| # | ProjectEuler ... DONE - WORKING - PLANNING | how |
| 1 | If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000. |
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| 2 | 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, ... Find the sum of all the even-valued terms in the sequence which do not exceed four million. |
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| 3 | 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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| 4 | A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.
Find the largest palindrome made from the product of two 3-digit numbers. |
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| 5 | 2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest number that is evenly divisible by all of the numbers from 1 to 20? |
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| 6 | The sum of the squares of the first ten natural numbers is,
1^(2) + 2^(2) + ... + 10^(2) = 385
The square of the sum of the first ten natural numbers is, (1 + 2 + ... + 10)^(2) = 55^(2) = 3025 Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640. Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum. |
excel |
| 7 | By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6^(th) prime is 13.
What is the 10001^(st) prime number? |
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| 8 | Find the greatest product of five consecutive digits in the 1000-digit number.
73167176531330624919225119674426574742355349194934 96983520312774506326239578318016984801869478851843 85861560789112949495459501737958331952853208805511 12540698747158523863050715693290963295227443043557 66896648950445244523161731856403098711121722383113 62229893423380308135336276614282806444486645238749 30358907296290491560440772390713810515859307960866 70172427121883998797908792274921901699720888093776 65727333001053367881220235421809751254540594752243 52584907711670556013604839586446706324415722155397 53697817977846174064955149290862569321978468622482 83972241375657056057490261407972968652414535100474 82166370484403199890008895243450658541227588666881 16427171479924442928230863465674813919123162824586 17866458359124566529476545682848912883142607690042 24219022671055626321111109370544217506941658960408 07198403850962455444362981230987879927244284909188 84580156166097919133875499200524063689912560717606 05886116467109405077541002256983155200055935729725 71636269561882670428252483600823257530420752963450 |
eyes |
| 9> | A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
a^(2) + b^(2) = c^(2)
For example, 3^(2) + 4^(2) = 9 + 16 = 25 = 5^(2). There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc. |
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| 10 | The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million. |
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| 11 | In the 20×20 grid below, four numbers along a diagonal line have been [made bold].
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08 49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00 81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65 52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91 22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80 24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50 32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70 67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21 24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72 21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95 78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92 16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57 86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58 19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40 04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66 88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69 04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36 20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16 20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54 01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48The product of these numbers is 26 × 63 × 78 × 14 = 1788696. What is the greatest product of four adjacent numbers in any direction (up, down, left, right, or diagonally) in the 20×20 grid? |
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| 14 | The following iterative sequence is defined for the set of positive integers:
n -> n/2 (n is even) n -> 3n + 1 (n is odd) Using the rule above and starting with 13, we generate the following sequence: 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1. Which starting number, under one million, produces the longest chain? |
asm |
| 16 | 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)? |
asm |
| 19 | 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 century unless it is divisible by 400.
How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)? |
asm |
| 20 | n! means n * (n - 1) * ... * 3 * 2 * 1
Find the sum of the digits in the number 100! |
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| 22 | Using names.txt (right click and 'Save Link/Target As...'), 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 would obtain a score of 938 × 53 = 49714. What is the total of all the name scores in the file? |
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| 24 | 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 210What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9? |
asm |
| 25 | The Fibonacci sequence is defined by the recurrence relation:
F_(n) = F_(n-1) + F_(n-2), where F_(1) = 1 and F_(2) = 1.
Hence the first 12 terms will be:
F_(1) = 1
F_(2) = 1
F_(3) = 2
F_(4) = 3
F_(5) = 5
F_(6) = 8
F_(7) = 13
F_(8) = 21
F_(9) = 34
F_(10) = 55
F_(11) = 89
F_(12) = 144
The 12th term, F_(12), is the first term to contain three digits.
What is the first term in the Fibonacci sequence to contain 1000 digits? |
asm |
| 26 | A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:
1/2 = 0.5
1/3 = 0.(3)
1/4 = 0.25
1/5 = 0.2
1/6 = 0.1(6)
1/7 = 0.(142857)
1/8 = 0.125
1/9 = 0.(1)
1/10 = 0.1
Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.
Find the value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part. |
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| 28 | 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 13It 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? |
asm |
| 29 | Consider all integer combinations of a^(b) for 2 <= a <= 5 and 2 <= b = 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, 3125How many distinct terms are in the sequence generated by a^(b) for 2 = a = 100 and 2 = b = 100? |
pencil |
| 36 | The decimal number, 585 = 1001001001b (binary), is palindromic in both bases.
Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2. (Please note that the palindromic number, in either base, may not include leading zeros.) |
asm |
| 45 | Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
Triangle T_(n)=n(n+1)/2 1, 3, 6, 10, 15, ... Pentagonal P_(n)=n(3n-1)/2 1, 5, 12, 22, 35, ... Hexagonal H_(n)=n(2n-1) 1, 6, 15, 28, 45, ...It can be verified that T_(285) = P_(165) = H_(143) = 40755. Find the next triangle number that is also pentagonal and hexagonal. |
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| 55 | If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.
Not all numbers produce palindromes so quickly. For example, 349 + 943 = 1292, 1292 + 2921 = 4213 4213 + 3124 = 7337That is, 349 took three iterations to arrive at a palindrome. Although no one has proved it yet, it is thought that some numbers, like 196, never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, 10677 is the first number to be shown to require over fifty iterations before producing a palindrome: 4668731596684224866951378664 (53 iterations, 28-digits). Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is 4994. How many Lychrel numbers are there below ten-thousand? |
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| 68 | Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
 Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3. It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total. Total Solution Set 9 4,2,3; 5,3,1; 6,1,2 9 4,3,2; 6,2,1; 5,1,3 10 2,3,5; 4,5,1; 6,1,3 10 2,5,3; 6,3,1; 4,1,5 11 1,4,6; 3,6,2; 5,2,4 11 1,6,4; 5,4,2; 3,2,6 12 1,5,6; 2,6,4; 3,4,5 12 1,6,5; 3,5,4; 2,4,6By concatenating each group it is possible to form 9-digit strings; the maximum string for a 3-gon ring is 432621513. Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings. What is the maximum 16-digit string for a "magic" 5-gon ring?  |
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| 76 | It is possible to write five as a sum in exactly six different ways:
4 + 1 3 + 2 3 + 1 + 1 2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1How many different ways can one hundred be written as a sum of at least two positive integers? |
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| 89 | The rules for writing Roman numerals allow for many ways of writing each number (see FAQ: Roman Numerals). However, there is always a "best" 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 XVIThe last example being considered the most efficient, as it uses the least number of numerals. The 11K text file, roman.txt (right click and 'Save Link/Target As...'), 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. |
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| 92 | A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.
For example, 44 -> 32 -> 13 -> 10 -> 1 -> 1 85 -> 89 -> 145 -> 42 -> 20 -> 4 -> 16 -> 37 -> 58 -> 89Therefore any chain that arrives at 1 or 89 will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at 1 or 89. How many starting numbers below ten million will arrive at 89? |
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| 104 | The Fibonacci sequence is defined by the recurrence relation:
F(n) = F(n-1) + F(n-2), where F(1) = 1 and F(2) = 1. It turns out that F_(541), which contains 113 digits, is the first Fibonacci number for which the last nine digits are 1-9 pandigital (contain all the digits 1 to 9, but not necessarily in order). And F_(2749), which contains 575 digits, is the first Fibonacci number for which the first nine digits are 1-9 pandigital. Given that F_(k) is the first Fibonacci number for which the first nine digits AND the last nine digits are 1-9 pandigital, find k. |
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| 113 | Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420. We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, 155349. As n increases, the proportion of bouncy numbers below n increases such that there are only 12951 numbers below one-million that are not bouncy and only 277032 non-bouncy numbers below 10^(10). How many numbers below a googol (10^(100)) are not bouncy? |
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