md`# Problems 1-50`
[
{
"description": "If we list all the natural numbers below 10 that are multiples of 3 or 5,\nwe get 3, 5, 6 and 9. The sum of these multiples is 23.\n\nFind the sum of all the multiples of 3 or 5 below 1000.",
"answer": "233168",
"difficulty": 5,
"name": "Multiples of 3 and 5",
"id": 1
},
{
"description": "Each new term in the Fibonacci sequence is generated by adding the\nprevious two terms. By starting with 1 and 2, the first 10 terms will be:\n\n 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...\n\nFind the sum of all the even-valued terms in the sequence which do not\nexceed four million.",
"answer": "4613732",
"difficulty": 5,
"name": "Even Fibonacci numbers",
"id": 2
},
{
"description": "The prime factors of 13195 are 5, 7, 13 and 29.\n\nWhat is the largest prime factor of the number 600851475143?",
"answer": "6857",
"difficulty": 5,
"name": "Largest prime factor",
"id": 3
},
{
"description": "A palindromic number reads the same both ways. The largest palindrome made\nfrom the product of two 2-digit numbers is 9009 = 91 * 99.\n\nFind the largest palindrome made from the product of two 3-digit numbers.",
"answer": "906609",
"difficulty": 5,
"name": "Largest palindrome product",
"id": 4
},
{
"description": "2520 is the smallest number that can be divided by each of the numbers\nfrom 1 to 10 without any remainder.\n\nWhat is the smallest number that is evenly divisible by all of the numbers\nfrom 1 to 20?",
"answer": "232792560",
"difficulty": 5,
"name": "Smallest multiple",
"id": 5
},
{
"description": "The sum of the squares of the first ten natural numbers is,\n 1^2 + 2^2 + ... + 10^2 = 385\n\nThe square of the sum of the first ten natural numbers is,\n (1 + 2 + ... + 10)^2 = 55^2 = 3025\n\nHence the difference between the sum of the squares of the first ten\nnatural numbers and the square of the sum is 3025 - 385 = 2640.\n\nFind the difference between the sum of the squares of the first one\nhundred natural numbers and the square of the sum.",
"answer": "25164150",
"difficulty": 5,
"name": "Sum square difference",
"id": 6
},
{
"description": "By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see\nthat the 6th prime is 13.\n\nWhat is the 10001st prime number?",
"answer": "104743",
"difficulty": 5,
"name": "10001st prime",
"id": 7
},
{
"description": "Find the greatest product of thirteen consecutive digits in the 1000-digit\nnumber.\n\n 73167176531330624919225119674426574742355349194934\n 96983520312774506326239578318016984801869478851843\n 85861560789112949495459501737958331952853208805511\n 12540698747158523863050715693290963295227443043557\n 66896648950445244523161731856403098711121722383113\n 62229893423380308135336276614282806444486645238749\n 30358907296290491560440772390713810515859307960866\n 70172427121883998797908792274921901699720888093776\n 65727333001053367881220235421809751254540594752243\n 52584907711670556013604839586446706324415722155397\n 53697817977846174064955149290862569321978468622482\n 83972241375657056057490261407972968652414535100474\n 82166370484403199890008895243450658541227588666881\n 16427171479924442928230863465674813919123162824586\n 17866458359124566529476545682848912883142607690042\n 24219022671055626321111109370544217506941658960408\n 07198403850962455444362981230987879927244284909188\n 84580156166097919133875499200524063689912560717606\n 05886116467109405077541002256983155200055935729725\n 71636269561882670428252483600823257530420752963450",
"answer": "23514624000",
"difficulty": 5,
"name": "Largest product in a series",
"id": 8
},
{
"description": "A Pythagorean triplet is a set of three natural numbers, a < b < c, for\nwhich,\n a^2 + b^2 = c^2\n\nFor example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.\n\nThere exists exactly one Pythagorean triplet for which a + b + c = 1000.\nFind the product abc.",
"answer": "31875000",
"difficulty": 5,
"name": "Special Pythagorean triplet",
"id": 9
},
{
"description": "The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\n\nFind the sum of all the primes below two million.",
"answer": "142913828922",
"difficulty": 5,
"name": "Summation of primes",
"id": 10
},
{
"description": "In the 20 * 20 grid below, four numbers along a diagonal line have been\nmarked in red.\n\n 08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08\n 49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00\n 81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65\n 52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91\n 22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80\n 24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50\n 32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70\n 67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21\n 24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72\n 21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95\n 78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92\n 16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57\n 86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58\n 19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40\n 04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66\n 88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69\n 04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36\n 20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16\n 20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54\n 01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48\n\nThe product of these numbers is 26 * 63 * 78 * 14 = 1788696.\n\nWhat is the greatest product of four adjacent numbers in any direction\n(up, down, left, right, or diagonally) in the 20 * 20 grid?",
"answer": "70600674",
"difficulty": 5,
"name": "Largest product in a grid",
"id": 11
},
{
"description": "The sequence of triangle numbers is generated by adding the natural\nnumbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 =\n28. The first ten terms would be:\n\n 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...\n\nLet us list the factors of the first seven triangle numbers:\n\n 1: 1\n 3: 1,3\n 6: 1,2,3,6\n 10: 1,2,5,10\n 15: 1,3,5,15\n 21: 1,3,7,21\n 28: 1,2,4,7,14,28\n\nWe can see that 28 is the first triangle number to have over five\ndivisors.\n\nWhat is the value of the first triangle number to have over five hundred\ndivisors?",
"answer": "76576500",
"difficulty": 5,
"name": "Highly divisible triangular number",
"id": 12
},
{
"description": "Work out the first ten digits of the sum of the following one-hundred\n50-digit numbers.\n\n 37107287533902102798797998220837590246510135740250\n 46376937677490009712648124896970078050417018260538\n 74324986199524741059474233309513058123726617309629\n 91942213363574161572522430563301811072406154908250\n 23067588207539346171171980310421047513778063246676\n 89261670696623633820136378418383684178734361726757\n 28112879812849979408065481931592621691275889832738\n 44274228917432520321923589422876796487670272189318\n 47451445736001306439091167216856844588711603153276\n 70386486105843025439939619828917593665686757934951\n 62176457141856560629502157223196586755079324193331\n 64906352462741904929101432445813822663347944758178\n 92575867718337217661963751590579239728245598838407\n 58203565325359399008402633568948830189458628227828\n 80181199384826282014278194139940567587151170094390\n 35398664372827112653829987240784473053190104293586\n 86515506006295864861532075273371959191420517255829\n 71693888707715466499115593487603532921714970056938\n 54370070576826684624621495650076471787294438377604\n 53282654108756828443191190634694037855217779295145\n 36123272525000296071075082563815656710885258350721\n 45876576172410976447339110607218265236877223636045\n 17423706905851860660448207621209813287860733969412\n 81142660418086830619328460811191061556940512689692\n 51934325451728388641918047049293215058642563049483\n 62467221648435076201727918039944693004732956340691\n 15732444386908125794514089057706229429197107928209\n 55037687525678773091862540744969844508330393682126\n 18336384825330154686196124348767681297534375946515\n 80386287592878490201521685554828717201219257766954\n 78182833757993103614740356856449095527097864797581\n 16726320100436897842553539920931837441497806860984\n 48403098129077791799088218795327364475675590848030\n 87086987551392711854517078544161852424320693150332\n 59959406895756536782107074926966537676326235447210\n 69793950679652694742597709739166693763042633987085\n 41052684708299085211399427365734116182760315001271\n 65378607361501080857009149939512557028198746004375\n 35829035317434717326932123578154982629742552737307\n 94953759765105305946966067683156574377167401875275\n 88902802571733229619176668713819931811048770190271\n 25267680276078003013678680992525463401061632866526\n 36270218540497705585629946580636237993140746255962\n 24074486908231174977792365466257246923322810917141\n 91430288197103288597806669760892938638285025333403\n 34413065578016127815921815005561868836468420090470\n 23053081172816430487623791969842487255036638784583\n 11487696932154902810424020138335124462181441773470\n 63783299490636259666498587618221225225512486764533\n 67720186971698544312419572409913959008952310058822\n 95548255300263520781532296796249481641953868218774\n 76085327132285723110424803456124867697064507995236\n 37774242535411291684276865538926205024910326572967\n 23701913275725675285653248258265463092207058596522\n 29798860272258331913126375147341994889534765745501\n 18495701454879288984856827726077713721403798879715\n 38298203783031473527721580348144513491373226651381\n 34829543829199918180278916522431027392251122869539\n 40957953066405232632538044100059654939159879593635\n 29746152185502371307642255121183693803580388584903\n 41698116222072977186158236678424689157993532961922\n 62467957194401269043877107275048102390895523597457\n 23189706772547915061505504953922979530901129967519\n 86188088225875314529584099251203829009407770775672\n 11306739708304724483816533873502340845647058077308\n 82959174767140363198008187129011875491310547126581\n 97623331044818386269515456334926366572897563400500\n 42846280183517070527831839425882145521227251250327\n 55121603546981200581762165212827652751691296897789\n 32238195734329339946437501907836945765883352399886\n 75506164965184775180738168837861091527357929701337\n 62177842752192623401942399639168044983993173312731\n 32924185707147349566916674687634660915035914677504\n 99518671430235219628894890102423325116913619626622\n 73267460800591547471830798392868535206946944540724\n 76841822524674417161514036427982273348055556214818\n 97142617910342598647204516893989422179826088076852\n 87783646182799346313767754307809363333018982642090\n 10848802521674670883215120185883543223812876952786\n 71329612474782464538636993009049310363619763878039\n 62184073572399794223406235393808339651327408011116\n 66627891981488087797941876876144230030984490851411\n 60661826293682836764744779239180335110989069790714\n 85786944089552990653640447425576083659976645795096\n 66024396409905389607120198219976047599490197230297\n 64913982680032973156037120041377903785566085089252\n 16730939319872750275468906903707539413042652315011\n 94809377245048795150954100921645863754710598436791\n 78639167021187492431995700641917969777599028300699\n 15368713711936614952811305876380278410754449733078\n 40789923115535562561142322423255033685442488917353\n 44889911501440648020369068063960672322193204149535\n 41503128880339536053299340368006977710650566631954\n 81234880673210146739058568557934581403627822703280\n 82616570773948327592232845941706525094512325230608\n 22918802058777319719839450180888072429661980811197\n 77158542502016545090413245809786882778948721859617\n 72107838435069186155435662884062257473692284509516\n 20849603980134001723930671666823555245252804609722\n 53503534226472524250874054075591789781264330331690",
"answer": "5537376230",
"difficulty": 5,
"name": "Large sum",
"id": 13
},
{
"description": "The following iterative sequence is defined for the set of positive\nintegers:\n\nn->n/2 (n is even)\nn->3n+1 (n is odd)\n\nUsing the rule above and starting with 13, we generate the following\nsequence:\n 13->40->20->10->5->16->8->4->2->1\n\nIt can be seen that this sequence (starting at 13 and finishing at 1)\ncontains 10 terms. Although it has not been proved yet (Collatz Problem),\nit is thought that all starting numbers finish at 1.\n\nWhich starting number, under one million, produces the longest chain?\n\nNOTE: Once the chain starts the terms are allowed to go above one million.",
"answer": "837799",
"difficulty": 5,
"name": "Longest Collatz sequence",
"id": 14
},
{
"description": "Starting in the top left corner of a 2 * 2 grid, there are 6 routes\n(without backtracking) to the bottom right corner.\n\nHow many routes are there through a 20 * 20 grid?",
"answer": "137846528820",
"difficulty": 5,
"name": "Lattice paths",
"id": 15
},
{
"description": "2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.\n\nWhat is the sum of the digits of the number 2^1000?",
"answer": "1366",
"difficulty": 5,
"name": "Power digit sum",
"id": 16
},
{
"description": "If the numbers 1 to 5 are written out in words: one, two, three, four,\nfive, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.\n\nIf all the numbers from 1 to 1000 (one thousand) inclusive were written\nout in words, how many letters would be used?\n\nNOTE: Do not count spaces or hyphens. For example, 342 (three hundred and\nforty-two) contains 23 letters and 115 (one hundred and fifteen) contains\n20 letters. The use of \"and\" when writing out numbers is in compliance\nwith British usage.",
"answer": "21124",
"difficulty": 5,
"name": "Number letter counts",
"id": 17
},
{
"description": "By starting at the top of the triangle below and moving to adjacent\nnumbers on the row below, the maximum total from top to bottom is 23.\n\n 3\n 7 4\n 2 4 6\n 8 5 9 3\n\nThat is, 3 + 7 + 4 + 9 = 23.\n\nFind the maximum total from top to bottom of the triangle below:\n\n 75\n 95 64\n 17 47 82\n 18 35 87 10\n 20 04 82 47 65\n 19 01 23 75 03 34\n 88 02 77 73 07 63 67\n 99 65 04 28 06 16 70 92\n 41 41 26 56 83 40 80 70 33\n 41 48 72 33 47 32 37 16 94 29\n 53 71 44 65 25 43 91 52 97 51 14\n 70 11 33 28 77 73 17 78 39 68 17 57\n 91 71 52 38 17 14 91 43 58 50 27 29 48\n 63 66 04 68 89 53 67 30 73 16 69 87 40 31\n 04 62 98 27 23 09 70 98 73 93 38 53 60 04 23\n\nNOTE: As there are only 16384 routes, it is possible to solve this problem\n by trying every route. However, Problem 67, is the same challenge with\na triangle containing one-hundred rows; it cannot be solved by brute\nforce, and requires a clever method! ;o)",
"answer": "1074",
"difficulty": 5,
"name": "Maximum path sum I",
"id": 18
},
{
"description": "You are given the following information, but you may prefer to do some\nresearch for yourself.\n\n * 1 Jan 1900 was a Monday.\n * Thirty days has September,\n April, June and November.\n All the rest have thirty-one,\n Saving February alone,\n Which has twenty-eight, rain or shine.\n And on leap years, twenty-nine.\n * A leap year occurs on any year evenly divisible by 4, but not on a\n century unless it is divisible by 400.\n\nHow many Sundays fell on the first of the month during the twentieth\ncentury (1 Jan 1901 to 31 Dec 2000)?",
"answer": "171",
"difficulty": 5,
"name": "Counting Sundays",
"id": 19
},
{
"description": "n! means n * (n - 1) * ... * 3 * 2 * 1\n\nFind the sum of the digits in the number 100!",
"answer": "648",
"difficulty": 5,
"name": "Factorial digit sum",
"id": 20
},
{
"description": "Let d(n) be defined as the sum of proper divisors of n (numbers less than\nn which divide evenly into n).\nIf d(a) = b and d(b) = a, where a =/= b, then a and b are an amicable pair\nand each of a and b are called amicable numbers.\n\nFor example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22,\n44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1,\n2, 4, 71 and 142; so d(284) = 220.\n\nEvaluate the sum of all the amicable numbers under 10000.",
"answer": "31626",
"difficulty": 5,
"name": "Amicable numbers",
"id": 21
},
{
"description": "Using names.txt, a 46K text file containing over five-thousand first names,\nbegin by sorting it into alphabetical order. Then working out the\nalphabetical value for each name, multiply this value by its alphabetical\nposition in the list to obtain a name score.\n\nFor example, when the list is sorted into alphabetical order, COLIN, which\nis worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So,\nCOLIN would obtain a score of 938 * 53 = 49714.\n\nWhat is the total of all the name scores in the file?",
"answer": "871198282",
"resources": ["names.txt"],
"difficulty": 5,
"name": "Names scores",
"id": 22
},
{
"description": "A perfect number is a number for which the sum of its proper divisors is\nexactly equal to the number. For example, the sum of the proper divisors\nof 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect\nnumber.\n\nA number whose proper divisors are less than the number is called\ndeficient and a number whose proper divisors exceed the number is called\nabundant.\n\nAs 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the\nsmallest number that can be written as the sum of two abundant numbers is\n24. By mathematical analysis, it can be shown that all integers greater\nthan 28123 can be written as the sum of two abundant numbers. However,\nthis upper limit cannot be reduced any further by analysis even though it\nis known that the greatest number that cannot be expressed as the sum of\ntwo abundant numbers is less than this limit.\n\nFind the sum of all the positive integers which cannot be written as the\nsum of two abundant numbers.",
"answer": "4179871",
"difficulty": 5,
"name": "Non-abundant sums",
"id": 23
},
{
"description": "A permutation is an ordered arrangement of objects. For example, 3124 is\none possible permutation of the digits 1, 2, 3 and 4. If all of the\npermutations are listed numerically or alphabetically, we call it\nlexicographic order. The lexicographic permutations of 0, 1 and 2 are:\n\n 012 021 102 120 201 210\n\nWhat is the millionth lexicographic permutation of the digits 0, 1, 2, 3,\n4, 5, 6, 7, 8 and 9?",
"answer": "2783915460",
"difficulty": 5,
"name": "Lexicographic permutations",
"id": 24
},
{
"description": "The Fibonacci sequence is defined by the recurrence relation:\n\n F[n] = F[n[1]] + F[n[2]], where F[1] = 1 and F[2] = 1.\n\nHence the first 12 terms will be:\n\n F[1] = 1\n F[2] = 1\n F[3] = 2\n F[4] = 3\n F[5] = 5\n F[6] = 8\n F[7] = 13\n F[8] = 21\n F[9] = 34\n F[10] = 55\n F[11] = 89\n F[12] = 144\n\nThe 12th term, F[12], is the first term to contain three digits.\n\nWhat is the first term in the Fibonacci sequence to contain 1000 digits?",
"answer": "4782",
"difficulty": 5,
"name": "1000-digit Fibonacci number",
"id": 25
},
{
"description": "A unit fraction contains 1 in the numerator. The decimal representation of\nthe unit fractions with denominators 2 to 10 are given:\n\n 1/2 = 0.5\n 1/3 = 0.(3)\n 1/4 = 0.25\n 1/5 = 0.2\n 1/6 = 0.1(6)\n 1/7 = 0.(142857)\n 1/8 = 0.125\n 1/9 = 0.(1)\n 1/10 = 0.1\n\nWhere 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can\nbe seen that ^1/[7] has a 6-digit recurring cycle.\n\nFind the value of d < 1000 for which ^1/[d] contains the longest recurring\ncycle in its decimal fraction part.",
"answer": "983",
"difficulty": 5,
"name": "Reciprocal cycles",
"id": 26
},
{
"description": "Euler published the remarkable quadratic formula:\n\n n^2 + n + 41\n\nIt turns out that the formula will produce 40 primes for the consecutive\nvalues n = 0 to 39. However, when n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41\nis divisible by 41, and certainly when n = 41, 41^2 + 41 + 41 is clearly\ndivisible by 41.\n\nUsing computers, the incredible formula n^2 - 79n + 1601 was discovered,\nwhich produces 80 primes for the consecutive values n = 0 to 79. The\nproduct of the coefficients, 79 and 1601, is 126479.\n\nConsidering quadratics of the form:\n\n n^2 + an + b, where |a| < 1000 and |b| < 1000\n\n where |n| is the modulus/absolute value of n\n e.g. |11| = 11 and |-4| = 4\n\nFind the product of the coefficients, a and b, for the quadratic\nexpression that produces the maximum number of primes for consecutive\nvalues of n, starting with n = 0.",
"answer": "-59231",
"difficulty": 5,
"name": "Quadratic primes",
"id": 27
},
{
"description": "Starting with the number 1 and moving to the right in a clockwise\ndirection a 5 by 5 spiral is formed as follows:\n\n 21 22 23 24 25\n 20 7 8 9 10\n 19 6 1 2 11\n 18 5 4 3 12\n 17 16 15 14 13\n\nIt can be verified that the sum of both diagonals is 101.\n\nWhat is the sum of both diagonals in a 1001 by 1001 spiral formed in the\nsame way?",
"answer": "669171001",
"difficulty": 5,
"name": "Number spiral diagonals",
"id": 28
},
{
"description": "Consider all integer combinations of a^b for 2 a 5 and 2 b 5:\n\n 2^2=4, 2^3=8, 2^4=16, 2^5=32\n 3^2=9, 3^3=27, 3^4=81, 3^5=243\n 4^2=16, 4^3=64, 4^4=256, 4^5=1024\n 5^2=25, 5^3=125, 5^4=625, 5^5=3125\n\nIf they are then placed in numerical order, with any repeats removed, we\nget the following sequence of 15 distinct terms:\n\n 4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125\n\nHow many distinct terms are in the sequence generated by a^b for\n2 <= a <= 100 and 2 <= b <= 100?",
"answer": "9183",
"difficulty": 5,
"name": "Distinct powers",
"id": 29
},
{
"description": "Surprisingly there are only three numbers that can be written as the sum\nof fourth powers of their digits:\n\n 1634 = 1^4 + 6^4 + 3^4 + 4^4\n 8208 = 8^4 + 2^4 + 0^4 + 8^4\n 9474 = 9^4 + 4^4 + 7^4 + 4^4\n\nAs 1 = 1^4 is not a sum it is not included.\n\nThe sum of these numbers is 1634 + 8208 + 9474 = 19316.\n\nFind the sum of all the numbers that can be written as the sum of fifth\npowers of their digits.",
"answer": "443839",
"difficulty": 5,
"name": "Digit fifth powers",
"id": 30
},
{
"description": "In England the currency is made up of pound, -L-, and pence, p, and there\nare eight coins in general circulation:\n\n 1p, 2p, 5p, 10p, 20p, 50p, -L-1 (100p) and -L-2 (200p).\n\nIt is possible to make -L-2 in the following way:\n\n 1 * -L-1 + 1 * 50p + 2 * 20p + 1 * 5p + 1 * 2p + 3 * 1p\n\nHow many different ways can -L-2 be made using any number of coins?",
"answer": "73682",
"difficulty": 5,
"name": "Coin sums",
"id": 31
},
{
"description": "We shall say that an n-digit number is pandigital if it makes use of all\nthe digits 1 to n exactly once; for example, the 5-digit number, 15234,\nis 1 through 5 pandigital.\n\nThe product 7254 is unusual, as the identity, 39 * 186 = 7254, containing\nmultiplicand, multiplier, and product is 1 through 9 pandigital.\n\nFind the sum of all products whose multiplicand/multiplier/product\nidentity can be written as a 1 through 9 pandigital.\n\nHINT: Some products can be obtained in more than one way so be sure to\nonly include it once in your sum.",
"answer": "45228",
"difficulty": 5,
"name": "Pandigital products",
"id": 32
},
{
"description": "The fraction 49/98 is a curious fraction, as an inexperienced\nmathematician in attempting to simplify it may incorrectly believe that\n49/98 = 4/8, which is correct, is obtained by cancelling the 9s.\n\nWe shall consider fractions like, 30/50 = 3/5, to be trivial examples.\n\nThere are exactly four non-trivial examples of this type of fraction, less\nthan one in value, and containing two digits in the numerator and\ndenominator.\n\nIf the product of these four fractions is given in its lowest common\nterms, find the value of the denominator.",
"answer": "100",
"difficulty": 5,
"name": "Digit cancelling fractions",
"id": 33
},
{
"description": "145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.\n\nFind the sum of all numbers which are equal to the sum of the factorial of\ntheir digits.\n\nNote: as 1! = 1 and 2! = 2 are not sums they are not included.",
"answer": "40730",
"difficulty": 5,
"name": "Digit factorials",
"id": 34
},
{
"description": "The number, 197, is called a circular prime because all rotations of the\ndigits: 197, 971, and 719, are themselves prime.\n\nThere are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37,\n71, 73, 79, and 97.\n\nHow many circular primes are there below one million?",
"answer": "55",
"difficulty": 5,
"name": "Circular primes",
"id": 35
},
{
"description": "The decimal number, 585 = 1001001001[2] (binary), is palindromic in both\nbases.\n\nFind the sum of all numbers, less than one million, which are palindromic\nin base 10 and base 2.\n\n(Please note that the palindromic number, in either base, may not include\nleading zeros.)",
"answer": "872187",
"difficulty": 5,
"name": "Double-base palindromes",
"id": 36
},
{
"description": "The number 3797 has an interesting property. Being prime itself, it is\npossible to continuously remove digits from left to right, and remain\nprime at each stage: 3797, 797, 97, and 7. Similarly we can work from\nright to left: 3797, 379, 37, and 3.\n\nFind the sum of the only eleven primes that are both truncatable from left\nto right and right to left.\n\nNOTE: 2, 3, 5, and 7 are not considered to be truncatable primes.",
"answer": "748317",
"difficulty": 5,
"name": "Truncatable primes",
"id": 37
},
{
"description": "Take the number 192 and multiply it by each of 1, 2, and 3:\n\n 192 * 1 = 192\n 192 * 2 = 384\n 192 * 3 = 576\n\nBy concatenating each product we get the 1 to 9 pandigital, 192384576. We\nwill call 192384576 the concatenated product of 192 and (1,2,3)\n\nThe same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4,\nand 5, giving the pandigital, 918273645, which is the concatenated product\nof 9 and (1,2,3,4,5).\n\nWhat is the largest 1 to 9 pandigital 9-digit number that can be formed as\nthe concatenated product of an integer with (1,2, ... , n) where n > 1?",
"answer": "932718654",
"difficulty": 5,
"name": "Pandigital multiples",
"id": 38
},
{
"description": "If p is the perimeter of a right angle triangle with integral length\nsides, {a,b,c}, there are exactly three solutions for p = 120.\n\n {20,48,52}, {24,45,51}, {30,40,50}\n\nFor which value of p < 1000, is the number of solutions maximised?",
"answer": "840",
"difficulty": 5,
"name": "Integer right triangles",
"id": 39
},
{
"description": "An irrational decimal fraction is created by concatenating the positive\nintegers:\n\n 0.123456789101112131415161718192021...\n ^\n\nIt can be seen that the 12th digit of the fractional part is 1.\n\nIf d[n] represents the n-th digit of the fractional part, find the value\nof the following expression.\n\n d[1] * d[10] * d[100] * d[1000] * d[10000] * d[100000] * d[1000000]",
"answer": "210",
"difficulty": 5,
"name": "Champernowne's constant",
"id": 40
},
{
"description": "We shall say that an n-digit number is pandigital if it makes use of all\nthe digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital\nand is also prime.\n\nWhat is the largest n-digit pandigital prime that exists?",
"answer": "7652413",
"difficulty": 5,
"name": "Pandigital prime",
"id": 41
},
{
"description": "The n-th term of the sequence of triangle numbers is given by, t[n] =\n1/2n(n+1); so the first ten triangle numbers are:\n\n 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...\n\nBy converting each letter in a word to a number corresponding to its\nalphabetical position and adding these values we form a word value. For\nexample, the word value for SKY is 19 + 11 + 25 = 55 = t[10]. If the word\nvalue is a triangle number then we shall call the word a triangle word.\n\nUsing words.txt, a 16K text file containing nearly two-thousand common\nEnglish words, how many are triangle words?",
"answer": "162",
"resources": ["words.txt"],
"difficulty": 5,
"name": "Coded triangle numbers",
"id": 42
},
{
"description": "The number, 1406357289, is a 0 to 9 pandigital number because it is made\nup of each of the digits 0 to 9 in some order, but it also has a rather\ninteresting sub-string divisibility property.\n\nLet d[1] be the 1st digit, d[2] be the 2nd digit, and so on. In this\nway, we note the following:\n\n * d[2]d[3]d[4]=406 is divisible by 2\n * d[3]d[4]d[5]=063 is divisible by 3\n * d[4]d[5]d[6]=635 is divisible by 5\n * d[5]d[6]d[7]=357 is divisible by 7\n * d[6]d[7]d[8]=572 is divisible by 11\n * d[7]d[8]d[9]=728 is divisible by 13\n * d[8]d[9]d[10]=289 is divisible by 17\n\nFind the sum of all 0 to 9 pandigital numbers with this property.",
"answer": "16695334890",
"difficulty": 5,
"name": "Sub-string divisibility",
"id": 43
},
{
"description": "Pentagonal numbers are generated by the formula, P[n]=n(3n-1)/2. The first\nten pentagonal numbers are:\n\n 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...\n\nIt can be seen that P[4] + P[7] = 22 + 70 = 92 = P[8]. However, their\ndifference, 70 - 22 = 48, is not pentagonal.\n\nFind the pair of pentagonal numbers, P[j] and P[k], for which their sum\nand difference is pentagonal and D = |P[k] - P[j]| is minimised; what is\nthe value of D?",
"answer": "5482660",
"difficulty": 5,
"name": "Pentagon numbers",
"id": 44
},
{
"description": "Triangle, pentagonal, and hexagonal numbers are generated by the following\nformulae:\n\nTriangle T[n]=n(n+1)/2 1, 3, 6, 10, 15, ...\nPentagonal P[n]=n(3n-1)/2 1, 5, 12, 22, 35, ...\nHexagonal H[n]=n(2n-1) 1, 6, 15, 28, 45, ...\n\nIt can be verified that T[285] = P[165] = H[143] = 40755.\n\nFind the next triangle number that is also pentagonal and hexagonal.",
"answer": "1533776805",
"difficulty": 5,
"name": "Triangular, pentagonal, and hexagonal",
"id": 45
},
{
"description": "It was proposed by Christian Goldbach that every odd composite number can\nbe written as the sum of a prime and twice a square.\n\n9 = 7 + 2 * 1^2\n15 = 7 + 2 * 2^2\n21 = 3 + 2 * 3^2\n25 = 7 + 2 * 3^2\n27 = 19 + 2 * 2^2\n33 = 31 + 2 * 1^2\n\nIt turns out that the conjecture was false.\n\nWhat is the smallest odd composite that cannot be written as the sum of a\nprime and twice a square?",
"answer": "5777",
"difficulty": 5,
"name": "Goldbach's other conjecture",
"id": 46
},
{
"description": "The first two consecutive numbers to have two distinct prime factors are:\n\n14 = 2 * 7\n15 = 3 * 5\n\nThe first three consecutive numbers to have three distinct prime factors\nare:\n\n644 = 2^2 * 7 * 23\n645 = 3 * 5 * 43\n646 = 2 * 17 * 19.\n\nFind the first four consecutive integers to have four distinct primes\nfactors. What is the first of these numbers?",
"answer": "134043",
"difficulty": 5,
"name": "Distinct primes factors",
"id": 47
},
{
"description": "The series, 1^1 + 2^2 + 3^3 + ... + 10^10 = 10405071317.\n\nFind the last ten digits of the series, 1^1 + 2^2 + 3^3 + ... + 1000^1000.",
"answer": "9110846700",
"difficulty": 5,
"name": "Self powers",
"id": 48
},
{
"description": "The arithmetic sequence, 1487, 4817, 8147, in which each of the terms\nincreases by 3330, is unusual in two ways: (i) each of the three terms are\nprime, and, (ii) each of the 4-digit numbers are permutations of one\nanother.\n\nThere are no arithmetic sequences made up of three 1-, 2-, or 3-digit\nprimes, exhibiting this property, but there is one other 4-digit\nincreasing sequence.\n\nWhat 12-digit number do you form by concatenating the three terms in this\nsequence?",
"answer": "296962999629",
"difficulty": 5,
"name": "Prime permutations",
"id": 49
},
{
"description": "The prime 41, can be written as the sum of six consecutive primes:\n\n 41 = 2 + 3 + 5 + 7 + 11 + 13\n\nThis is the longest sum of consecutive primes that adds to a prime below\none-hundred.\n\nThe longest sum of consecutive primes below one-thousand that adds to a\nprime, contains 21 terms, and is equal to 953.\n\nWhich prime, below one-million, can be written as the sum of the most\nconsecutive primes?",
"answer": "997651",
"difficulty": 5,
"name": "Consecutive prime sum",
"id": 50
}
]
Purpose-built for displays of data
Observable is your go-to platform for exploring data and creating expressive data visualizations. Use reactive JavaScript notebooks for prototyping and a collaborative canvas for visual data exploration and dashboard creation.
