ProjectEuler.nb

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If we list all the natural numbers below 10 that are multiples of 3 or 5, we \
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Find the sum of all the multiples of 3 or 5 below 1000.\
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Each new term in the Fibonacci sequence is generated by adding the previous \
two terms. By starting with 1 and 2, the first 10 terms will be:

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

By considering the terms in the Fibonacci sequence whose values do not exceed \
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The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143?\
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 A palindromic number reads the same both ways. The largest palindrome made \
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Find the largest palindrome made from the product of two 3-digit numbers:\
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2520 is the smallest number that can be divided by each of the numbers from 1 \
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What is the smallest positive number that is evenly divisible by all of the \
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*Note: If a number is divisible by 20, it is also divisible by 5,4,2.. ...a \
trial and error was used to find the range of this number, between 220 \
million and 240 million.\
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ten natural numbers and the square of the sum is 3025 \[Minus] 385 = 2640.\n\n\
Find the difference between the sum of the squares of the first one hundred \
natural numbers and the square of the sum."
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The prime By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we \
can see that the 6th prime is 13.

What is the 10 001st prime number?\
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Find the greatest product of five consecutive digits in the following \
1000-digit number:\
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Cell["\<\
 In the 20\[Times]20 grid below, four numbers along a diagonal line have been \
marked in red.

The product of these numbers is 26 \[Times] 63 \[Times] 78 \[Times] 14 = \
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What is the greatest product of four adjacent numbers in any direction (up, \
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The sequence of triangle numbers is generated by adding the natural numbers. \
So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first \
ten terms would be:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

Let us list the factors of the first seven triangle numbers:

     1: 1
     3: 1,3
     6: 1,2,3,6
    10: 1,2,5,10
    15: 1,3,5,15
    21: 1,3,7,21
    28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred \
divisors?\
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Work out the first ten digits of the sum of the following one-hundred \
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2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.

What is the sum of the digits of the number 2^1000?\
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If the numbers 1 to 5 are written out in words: one, two, three, four, five, \
then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.

If all the numbers from 1 to 1000 (one thousand) inclusive were written out \
in words, how many letters would be used?

NOTE: Do not count spaces or hyphens. For example, 342 (three hundred and \
forty-two) contains 23 letters and 115 (one hundred and fifteen) contains 20 \
letters. The use of \[OpenCurlyDoubleQuote]and\[CloseCurlyDoubleQuote] when \
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By starting at the top of the triangle below and moving to adjacent numbers \
on the row below, the maximum total from top to bottom is 23.

3
7 4
2 4 6
8 5 9 3

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:

75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
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70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23

NOTE: As there are only 16384 routes, it is possible to solve this problem by \
trying every route. However, Problem 67, is the same challenge with a \
triangle containing one-hundred rows; it cannot be solved by brute force, and \
requires a clever method! ;o)\
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You are given the following information, but you may prefer to do some \
research for yourself.

    * 1 Jan 1900 was a Monday.
    * Thirty days has September,
      April, June and November.
      All the rest have thirty-one,
      Saving February alone,
      Which has twenty-eight, rain or shine.
      And on leap years, twenty-nine.
    * A leap year occurs on any year evenly divisible by 4, but not on a \
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How many Sundays fell on the first of the month during the twentieth century \
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Let d(n) be defined as the sum of proper divisors of n (numbers less than n \
which divide evenly into n).
If d(a) = b and d(b) = a, where a \[NotEqual] b, then a and b are an amicable \
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For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, \
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Evaluate the sum of all the amicable numbers under 10000.\
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8128,496, 6 and 28 are not amicable pairs. 8128,496, 6, and 128 are perfect \
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Using names.txt (right click and \[OpenCurlyQuote]Save Link/Target As...\
\[CloseCurlyQuote]), a 46K text file containing over five-thousand first \
names, begin by sorting it into alphabetical order. Then working out the \
alphabetical value for each name, multiply this value by its alphabetical \
position in the list to obtain a name score.

For example, when the list is sorted into alphabetical order, COLIN, which is \
worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN \
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What is the total of all the name scores in the file?\
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A perfect number is a number for which the sum of its proper divisors is \
exactly equal to the number. For example, the sum of the proper divisors of \
28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

A number n is called deficient if the sum of its proper divisors is less than \
n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest \
number that can be written as the sum of two abundant numbers is 24. By \
mathematical analysis, it can be shown that all integers greater than 28123 \
can be written as the sum of two abundant numbers. However, this upper limit \
cannot be reduced any further by analysis even though it is known that the \
greatest number that cannot be expressed as the sum of two abundant numbers \
is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum \
of two abundant numbers.\
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6 is not a perfect number, it will have to be subtracted from the final sum\
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Any multiple of a perfect number or abundant number is also abundant.\
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A permutation is an ordered arrangement of objects. For example, 3124 is one \
possible permutation of the digits 1, 2, 3 and 4. If all of the permutations \
are listed numerically or alphabetically, we call it lexicographic order. The \
lexicographic permutations of 0, 1 and 2 are:

012   021   102   120   201   210

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, \
5, 6, 7, 8 and 9?\
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The Fibonacci sequence is defined by the recurrence relation:

    Fn = Fn\[Minus]1 + Fn\[Minus]2, where F1 = 1 and F2 = 1.

Hence the first 12 terms will be:

    F1 = 1
    F2 = 1
    F3 = 2
    F4 = 3
    F5 = 5
    F6 = 8
    F7 = 13
    F8 = 21
    F9 = 34
    F10 = 55
    F11 = 89
    F12 = 144

The 12th term, F12, is the first term to contain three digits.

What is the first term in the Fibonacci sequence to contain 1000 digits?\
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Cell["\<\
Euler published the remarkable quadratic formula:

n\.b2 + n + 41

It turns out that the formula will produce 40 primes for the consecutive \
values n = 0 to 39. However, when n = 40, 402 + 40 + 41 = 40(40 + 1) + 41 is \
divisible by 41, and certainly when n = 41, 41\.b2 + 41 + 41 is clearly \
divisible by 41.

Using computers, the incredible formula  n\.b2 \[Minus] 79n + 1601 was \
discovered, which produces 80 primes for the consecutive values n = 0 to 79. \
The product of the coefficients, \[Minus]79 and 1601, is \[Minus]126479.

Considering quadratics of the form:

    n\.b2 + an + b, where |a| < 1000 and |b| < 1000

    where |n| is the modulus/absolute value of n
    e.g. |11| = 11 and |\[Minus]4| = 4

Find the product of the coefficients, a and b, for the quadratic expression \
that produces the maximum number of primes for consecutive values of n, \
starting with n = 0.\
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Cell["\<\
Starting with the number 1 and moving to the right in a clockwise direction a \
5 by 5 spiral is formed as follows:

21 22 23 24 25
20  7  8  9 10
19  6  1  2 11
18  5  4  3 12
17 16 15 14 13

It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral \
formed in the same way?\
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 " to create a spiral of 1001x1001, with the command spiral(1001). This also \
generated the same 5x5 matrix as above with spiral(5), used: \
\[OpenCurlyDoubleQuote]save p28.xls d -ascii\[OpenCurlyDoubleQuote] to save \
data.."
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Cell["\<\
Consider all integer combinations of a^d for 2 \[LessEqual] a \[LessEqual] 5 \
and 2 \[LessEqual] d \[LessEqual] 5:

    2^2=4, 2^3=8, 2^4=16, 2^5=32
    3^2=9, 3^3=27, 3^4=81, 3^5=243
    4^2=16, 4^3=64, 4^4=256, 4^5=1024
    5^2=25, 5^3=125, 5^4=625, 5^5=3125

If they are then placed in numerical order, with any repeats removed, we get \
the following sequence of 15 distinct terms:

4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125

How many distinct terms are in the sequence generated by a^d for 2 \
\[LessEqual] a \[LessEqual] 100 and 2 \[LessEqual] d \[LessEqual] 100?\
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Cell["\<\
Surprisingly there are only three numbers that can be written as the sum of \
fourth powers of their digits:

    1634 = 1^4 + 6^4 + 3^4 + 4^4
    8208 = 8^4 + 2^4 + 0^4 + 8^4
    9474 = 9^4 + 4^4 + 7^4 + 4^4

As 1 = 14 is not a sum it is not included.

The sum of these numbers is 1634 + 8208 + 9474 = 19316.

Find the sum of all the numbers that can be written as the sum of fifth \
powers of their digits.\
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In England the currency is made up of pound, \[Sterling], and pence, p, and \
there are eight coins in general circulation:

    1p, 2p, 5p, 10p, 20p, 50p, \[Sterling]1 (100p) and \[Sterling]2 (200p).

It is possible to make \[Sterling]2 in the following way:

    1\[Times]\[Sterling]1 + 1\[Times]50p + 2\[Times]20p + 1\[Times]5p + 1\
\[Times]2p + 3\[Times]1p

How many different ways can \[Sterling]2 be made using any number of coins?\
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Note: by coins they are including both \[OpenCurlyDoubleQuote]pound\
\[CloseCurlyDoubleQuote] denominations.\
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Cell["\<\
The fraction 49/98 is a curious fraction, as an inexperienced mathematician \
in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which \
is correct, is obtained by cancelling the 9s.

We shall consider fractions like, 30/50 = 3/5, to be trivial examples.

There are exactly four non-trivial examples of this type of fraction, less \
than one in value, and containing two digits in the numerator and denominator.

If the product of these four fractions is given in its lowest common terms, \
find the value of the denominator.\
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Cell["\<\
145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.

Find the sum of all numbers which are equal to the sum of the factorial of \
their digits.

Note: as 1! = 1 and 2! = 2 are not sums they are not included.\
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The number, 197, is called a circular prime because all rotations of the \
digits: 197, 971, and 719, are themselves prime.

There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, \
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How many circular primes are there below one million?\
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The number 3797 has an interesting property. Being prime itself, it is \
possible to continuously remove digits from left to right, and remain prime \
at each stage: 3797, 797, 97, and 7. Similarly we can work from right to \
left: 3797, 379, 37, and 3.

Find the sum of the only eleven primes that are both truncatable from left to \
right and right to left.

NOTE: 2, 3, 5, and 7 are not considered to be truncatable primes.\
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Cell["\<\
Obviously this is 840 since it is listed with 8 solutions. Can write code to \
select that 840.\
\>", "Text",
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An irrational decimal fraction is created by concatenating the positive \
integers:

0.123456789101112131415161718192021...

It can be seen that the 12th digit of the fractional part is 1.

If dn represents the nth digit of the fractional part, find the value of the \
following expression.

d1 \[Times] d10 \[Times] d100 \[Times] d1000 \[Times] d10000 \[Times] d100000 \
\[Times] d1000000\
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The nth term of the sequence of triangle numbers is given by, tn = \
\.bdn(n+1); so the first ten triangle numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

By converting each letter in a word to a number corresponding to its \
alphabetical position and adding these values we form a word value. For \
example, the word value for SKY is 19 + 11 + 25 = 55 = t10. If the word value \
is a triangle number then we shall call the word a triangle word.

Using words.txt (right click and \[OpenCurlyQuote]Save Link/Target As...\
\[CloseCurlyQuote]), a 16K text file containing nearly two-thousand common \
English words, how many are triangle words?\
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Triangle, pentagonal, and hexagonal numbers are generated by the following \
formulae:
Triangle \t  \tTn=n(n+1)/2 \t  \t1, 3, 6, 10, 15, ...
Pentagonal \t  \tPn=n(3n\[Minus]1)/2 \t  \t1, 5, 12, 22, 35, ...
Hexagonal \t  \tHn=n(2n\[Minus]1) \t  \t1, 6, 15, 28, 45, ...

It can be verified that T285 = P165 = H143 = 40755.

Find the next triangle number that is also pentagonal and hexagonal.\
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The prime 41, can be written as the sum of six consecutive primes:
41 = 2 + 3 + 5 + 7 + 11 + 13

This is the longest sum of consecutive primes that adds to a prime below \
one-hundred.

The longest sum of consecutive primes below one-thousand that adds to a \
prime, contains 21 terms, and is equal to 953.

Which prime, below one-million, can be written as the sum of the most \
consecutive primes?\
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this was one was solved using a for-loop and intersection of the \
map[permutation[ command but was not saved sadface.\
\>", "Text",
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the notation, 5C3 = 10.\n\nIn general,\n",
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\[Times]2\[Times]1, and 0! = 1.\n\nIt is not until n = 23, that a value \
exceeds one-million: 23C10 = 1144066.\n\nHow many, not necessarily distinct, \
values of  nCr, for 1 \[LessEqual] n \[LessEqual] 100, are greater than \
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It is possible to show that the square root of two can be expressed as an \
infinite continued fraction.

\[Sqrt] 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...

By expanding this for the first four iterations, we get:

1 + 1/2 = 3/2 = 1.5
1 + 1/(2 + 1/2) = 7/5 = 1.4
1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...
1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...

The next three expansions are 99/70, 239/169, and 577/408, but the eighth \
expansion, 1393/985, is the first example where the number of digits in the \
numerator exceeds the number of digits in the denominator.

In the first one-thousand expansions, how many fractions contain a numerator \
with more digits than denominator?\
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Starting with 1 and spiralling anticlockwise in the following way, a square \
spiral with side length 7 is formed.

37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18  5  4  3 12 29
40 19  6  1  2 11 28
41 20  7  8  9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49

It is interesting to note that the odd squares lie along the bottom right \
diagonal, but what is more interesting is that 8 out of the 13 numbers lying \
along both diagonals are prime; that is, a ratio of 8/13 \[TildeTilde] 62%.

If one complete new layer is wrapped around the spiral above, a square spiral \
with side length 9 will be formed. If this process is continued, what is the \
side length of the square spiral for which the ratio of primes along both \
diagonals first falls below 10%?
\
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Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers \
are all figurate (polygonal) numbers and are generated by the following \
formulae:
Triangle \t  \tP3,n=n(n+1)/2 \t  \t1, 3, 6, 10, 15, ...
Square \t  \t\tP4,n=n^2 \t  \t1, 4, 9, 16, 25, ...
Pentagonal \t  \tP5,n=n(3n\[Minus]1)/2 \t  \t1, 5, 12, 22, 35, ...
Hexagonal \t  \tP6,n=n(2n\[Minus]1) \t  \t1, 6, 15, 28, 45, ...
Heptagonal \t  \tP7,n=n(5n\[Minus]3)/2 \t  \t1, 7, 18, 34, 55, ...
Octagonal \t  \tP8,n=n(3n\[Minus]2) \t  \t1, 8, 21, 40, 65, ...

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three \
interesting properties.

   1. The set is cyclic, in that the last two digits of each number is the \
first two digits of the next number (including the last number with the first).
   2. Each polygonal type: triangle (P3,127=8128), square (P4,91=8281), and \
pentagonal (P5,44=2882), is represented by a different number in the set.
   3. This is the only set of 4-digit numbers with this property.

Find the sum of the only ordered set of six cyclic 4-digit numbers for which \
each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and \
octagonal, is represented by a different number in the set.\
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The 5-digit number, 16807=7^5, is also a fifth power. Similarly, the 9-digit \
number, 134217728=8^9, is a ninth power.

How many n-digit positive integers exist which are also an nth power?\
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Euler\[CloseCurlyQuote]s Totient function, \[CurlyPhi](n) [sometimes called \
the phi function], is used to determine the number of numbers less than n \
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n \tRelatively Prime \t\[CurlyPhi](n) \tn/\[CurlyPhi](n)
2 \t1 \t                        1 \t 2
3 \t1,2 \t                        2 \t 1.5
4 \t1,3 \t\t\t2 \t 2
5 \t1,2,3,4 \t\t4 \t 1.25
6 \t1,5 \t\t\t2 \t3
7 \t1,2,3,4,5,6 \t\t6 \t1.1666...
8 \t1,3,5,7 \t\t4 \t2
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10 \t1,3,7,9 \t\t4 \t2.5

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Find the value of n \[LessEqual] 1,000,000 for which n/\[CurlyPhi](n) is a \
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For a prime p, phi(p)=p-1, since all numbers less than p are relatively prime \
to p\
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The rules for writing Roman numerals allow for many ways of writing each \
number (see FAQ: Roman Numerals). However, there is always a \
\[OpenCurlyDoubleQuote]best\[CloseCurlyDoubleQuote] way of writing a \
particular number.

For example, the following represent all of the legitimate ways of writing \
the number sixteen:

IIIIIIIIIIIIIIII
VIIIIIIIIIII
VVIIIIII
XIIIIII
VVVI
XVI

The last example being considered the most efficient, as it uses the least \
number of numerals.

The 11K text file, roman.txt (right click and \[OpenCurlyQuote]Save \
Link/Target As...\[CloseCurlyQuote]), contains one thousand numbers written \
in valid, but not necessarily minimal, Roman numerals; that is, they are \
arranged in descending units and obey the subtractive pair rule (see FAQ for \
the definitive rules for this problem).

Find the number of characters saved by writing each of these in their minimal \
form.
Note: You can assume that all the Roman numerals in the file contain no more \
than four consecutive identical units.\
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Cell["\<\
It turns out that 12 cm is the smallest length of wire that can be bent to \
form an integer sided right angle triangle in exactly one way, but there are \
many more examples.

12 cm: (3,4,5)
24 cm: (6,8,10)
30 cm: (5,12,13)
36 cm: (9,12,15)
40 cm: (8,15,17)
48 cm: (12,16,20)

In contrast, some lengths of wire, like 20 cm, cannot be bent to form an \
integer sided right angle triangle, and other lengths allow more than one \
solution to be found; for example, using 120 cm it is possible to form \
exactly three different integer sided right angle triangles.

120 cm: (30,40,50), (20,48,52), (24,45,51)

Given that L is the length of the wire, for how many values of L \[LessEqual] \
1,500,000 can exactly one integer sided right angle triangle be formed?

Note: This problem has been changed recently, please check that you are using \
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Given the positive integers, x, y, and z, are consecutive terms of an \
arithmetic progression, the least value of the positive integer, n, for which \
the equation, x2 \[Minus] y2 \[Minus] z2 = n, has exactly two solutions is n \
= 27:

342 \[Minus] 272 \[Minus] 202 = 122 \[Minus] 92 \[Minus] 62 = 27

It turns out that n = 1155 is the least value which has exactly ten solutions.

How many values of n less than one million have exactly ten distinct \
solutions?\
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Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, \
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Colin has six six-sided (cubic) dice, each with faces numbered 1, 2, 3, 4, 5, \
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Peter and Colin roll their dice and compare totals: the highest total wins. \
The result is a draw if the totals are equal.

What is the probability that Pyramidal Pete beats Cubic Colin? Give your \
answer rounded to seven decimal places in the form 0.abcdefg\
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A triplet of positive integers (a,b,c) is called a Cardano Triplet if it \
satisfies the condition:

For example, (2,1,5) is a Cardano Triplet.

There exist 149 Cardano Triplets for which a+b+c \[LessEqual] 1000.

Find how many Cardano Triplets exist such that a+b+c \[LessEqual] \
110,000,000. \
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