NUMBER MAGIC
Eric Lord
for Janet
Twelve articles first published in the magazine hoop-la* in 2008
*http://www.hooplaclub.com/member_schools.htm
Number Magic 1
Fig.1
When your editor asked if I could do something for
hoop-la my first thought was to do some silly
drawings of the kind that made my classmates laugh
when I was at school (Fig.1). Then I decided to
write a maths column instead, because I’m a
mathematician. But it should be interesting for all of
you. That’s not so easy. Some people don’t like
maths − they think it’s hard and boring. If you’re
one of those, be patient with me and I’ll try to show
that maths is really about seeing patterns, and about
how one idea can lead to another (see Fig.1) and
that it can be fun. On the other hand, a lot of Indian
children are ‘wizards’ at maths − better than I am. If
you’re one of those, be patient with me. Some of
what I write you’ll already know, but I hope that
now and then I’ll show you something you’ll find
challenging.
1089: a Special Number
Write 1089 secretly on a piece of paper. Tell a friend to write a three-figure number (the
first and last digits must be different). Then tell them to reverse the digits − turn the
number back to front − and subtract the smaller of the two numbers from the larger. If
the 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 doesn’t matter which 3-digit number you start with. Your
friend 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 out
why this works.
Another curious property of 1089 is that the number got by writing it backwards is a
multiple of it:
9801 = 1089 × 9
There’s 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 until
I run out of ideas! Please write in − especially if you know any interesting number facts
you’d like to share with other readers. Next time, I’ll write about another very interesting
and peculiar number, 142857.
Number Magic 2
142857: a Special Number
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
et cetera. Notice what’s happening. For each multiplication, the answer is the same string
of six digits, in the same order − only the starting place changes (in mathematicians’
jargon, the six digits are ‘cyclically permuted’). This goes on if we multiply by 4, 5 and
6. 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 the
number of digits to six by splitting the answer into groups of six, from the right, and
adding. 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 groups
of 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
= 0.1428571428571428571428…
7
Again, we have the same string of six digits. All these strange properties of 142857 are
related to the fact that this number is a factor of 999999. One idea leads to another. Are
there 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
(a prime number is one that has no factors, so when a number has been split into its prime
factors it can’t be factorised any further). We have seen what happens when 999999 is
divided by its factor 7, so it’s worth dividing it by other factors (not necessarily prime) to
see 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 and
047619 (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 two
strings 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 now
there are three different strings of six digits. And surprises when you multiply by 7 and
by 14. Try it (better use a calculator or you’ll get fed up...). A further surprise comes
when we split 047619 into groups of three, and add. We don’t get 999.
For some mysterious reason, we get the ‘number of the beast’
047 + 619 = 666 (!)
[In the very strange book Revelation (sometimes called the Apocalypse), the final book
of the Christian Bible, it says “Let him that hath understanding count the number of the
beast: for it is the number of a man; and his number is six hundred threescore and six”.
Nobody knows what this means.]
Number Magic 3
The Tower of Brahma
The French mathematician Edouard Lucas invented an interesting puzzle-toy. It consists
of 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 whole
conical tower has been transferred to a different stick. The rule is that no disc must ever
be 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 in
the picture. It’s not difficult to figure out that, if it takes n7 moves to solve the 7- disc
puzzle, it will take n8 = 2 × n7 + 1 moves to solve the 8-disc puzzle. [Before the largest
disc (the eighth) can be moved we need to have the 7 discs above it transferred to another
stick, with one stick free to receive the eighth disc. That takes n7 moves. One more move
transfers the eighth disc. Then n7 more moves are needed to get the 7-disc tower on top
of it.] More generally, if the puzzle with k discs needs nk moves, the puzzle with k + 1
discs will need nk+1 = 2 × nk + 1 moves. We know that n1 = 1, so we can calculate:
n2 = 2 × 1 + 1 = 3, n 3 = 2 × 3 + 1 = 7, n 4 = 2 × 7 + 1 = 15,
n5 = 2 × 15 + 1 = 31, n 6 = 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 haven’t proved this, we’ve only
noticed it’s a reasonable guess. That’s how mathematical discoveries are made − by
making 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 story
about it (which isn’t true!): in Banaras there’s a temple containing a version with three
diamond rods and sixty-four golden discs. Brahma himself put it there when the world
was created. The priests work night day moving the discs. When their task is completed
the world will end!
The number of moves the priests need to complete the task is 264 − 1. This is an
enormous number! Even if the priests work at lightning speed it’s likely that the universe
itself really will end long before they’ve finished!
Number Magic 4
Number Symbols: part 1
All over the world it’s now taken for
granted that any (whole) number can
be written down using just ten
symbols, and that arithmetic can be
done on a piece of paper by methods
everyone learns at school. Different
cultures use different shapes for the
symbols, but the idea is the same.
[Just a few examples: the rows in the
table show the modern ten numerals
we are all familiar with, Arabic numerals, Hindi (Devanagari) numerals, and Kannada
numerals.]
The beauty and simplicity of this way of writing and working with numbers was the
invention of an unknown Indian, probably about 1700 years ago. The methods were
adopted by the Arabs, entered Italy in the thirteenth century, and slowly through the
centuries spread throughout Europe and eventually throughout the whole world. The
brilliance of the idea can only be fully appreciated if we compare it with other ways of
writing 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 thousand
can 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 with
the 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 Roman
numerals! (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 like
this? The answer is that they didn’t. They used an abacus, and used written numbers
only to record the answer. The earliest kind of abacus was just lines drawn in the
sand, 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 word
‘calculate’ 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 ‘IndoArabic’ way of writing numbers comes from having a symbol (0 = ‘zero’) to represent an
empty column on the abacus. Then it’s no longer necessary to use different symbols (eg.
I, X, C, M,...) for units, tens, hundreds, etc. The position of a symbol in the written
number will tell you whether it’s unit, tens, or hundreds, etc. You can then throw your
abacus away − you can calculate with the written numbers in the way it’s done today.
Abaci are still used in China and Japan by shopkeepers. An expert on the abacus can
move the beads very fast, in intricate ways, to add, subtract, multiply and divide, and
even to find square roots and cube roots! The picture shows a Japanese abacus. Each
bead above the horizontal bar represents five and each bead below represents one.
Numbers are represented by sliding the beads into contact with the bar.
Number Magic 5
Number Symbols: part 2
In the previous issue I wrote about the Roman number system and about how it was
gradually replaced by the Indo-Arabic system used today throughout the world. In both
these systems, the number ten plays a very special role; we use a ‘base ten’ or ‘decimal’
system for naming and writing numbers and for doing arithmetic. That’s not because
there’s anything mathematically special about the number ten. It came about because we
happen to have ten fingers, so ‘counting in tens’ comes naturally to us when we give
names to numbers in any language and when we want to write numbers down. When we
write 365 we mean 3 × 102 + 6 × 10 + 5. If human beings had evolved with sixteen
fingers (eight on each hand) they might have been writing ‘365’ to mean the number that
we call ‘eight hundred and sixty-nine’! [because 3 × 162 + 6 × 16 + 5 = 869. ] This is the
‘base sixteen’ or ‘hexadecimal’ system of writing numbers. Computer specialists
sometimes use this system (even though they have only ten fingers like everybody
else...), 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 decimal
system, 12 × 162 + 10 × 16 + 11 = 3243.]
Computers do arithmetic using the ‘base two’ or
‘binary’ system. Their silicon chips are circuits with
switches that are either off or on, which we can
represent by 0 and 1. Any number can be written using
just these two symbols. For example, 10001000001 in
binary means, in our decimal system, 1089 [because
210 + 26 +1 = 1024 + 64 +1 = 1089]. It’s surprising to
realise that, by using the binary system and agreeing
that the various fingers shall mean powers of two as in
the picture, any number up to 1023 can be indicated by
holding up fingers!
The Mayan civilization of Mexico used base
twenty (the ‘vigesimal’ system). Numbers up to
19 were written with dots representing ones and
bars representing fives. They also had a zero
symbol, so any number could be written
unambiguously.
The Babylonian civilization (from about 4300 years ago to 2500 years ago, in what is
now Iraq) used a base sixty number system (the ‘hexagesimal’ system) . They wrote on
clay by making marks with a triangular stick. Numbers were written with two kinds of
marks , representing 1 and 10, and they wrote all the numbers up to 59 by combining
them . For bigger numbers they used the same marks again: the symbol for 1 could also
mean 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 didn’t have a ‘zero’ mark, but they
seemed to manage fairly well. Here, I’ve shown how they’d have written the number that
we’d write as 142857 − expressing it in powers of sixty: 39 × 602 + 40 × 60 + 57.
The division of an hour into sixty minutes and a minute into sixty seconds, and the
division of a circle into 360 degrees, comes down to us all the way from this ancient
Babylonian way of reckoning in sixties.
Number Magic 6
987654321: a Special Number
Write the number 987654321 backwards and subtract. The answer contains just the same
nine 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 of
surprising number patterns:
More Number Patterns
In ‘Number Magic 1’ I mentioned that 1089 reversed is a multiple of 1089 (9801 = 1089
× 9) and I said there was only one other four-figure number with this property. It’s 2178
(8712 = 2178 × 4). Each of these numbers is the first of an infinite sequence of numbers
whose reversal is a multiple of the original number:
Next time, I’ll show you some more of these strange multiplication patterns.
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 won’t try to explain it! I’ll just give you a few more such things for you to think about:
Finally, two oddities:
Number Magic 8
3816547290: a Special Number
Notice that this ten-digit number contains all the digits of the decimal system, in a funny
order. It’s 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 is
exactly divisible (i.e., without a remainder) by a given small number, without actually
doing the division. A well-known trick is to add up all the digits. If the result is divisible
by 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 also
divisible 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 it’s divisible by 2 and also by 3. A number is
divisible by 5 if its final digit is 0 or 5, and by 10 if its final digit is 0. To test whether a
large number is divisible by 11, add alternate digits and subtract. For example, is
987563201 is divisible by 11?
9 + 7 + 6 + 2 + 1 = 25; 8 + 5 + 3 + 0 = 14; 25 − 14 = 11. So the answer is yes, 987563201
is divisible by 11.
Obviously, a number is divisible by 12 if it’s 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 sometimes
take longer than actually going ahead and dividing! But they are interesting curiosities
anyway. 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 divisible
by 7.
For a large number (more than three digits) the work can be cut short by first reducing it
to a three-figure number. Split off the last three digits (think of 3816547 as 3816 and
547). 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?....
Number Magic 9
Fibonacci Numbers
In the year 1202 a book called Liber Abaci (the ‘Book of the Abacus’) was published in
Italy by Leonardo of Pisa, now better known as ‘Fibonacci’ which is short for filius
Bonacci − the son of Bonaccio. The book introduced into Europe, for the first time, the
‘Hindu-Arabic numerals’ that Fibonacci had learned about from Arab mathematicians
during his travels. In Liber Abaci Fibonacci demonstrated that this way of writing
numbers and doing calculations was far superior to the clumsy Roman numbers then in
use. The idea caught on and spread, very slowly and gradually over the next few
centuries, and is now the number system we all use and take for granted.
In Liber Abaci the sequence of numbers now called ‘Fibonacci numbers’ was used as one
of the examples. (Fibonacci didn’t discover it. The sequence is discussed in the Chandras
Shastra, a treatise on rhythms in Sanskrit poetry written by Pingala, perhaps as long ago
as 400BC.) It is very simple and easy to construct, but turns out to be have many quite
fascinating 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 we
call the number in the nth position in the sequence Fn, this rule is
Fn+1 = Fn−1 + Fn
Fibonacci explained the sequence in terms of the increase of a population of rabbits. Start
with a pair of baby rabbits, denoted by B . We suppose that, after a month, they have
become adults: B → A. At the end of the next month we suppose that the adult pair has
produced a new pair of baby rabbits: A→ AB. The rabbits are supposed to be immortal, so
that, 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 Fibonacci
number 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 of
the numbers that come before and after it:
12 = 1 × 2 − 1
22 = 1 × 3 + 1
32 = 2 × 5 − 1
52 = 3 × 8 + 1
82 = 8 × 13 – 1
and so on.... [The general formula Fn2 = F n−1 ×F n+1 − (−1)n ]
(2) The sum of the first n numbers of the sequence is always one less than a
Fibonacci number :
1=2−1
1+1=3−1
1+1+2=5−1
1+1+2+3=8−1
1 + 1 + 2 + 3 + 5 = 13 − 1
1 + 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 two
consecutive Fibonacci numbers:
12 + 12 = 1 × 2
12 + 12 + 22 = 2 × 3
12 + 12 + 22 + 32 = 3 × 5
12 + 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 = 5
22 + 32 = 13
32 + 52 = 34
52 + 82 = 89, etc.
[The general formula is Fn2 + Fn+12 = F2n+1]
(5) The difference of the squares of the two Fibonacci numbers on either side of a term in
the sequence is a Fibonacci number:
22 − 12 = 3
32 − 12 = 8
52 − 22 = 21
82 − 32 = 55,
etc.
[General formula Fn+12 − Fn−12 = F2n]
That’s just a small sample. There’s a journal, the Fibonacci Quarterly, that’s devoted
entirely to new properties of the Fibonacci sequence and its application in various
branches 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 the
arrangement 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-handed
helices, 8 rows going around right-handedly and 13 going lefthandedly − three Fibonacci numbers. In the close-packed
arrangements of florets in the head of a daisy or a sunflower we see three
prominent sets of intersecting spirals, Fn winding one way, and Fn−1 and
Fn+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 from
the centre). Quite big Fibonacci numbers, such as 21, 55 and 89 (or even
55, 89 and 144) are sometimes plainly visible in a well-developed
sunflower head.
Number Magic 10
The Golden Number
A problem dealt with in Euclid’s geometry is the division of a line in ‘extreme and mean
ratio’ (also called the ‘golden section’). What this means is that the line must be cut in
two so that the ratio of the smaller piece to the larger piece is equal to the ratio of the
larger piece to the whole line. If the ratio is 1 : τ (the length of the whole line being 1 +
τ), then
1
τ
=
,
τ
1+τ
and we get τ2 − τ − 1 = 0. The positive root is
τ=
1+ 5
= 1.61803398874989...
2
This number is called the golden number. It is irrational; it can’t be expressed as a
fraction and the decimal expression goes on forever without repeating.
Because τ =
1
, we can keep on replacing τ by
1+τ
1
on the right hand
1+τ
side to get the continued fraction
1
τ=
1
1+
1
1+
1+
1
1+....
Another strange expression comes in a similar way from τ = √(1 + t ):
τ = 1 + 1 + 1 + 1 + ....
The golden ratio has a place in the history of art and architecture, because its use in
design seems to lead to aesthetically pleasing proportions. Luca Pacioli’s book De Divina
Proportione (‘The Divine Proportion’) described its fascinating properties. It was
published in 1509 with illustrations by Leonardo da Vinci.
A golden rectangle is a rectangle whose
sides are in golden ratio. If a square is cut
off from a golden rectangle the remaining
piece is a smaller golden rectangle. The
picture shows what happens if you keep on
doing this ad infinitum.
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 the
two previous numbers. ) A rectangle can be built up
from squares whose sides are Fibonacci numbers.
Put two equal squares together, then place a square
with twice the edge length alongside them, then a
square of edge length 3, and so on. We get
rectangles of ever-increasing size, that get more and
more like a golden rectangle. This tells us that the
fractions
1
2
3
5
8
13
21
1
1
2
3
5
8
13
….
built from terms of the Fibonacci sequence keep on getting closer and closer to the
golden number. (Try it on a calculator and see.) Another connection between the
Fibonacci numbers and the golden number is revealed if we look at the powers of the
golden number:
τ2 = τ + 1
τ3 = 2τ + 1
τ4 = 3τ + 2
τ5 = 5τ + 3
τ6 = 8τ + 5
τ7 = 13τ + 8
and so on....
If we call the nth Fibonacci number Fn the general formula is
τn = Fnτ + Fn−1. A formula for the nth Fibonacci number can be got from this after some
rather tricky algebra (that I don’t want to get into here!) :
Fn =
τ
n
− ( − τ)
−n
5
It’s truly surprising that this expression, involving irrational numbers, gives an integer.
Here, to conclude, are some of the geometrical properties of the golden number τ that so
fascinated Pacioli:
The circumradius of a regular decagon is τ times the length
of its edge.
The lengths of the diagonals of a regular pentagon are τ
times the length of its edge − and these diagonals
(forming the five-pointed star in the picture) intersect
each other in golden ratio.
Because of this property of the regular pentagon, the twelve vertices of a regular
icosahedron (a solid with 20 equilateral triangular faces) are vertices of three golden
rectangles intersecting each other at right angles:
Number Magic 11
Pythagorean Triplets
The white triangle in this figure has a right angle. The
famous theorem of Pythagoras says that a2 + b2 = c2. There
are very many ways to prove it. Euclid’s proof, which I was
taught at school and rapidly forgot, is clumsy and tedious.
The simplest way I know, which you may not have come
across, is just to look at the two large squares in the next
figure. They are obviously the same size. The first is made
up of four copies of the triangle and ‘the square on the
hypotenuse’ (area c2) and the second is made up of four
copies of the triangle and the squares on the other two sides
(areas a2 and b2). So a2 + b2 = c2. QED [which stands for
quod erat demonstrandum (Latin), meaning ‘which was to
be demonstrated’, and is the traditional thing to write at the
end of a proof of a geometrical theorem. It could also stand
for ‘quite easily done’...].
A Pythagorian triplet is a set of three integers (a, b, c) satisfying a2 + b2 = c2. The
simplest 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’ has
been used for thousands of years. The Egyptians used a long rope with knots on it to
mark off lengths in the proportion 3 : 4 : 5, when laying out the plans for pyramids and
temples.
Another Pythagorean triplet is (5, 12, 13) [64 + 144 = 169]. The question arises: are there
systematic methods for finding Pythagorean triplets? Of course, knowing that (3, 4, 5) is
a Pythagorean triplet tells us that (6, 8, 10), (9, 12, 15) etc. will also do the trick, but
that’s not very interesting. It just means that the size of the triangle can be multiplied by
any number without changing its shape . So we only need to pay attention to ‘primitive’
Pythagorean triplets − those for which a and b have no common factor. Euclid himself
had a method that will give all possible primitive Pythagorean triplets. If p and q are any
two 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 know
more, look them up in Wikipedia. That’s what I did...).
One, and only one, of the numbers a and b is divisible by 2. Therefore the area (ab/2) of
the 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 r =
ab
and is always an integer.
a+b+c
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(a − b + 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 apply
Euclid’s method of generating primitive triplets, using for p and q a pair of consecutive
Fibonacci numbers. We get
and so on. Notice that c is always a Fibonacci number. The column of cs is in fact the
sequence of alternate numbers in the Fibonacci sequence (c = Fn with n odd).
By using the Fibonacci sequence the primitive triplet in any row can be calculated from
the triplet in the row above. Suppose (a, b, c = Fn) is a triplet derived from two
consecutive 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 (8−3, 3+4+5, 13)
= (5, 12, 13). Then (21−5, 5+12+13, 34) = (16, 30, 34). Then (55−16, 16+30+34, 89) =
(39, 80, 89). Et cetera!
Number Magic 12
Pascal’s Triangle
The array of numbers shown here is easy to write down. Each row
is got from the row immediately above it by adding pairs of
number s. Apart from the 1s, every number is just the sum of the
two numbers immediately above it. For example, from the third
row 1, 3, 3, 1 we get the fourth row 1, 1 + 3 = 4, 3 + 3 = 6,
3 + 1 = 4, 1. Here are just the first few rows. The pattern can
obviously be continued indefinitely.
The properties of this array of numbers were investigated by the 17th century French
mathematician and philosopher Blaise Pascal, so it’s called ‘Pascal’s triangle’, but the
earliest known discussion of it is in the Chandas Shastra, a treatise on the rhythms of
Sanskrit poetry, written by Pingala more than 2000 years ago.
Adding up the numbers in a horizontal row gives a power of 2:
1+1=2
1 + 2 + 1 = 4 = 22
1 + 3 + 3 + 1 = 8 = 23
1 + 4 + 6 + 4 + 1 = 16 = 24
1 + 5 +10 + 10 + 5 + 1 = 32 = 25
etc. This is just a special case of
x + y = (x + y)
x2 + 2xy + y2 = (x + y)2
x3 + 3x2y + 3xy2 + y3 = (x + y)3
x4 + 4x3y + 6x2y2 + 4xy3 + y 4 = (x + y)4
etc., where x and y are any two numbers. This is the ‘binomial theorem’ and this is why
the numbers in Pascal’s triangle are called ‘binomial coefficients’. In China the binomial
theorem has been known for a very long time. Yanghui wrote about it in 1261, and in
China Pascal’s triangle is still called Yanghui’s triangle.
Now look at the diagonal rows. The first diagonal is all 1s, which is not very interesting.
The second diagonal just gives the ‘natural numbers’ 1 2 3 4 5.... The third diagonal lists
the triangular numbers 1 3 6 10 15 21.... A triangular number is so called because it tells
us the number of things in a triangular array:
The way these patterns can be built up row by row corresponds to the fact that the
triangular numbers are the sums of the natural numbers:
1=1
1+2=3
1+2+3=6
1 + 2 + 3 + 4 = 10, etc.
The squares of the triangular numbers are sums of cubes:
1 3 = 12
13 + 2 3 = 3 2
13 + 23 +33 = 62
13 + 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 = 22
3 + 6 = 9 = 32
6 + 10 = 16 = 42
10 +14 = 25 = 52, etc!
The third diagonal lists the tetrahedral numbers 1 4 10 20 35.... These numbers can be
visualised by thinking about packing spheres:
The way these stacks can be built up layer by layer corresponds to the fact that the
tetrahedral numbers are sums of triangular numbers:
1=1
1+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 the
sum 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 be
indicated by a number n that identifies the row it’s in
and a number r that identifies its position in the row.
(Notice that the 1 right at the top is the ‘zeroth’ row and
the 1 at the far left of a row is in the ‘zeroth’ position).
For example, at n = 7, r = 3 we find 35.
n
= 35.
This is usually written
r
()
()
n
r
is the number of ways of choosing r objects from a set of n objects.
()
These numbers are important in many branches of mathematics. When n
n
would be very troublesome if
and r are large finding the number
r
one had to work through Pascal's triangle row by row to find it. Luckily, there’s an easier
way. ‘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
()
n
r
=
n!
r !( n − r ) !
(Check it out for the some small ns and rs.) For example,
()
6
4
=
6×5×3×2
(4 × 3 × 2) × 2
=
6×5
= 15
2
Fibonacci numbers
The Fibonacci numbers are 1 1 2 3 8 13 21 34 89 144.... (Each number in the
sequence − except the first two − is the sum of the previous two). Surprisingly, this
sequence is hidden in Pascal’s triangle, like this: