© Excerpted from Adam Spencer’s Book of Numbers
1
1 is called the multiplicative identity.
There’s a 1st for everything.
2
Two is the only even prime number.
of course, 2 is the basis of binary system.
3
Three is a Fibonacci number.
Our spatial world has 3 dimensions.
Christianity also has as its Big Three the Father, Son and Holy Spirit.
4
It is the first square number.
Four is the 1st composite number.
5
In western and eastern cultures 5 is often the number of love, being an indivisible combination of the masculine 3 and the feminine 2.
6
Six is a perfect number. \( 1 + 2 + 3 = 6 \)
The 6-sided figure, the hexagon, is an important building principle in nature: beehives and snowflakes are constructed of hexagons.
7
Seven is a Mersenne prime number.
\( 7 = 2^3 - 1 \)
8
Eight is a Fibonacci number.
\( 8 = 2^3 \)
9
\( 9 = 3^2 \)
10
Our number system is base 10.
11
\( 1^2 = 1, 11^2 = 121, 111^2 = 12321, 111111111^2 = 12345678987654321 \)
12
Twelve is the 1st abundant number.
13
Thirteen is a Fibonacci number.
14
\( 10^2 + 11^2 + 12^2 = 13^2 + 14^2 \)
15
Fifteen is the 5th triangular number.
\( T_5 = 15 \)
16
\( 16 = 2^4 = 4^2 \)
17
\( 17 = 2^{2^2} + 1 \) is the 2nd Fermat number.
18
Eighteen is the 6th Lucas number.
Eighteen is also an abundant number.
19
Nineteen is a prime number.
20
In case you hadn’t noticed, fingers and toes add up to 20, and so 20 was the basis for counting in many cultures.
Twenty is an abundant number.
21
Twenty-one is the 8th Fibonacci number. It is also the 6th triangular number. \( T_6 = 21 \)
22
The square of 22 is a palindrome: \( 22^2 = 484 \)
23
The strange-looking 23! is 23 digits long.
24
\( 24 = 4 \times 3 \times 2 \times1 = 4! \)
25
Twenty-five is a lucky number.
Twenty-five is also a perfect square: \( 5^2 = 25 \)
26
There are 26 characters in the alphabet used in English, which is called the modern Roman alphabet. There used to be only 25, until J came along in the 14th century.
27
\( 27 = 3^3 \)
28
Twenty-eight is the 2nd perfect number after 6 and 7th triangular number.
Twenty-eight is also the 12th Ulam unmber.
29
Seven cuts through a pizza can create up to 29 pieces.
30
Thirty is a primorial. A primorial is where you multiply a prime number by all the prime numbers less than itself.
\( 5 \times 3 \times 2 = 30 \)
31
Because \( 31 = 2^5 -1 \), it is the 3rd Mersenne prime.
32
\( 32 = 2^5 \)
33
\( 33 = 1! + 2! + 3! + 4! \)
Thirty-three is the 10th lucky number.
34
In a 4 x 4 magic square, the number 1 - 16 occur once each and all the rows, columns and diagonals add up to 34.
35
A polyomino with 6 squares is called a hexomino. A fun way to spend an afternoon is to try and find all 35 hexominoes.
36
If you add all the numbers between and including 1 and 36, which are all the numbers on a roulette wheel, you get 666, a popular number with the devil.
Thirty-six is the 1st number that can be written as the sum of 3 separete cubes: \( 36 = 1^3 + 2^3 + 3^3 \).
Thirty-six is what’s known as a highly composite number. 36 has divisors 1, 2, 3, 4, 6, 9, 12, 18 and 36. No number from 1 to 35 has 9 divisors.
Thirty-six is the 8th triangular number, and an abundant number.
And finally, \( 36 = 6^2 \)
37
Thirty-seven is the 4th centred hexagonal number.
If 37 divides a number abc, then it divides the number bca. For example, 37 divides into 259 \( 37 \times 7 = 259 \), so it divides into 925 \( 37 \times 25 = 925 \)
38
There is only 1 magic hexagon. It uses the number 1 through to 19, and each side and each diagonal adds up to 38.
39
Thirty-nine has been discribed as ‘the 1st uninteresting number’. For example: \( 39 = 3 \times 9 + 3 + 9 \). Not impressed? Well, this pattern holds for any number ending in 9. Why? It’s simple.
- A number with digits \( ab \) is equal to \( 10a + b \).
- \( 10a + b = a \times b + a + b \)
- \( 10a = a \times b + a \)
- \( 9a = a \times b \)
- \( b = 9 \)
So:
\( 79 = 7 \times 9 + 7 + 9 \)
\( 159 = 15 \times 9 + 15 + 9 \) etc.
40
What are the 4 weights that can be used on a scale pan to weigh any whole number of grams from 1 to 40 inclusive, if the weights can be placed in either side of the scale pan? This is the famous Bachet’s problem from 1612. The answer is 1, 3, 9 and 27.
Forty is an abundant number.
41
Leonhard Euler discoverd the formula \( n^2 + n + 41 \) gives a prime number as its value for whole numbers \( n = 0, n = 1, n = 2 \) up to \( n = 39 \)
Wolfgang Amadeus Mozart churned out 41 symphonies, the last of which was the Jupiter Symphony in C major.
42
42 is a very superstitious number for some Japanese.
43
Forty-three is the 12th lucky number. It’s also a prime number.
44
If you have 6 objects in a row, there are 44 ways of rearranging those objects so that none are in their original spot.
45
Forty-five is the 3rd Kaprekar number.
\( 45^2 = 2025 \) and \( 20 + 25 = 45 \)
46
Nine cuts through a pizza can create up to 46 pieces.
47
Forty-seven is a Ulam number.
48
Forty-eight and 75 are betrothed. This happy couple have a relationship because the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. The sum of the factors of 48, except 1 and itself, is \( 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 = 75 \). The sum of the factors of 75, excepting 1 and itself is, you guessed it, 48. So 48 and 75 are betrothed. How cute!
Forty-eight is an abundant number.
49
Forty-nine is the 13th lucky number. It’s also a perfect square: \( 7 \times 7 = 49 \)
50
Fifty is the smallest number that is also the sum of 2 squares in 2 different ways. So \( 5^2 + 5^2 = 50 \) and \( 7^2 + 1^2 = 50 \).
51
Fifty-one is the 14th lucky number.
52
Fifty-two is an untouchable number: it is never the sum of the proper divisors of any other number. The only other untouchable numbers under 100 are 2, 5, 88 and 96.
The standard deck of cards used in the west has 52 cards, excluding jokers.
53
\( 53 = 2^2 + 7^2 = 1^2 + 4^2 + 6^2 \)
54
What is the most annoying object ever invented? The answer, of course, is Rubik’s cube, a little plastic cube consisting of several smaller cubes in six colours. Erno Rubik’s 1st working prototype was made in 1974. After a lot of experientaion, Erno settled on the \( 3 \times 3 \) unit cube, which resulted in 54 outer surfaces. Which makes it bloody difficult to solve.
Fifty-four is an abundant number.
55
55 is the 5th square pyramidal number.
Fifty-five is the 10th Fibonacci number and the 4th Kaprekar number. It’s also the sum of all the numbers from 1 to 10. So:
\( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 \)
Which makes 55 the 10th triangular number.
56
56 is the 6th tetrahedral number.
Fifty-six is an abundant number.
57
In New York in the 1890s, German-American chef and entrepreneur Henry Heinz coined one of the advertising’s most famous catchphrases to describe the wide range of ketchups, sauces and relishes he’d created. The slogan ‘57 varieties’ was a giant hit, despite the fact that Heinz actually produced 65 products at the time. Henry allegedly liked the look of 57.
58
The sum of the 1st 7 prime numbers is 58.
59
Leonhard Euler proved that:
\( 635318657 = 59^4 + 158^4 = 133^4 + 134^4 \)
60
The Babylonicans divided a circle into 369 degrees and today we still divide hours into 60 minutes and minutes into 60 seconds.
Sixty is an abundant number, and 60 degrees is the interior angle of an equilateral triangle.
61
Sixty-one and 59 are twin primes.
Pell’s equation \( x^2 - 61y^2 = 1 \), a Diophantine equation, the answer is \( x = 1766319049 \) and \( y = 226153980 \).
62
61 and 62 are Maris-McGwire-Sosa pairs.
Adding the digits of 62 and its prime factors, we get \( 6 + 1 + 6 + 1 = 14 \)
Adding the digits of 62 and its prime factors, we get \( 6 + 2 + 2 + 3 + 1 = 14 \).
63
Sixty-three is the 15th lucky number.
64
The second 6th power after 1, 64 is also a square and a cube, making it a handy number: it equals \( 8^2 = 4^3 = 2^6 \).
There are 64 squares on a chess board.
65
Sixty-five is the constant of \( 5 \times 5 \) magic square.
\( 65 = 1^2 + 8^2 = 4^2 + 7^2 \)
66
Sixty-six is an abundant number. It’s also a palindromic triangular number.
67
Sixty-seven is the 16th lucky number.
Eleven cuts through a pizza can create up to 67 pieces.
68
The last 68 of 80 episodes of TV’s Dad’s Army were made in color.
69
Sixty-nine is the 17th lucky number. It’s also the only number whose square and cube between them use all digits 0-9 once each: \( 69^2 = 4761 \) and \( 69^3 = 328509 \).
70
Seventy is a weird number. A weird number is one that is abundant but isn’t equal to the sum of any of its divisors. Weird numbers are very rare, and 70 is the only below 100. See, 12 is abundant because \( 1 + 2 + 3 + 4 + 6 = 16 \), which is greater than 12. But \( 2 + 4 + 6 = 12 \), so 12 isn’t weird.
71
Seventy-one is prime.
72
\( 72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5 \)
73
Seventy-three is the 18th lucky number.
74
One theory that is constantly being proposed and refuted on the Internet and elsewhere is that, when jointly developing the compact disc, Sony and Philips agreed on it beling 74 minutes long so as to accomodate Beethoven’s Ninth Symphony.
75
Seventy-five is the 19th lucky number.
76
Seventy-six is the 9th Lucas number. It is also automorphic: \( 76^2 = 5776 \), which ends in the number 76, making it automorphic.
77
It turns out that \( 77! + 1 \) is a 115 digits prime number.
The sum of the 1st 8 prime numbers is 77.
78
Seventy-eight is an abundant number. It’s also the 12th triangular number.
79
Seventy-nine is the 20th lucky number.
Twelve cuts to a pizza can create up to 79 pieces.
80
A variation on the magic square is the magic star of David. There are 80 magic stars of David, and each row must add up to 26.
81
Eight-one is the only number whose square root is equal to the sum of its digits. \( 8 + 1 = 9 = \sqrt{81} \). It’s both square and heptagonal.
\( \frac{1}{81} = 0.012345679 012345679 012 … \)
82
\( 82 = 33 + 49 = 27 + 55 = 57 + 25 \). notice that none of the odd numbers used here are prime.
83
Eighty-three per cent of people hit by lightening are men.
84
Eighty-four is an abundant number.
85
\( 85 = 2^2 + 9^2 = 6^2 + 7^2 \)
86
\( 2^{86} \) does not have a zero in it.
87
Eighty-seven is the 21th lucky number. Ironically, 13 is also a lucky number.
88
With 88 or more people in a room, there’s a better than even chance that 3 of them share a birthday.
89
Eighty-nine is the 11th Fibonacci number.
90
Ninety is an abundant number. And the number of degrees in a right angle, which must count for something.
A bingo card usually has 90 numbers.
91
Ninety-one is triangular, square pyramidal and centred hexagonal.
92
Ninety-two is the atomic number of uranium.
93
Ninety-three is the 22nd lucky number.
94
\( 94 = 41 + 53 = 5 + 89 = 47 + 47 \)
95
Saturn is 95 times heavier than Earth.
96
In Measurement of the Circle, the Greek mathematician Archimedes shows that the exact value of \( \pi \) lies between the values \( 3\frac{10}{71} \) and \( 3\frac{1}{7} \). He worked this out by circuscribing and inscribing a circle with regular polygons having 96 sides. Eureka!
Ninety-six is an abundant number.
97
The strongest any liquor can be is 190 proof. This means it’s a little more than 97 per cent alcohol.
98
Humans seem to share pretty much 98 per cent of their DNA with - wait for it - chimpanzees.
The atmoshpere on Mars is about 98 per cent carbon dioxide.
99
Ninety-nine is the 5th Kaprekar number: \( 99^2 = 9801 \) and \( 98 + 01 = 99 \)
\( 99 = 2^3 + 3^3 + 4^3 \)
100
\( 100 = 1^3 + 2^3 + 3^3 + 4^3 \)
\( 123 - 45 - 67 + 89 = 100 \)
- \( 2025 = 45^2 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)^2 = 27^2 + 36^2 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 \)
