n numbers of the sequence is always one less than aFibonacci number :

1 = 2 11 + 1 = 3 1

1 + 1 + 2 = 5 11 + 1 + 2 + 3 = 8 1

1 + 1 + 2 + 3 + 5 = 13 11 + 1 + 2 + 3 + 5 + 8 = 21 1

et cetera

(3) The sum of the squares of the first n numbers of the sequence is the product of twoconsecutive Fibonacci numbers:

12 + 12 = 1 212 + 12 + 22 = 2 3

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12 + 12 + 22 + 32 = 3 512 + 12 + 22 + 32 + 52 = 5 8

12 + 12 + 22 + 32 + 52 + 132 = 8 13

(4) The sum of the squares of two consecutive Fibonacci numbers is a Fibonacci number.For example:

12 + 22 = 522 + 32 = 1332 + 52 = 3452 + 82 = 89, etc.

[The general formula isFn2 +Fn+1

2 =F2n+1]

(5) The difference of the squares of the two Fibonacci numbers on either side of a term inthe sequence is a Fibonacci number:

22 12 = 332 12 = 852 22 = 2182 32 = 55,

etc.

[General formulaFn+12Fn1

2 =F2n]

Thats just a small sample. Theres a journal, theFibonacci Quarterly, thats devotedentirely to new properties of the Fibonacci sequence and its application in variousbranches of mathematics and science. It has been published regularly since 1963.

Fibonacci numbers in Nature

Fibonacci numbers can be seen in the structure of many plants, in thearrangement of leaves on a stem or in the structure of blossoms and fruits.Next time you look at a pineapple, notice how the (roughly hexagonal)units spiral around it. There are 5 rows going around in left-handedhelices, 8 rows going around right-handedly and 13 going left-handedly three Fibonacci numbers. In the close-packedarrangements of florets in the head of a daisy or a sunflower we see threeprominent sets of intersecting spirals,Fn winding one way, andFn1 andFn+1 winding the other way. Try counting the spirals in the pictures below(but beware the numbers can change as the spirals get further out fromthe centre). Quite big Fibonacci numbers, such as 21, 55 and 89 (or even55, 89 and 144) are sometimes plainly visible in a well-developedsunflower head.

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1

= 11+

11+

11+

1+....

1 1 1 1 ....= + + + +

Number Magic 10

The Golden Number

A problem dealt with in Euclids geometry is the division of a line in extreme and meanratio (also called the golden section). What this means is that the line must be cut intwo so that the ratio of the smaller piece to the larger piece is equal to the ratio of thelarger piece to the whole line. If the ratio is 1 : (the length of the whole line being 1 +), then

1

1+= ,

and we get 2 1 = 0. The positive root is

1+ 5 =

2

= 1.61803398874989...

This number is called thegolden number. It is irrational; it cant be expressed as afraction and the decimal expression goes on forever without repeating.

Because =1

1+

, we can keep on replacing by1

1+

on the right hand

side to get the continued fraction

Another strange expression comes in a similar way from = (1 + t ):

The golden ratio has a place in the history of art and architecture, because its use indesign seems to lead to aesthetically pleasing proportions. Luca Paciolis bookDe Divina

Proportione (The Divine Proportion) described its fascinating properties. It waspublished in 1509 with illustrations by Leonardo da Vinci.

A golden rectangle is a rectangle whosesides are in golden ratio. If a square is cutoff from a golden rectangle the remainingpiece is a smaller golden rectangle. Thepicture shows what happens if you keep ondoing this ad infinitum.

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The golden number is intimately related to the Fibonacci numbers

1 1 2 3 5 8 13 21 34 55 89 144 ....

(After the first two 1s each number is the sum of thetwo previous numbers. ) A rectangle can be built upfrom squares whose sides are Fibonacci numbers.Put two equal squares together, then place a squarewith twice the edge length alongside them, then asquare of edge length 3, and so on. We getrectangles of ever-increasing size, that get more andmore like a golden rectangle. This tells us that thefractions

1

1

2

1

3

2

5

3

8

5

13

8

21

13

.

built from terms of the Fibonacci sequence keep on getting closer and closer to thegolden number. (Try it on a calculator and see.) Another connection between theFibonacci numbers and the golden number is revealed if we look at the powers of the

golden number:

2 = + 13 = 2 + 14 = 3 + 25 = 5 + 36 = 8 + 57 = 13 + 8

and so on....

If we call the nth Fibonacci numberFn the general formula isn =Fn +Fn1. A formula for the nth Fibonacci number can be got from this after some

rather tricky algebra (that I dont want to get into here!) :

( )

5

n n

Fn

=

Its truly surprising that this expression, involving irrational numbers, gives an integer.

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Here, to conclude, are some of the geometrical properties of the golden number that sofascinated Pacioli:

The circumradius of a regular decagon is times the lengthof its edge.

The lengths of the diagonals of a regular pentagon are times the length of its edge and these diagonals(forming the five-pointed star in the picture) intersecteach other in golden ratio.

Because of this property of the regular pentagon, the twelve vertices of a regularicosahedron (a solid with 20 equilateral triangular faces) are vertices of three goldenrectangles intersecting each other at right angles:

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Number Magic 11

Pythagorean Triplets

The white triangle in this figure has a right angle. Thefamous theorem of Pythagoras says that a2 + b2 = c2. There

are very many ways to prove it. Euclids proof, which I wastaught at school and rapidly forgot, is clumsy and tedious.The simplest way I know, which you may not have comeacross, is just to look at the two large squares in the nextfigure. They are obviously the same size. The first is madeup of four copies of the triangle and the square on thehypotenuse (area c2) and the second is made up of fourcopies of the triangle and the squares on the other two sides(areas a2 and b2). So a2 + b2 = c2. QED [which stands forquod erat demonstrandum (Latin), meaning which was tobe demonstrated, and is the traditional thing to write at theend of a proof of a geometrical theorem. It could also standfor quite easily done...].

A Pythagorian triplet is a set of three integers (a, b, c) satisfying a2 + b2 = c2. Thesimplest and most well known example is (3, 4, 5) [9 + 16 = 25]. Being able to make

accurate right angles is a skill needed by architects and builders. The 3, 4, 5 method hasbeen used for thousands of years. The Egyptians used a long rope with knots on it tomark off lengths in the proportion 3 : 4 : 5, when laying out the plans for pyramids andtemples.

Another Pythagorean triplet is (5, 12, 13) [64 + 144 = 169]. The question arises: are theresystematic methods for finding Pythagorean triplets? Of course, knowing that (3, 4, 5) isa Pythagorean triplet tells us that (6, 8, 10), (9, 12, 15) etc. will also do the trick, butthats not very interesting. It just means that the size of the triangle can be multiplied byany number without changing its shape . So we only need to pay attention to primitivePythagorean triplets those for which a and b have no common factor. Euclid himselfhad a method that will give all possible primitive Pythagorean triplets. Ifp and q are anytwo integers with no common factor, one odd and one even, and q >p, then

a = q2 p2 , b = 2pq, c = p2 + q2

is a primitive Pythagorean triplet. The method gives all the primitive Pythagorean triplets.There are an infinite number of them. Here are the first few:

The triplets have many curious properties. Here are a few examples. (If you want to knowmore, look them up in Wikipedia. Thats what I did...).

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One, and only one, of the numbers a and b is divisible by 2. Therefore the area (ab/2) ofthe right angle triangle is an integer.

One, and only one, of the numbers a and b is divisible by 3.One, and only one, of the numbers a and b is divisible by 4.One, and only one, of the numbers a, b and c is divisible by 5.At most one of the numbers a and b is a square.For every right angle triangle with integer sides a, b and c the radius of its

inscribed circle is abra b c

=+ +

and is always an integer.

The radii of its three excircles are also integers.

If (a, b, c) is a primitive Pythagorean triplet, then so are

(s a, s b,s + c) with s = 2(a + b + c),(s + a,s b,s + c) with s = 2(a + b + c) , and(s a,s + b,s + c) with s = 2(ab + c).

Try it! For example: from (3, 4, 5) we get

s = 24, (21, 20, 29), 212

+ 202

= 292

,s = 12, (15, 8, 17), 152 + 82 = 172,s = 8, (5, 12, 13), 52 + 122 = 132.

Applying the processes again to each of these three triples, we get nine triples (119, 120,169), (77, 36, 85), (39, 80, 89); (65, 72, 97), (35, 12, 37), (33, 56, 65); (55, 48, 73), (45,28, 53), (7, 24, 25). In the next generation there will be twenty-seven triplets. And so on...

Pythagorean Triplets and Fibonacci Numbers

Pythagorean triplets can be constructed from the Fibonacci numbers

1 1 2 3 5 8 13 21 34 55 89 144 233 ...

(The sum of any two consecutive terms gives the next term in the sequence.) Let us applyEuclids method of generating primitive triplets, using forp and q a pair of consecutiveFibonacci numbers. We get

and so on. Notice that c is always a Fibonacci number. The column ofcs is in fact thesequence ofalternate numbers in the Fibonacci sequence (c =Fn with n odd).

By using the Fibonacci sequence the primitive triplet in any row can be calculated fromthe triplet in the row above. Suppose (a, b, c =Fn) is a triplet derived from twoconsecutive Fibonacci numbers by the above method. Then the next triplet will be(Fn+1 a , a + b + c,Fn+2). For example, starting from (3, 4, 5) we get (83, 3+4+5, 13)= (5, 12, 13). Then (215, 5+12+13, 34) = (16, 30, 34). Then (5516, 16+30+34, 89) =(39, 80, 89).Et cetera!

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13 = 1213 + 23 = 32

13 + 23 +33 = 6213 + 23 +33 + 43 = 102

13 + 23 +33 + 43 +53 = 152, etc.

Look what happens when consecutive pairs of triangular numbers are added:

1 + 3 = 4 = 223 + 6 = 9 = 32

6 + 10 = 16 = 4210 +14 = 25 = 52, etc!

The third diagonal lists the tetrahedral numbers 1 4 10 20 35.... These numbers can bevisualised by thinking about packing spheres:

The way these stacks can be built up layer by layer corresponds to the fact that thetetrahedral numbers are sums of triangular numbers:

1 = 11 + 3 = 4

1 + 3 + 6 = 10,1 + 3 + 6 + 10 = 20, etc.

These properties of Pascal's triangle are very general. Any number in the triangle is thesum of all the numbers in a diagonal row next to it, and above it. For example 35 = 1 + 3+ 6 + 10 + 15 and also 35 = 1 + 4 + 10 + 20.

The location of any number in the triangle can beindicated by a numbern that identifies the row its inand a numberrthat identifies its position in the row.(Notice that the 1 right at the top is the zeroth row andthe 1 at the far left of a row is in the zeroth position).For example, at n = 7, r= 3 we find 35.

This is usually written ( )nr = 35.

( )nr is the number of ways of choosing robjects from a set ofn objects.These numbers are important in many branches of mathematics. When n

and rare large finding the number ( )nr would be very troublesome ifone had to work through Pascal's triangle row by row to find it. Luckily, theres an easierway. Factorial n (written as n!) is the product of all the integers up to (and including) n.For example 5! = 5 4 3 2 1 = 120. The formula for working out binomial

coefficients is

( ) !!( ) !

nn

r r n r=

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(Check it out for the some small ns and rs.) For example,

( ) 6 5 3 2 6 56 154 (4 3 2) 2 2

= = =

Fibonacci numbers

The Fibonacci numbers are 1 1 2 3 8 13 21 34 89 144.... (Each number in thesequence except the first two is the sum of the previous two). Surprisingly, thissequence is hidden in Pascals triangle, like this:

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Number Magic 9982120