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Problem 69Totient maximumEuler’s Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For exa

Problem 70Totient permutationEuler’s Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For

Problem 68Magic 5-gon ringConsider 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

Problem 71Ordered fractionsConsider the fraction, n/d, where n and d are positive integers. If n If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we

Problem 74Digit factorial chainsThe number 145 is well known for the property that the sum of the factorial of its digits is equal to 145:1! + 4! + 5! = 1 + 24 + 120 = 145Perhaps l

Problem 73Counting fractions in a rangeConsider the fraction, n/d, where n and d are positive integers. If n If we list the set of reduced proper fractions for d ≤ 8 in ascending order

Problem 76Counting summationsIt is possible to write five as a sum in exactly six different ways:4 + 13 + 23 + 1 + 12 + 2 + 12 + 1 + 1 + 11 + 1 + 1 + 1 + 1How many differ

Problem 75Singular integer right trianglesIt 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 ther

Problem 72Counting fractionsConsider the fraction, n/d, where n and d are positive integers. If n If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, w

Problem 77Prime summationsIt is possible to write ten as the sum of primes in exactly five different ways:7 + 35 + 55 + 3 + 23 + 3 + 2 + 22 + 2 + 2 + 2 + 2What is the first

Problem 78Coin partitionsLet p(n) represent the number of different ways in which n coins can be separated into piles. For example, five coins can separated into piles in exactly seven dif

Problem 79Passcode derivationA common security method used for online banking is to ask the user for three random characters from a passcode. For example, if the passcode was 531278, they

Problem 81Path sum: two waysIn the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and down, is indicated in bold red and is eq

Problem 82Path sum: three waysNOTE: This problem is a more challenging version of Problem 81.The minimal path sum in the 5 by 5 matrix below, by starting in any cell in the left colu

Problem 80Square root digital expansionIt is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is

Problem 83Path sum: four waysNOTE: This problem is a significantly more challenging version of Problem 81.In the 5 by 5 matrix below, the minimal path sum from the top left to the bo

Problem 85Counting rectanglesBy counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles:Although there exists no rectangular grid th

Problem 87Prime power triplesThe smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28. In fact, there are exactly four numbers below fifty tha

Problem 84Monopoly oddsIn the game, Monopoly, the standard board is set up in the following way:           GOA1CC1

Problem 86Cuboid routeA spider, S, sits in one corner of a cuboid room, measuring 6 by 5 by 3, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shor

Diophantine reciprocals IIn the following equation x, y, and n are positive integers.1x+1y=1nFor n = 4 there are exactly three distinct solutions:15+120=1416+112=14

Diophantine reciprocals IIIn the following equation x, y, and n are positive integers.1x+1y=1nFor n = 4 there are exactly three distinct solutions:15+120=1416+112=14

Bouncy numbersWorking 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

Primes with runsConsidering 4-digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by 11, 2222 is divisible by 22, and so on. But there are nine

Non-bouncy numbersWorking 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 di

Problem 88Product-sum numbersA natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, {a1, a2, … , ak} is called a product-sum numb

Problem 89Roman numeralsFor a number written in Roman numerals to be considered valid there are basic rules which must be followed. Even though the rules allow some numbers to be expressed

Red, green, and blue tilesUsing a combination of black square tiles and oblong tiles chosen from: red tiles measuring two units, green tiles measuring three units, and blue tiles measuring four un

Pandigital prime setsUsing all of the digits 1 through 9 and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set {2,5,47,89,631}, all of th

Digit power sumThe number 512 is interesting because it is equal to the sum of its digits raised to some power: 5 + 1 + 2 = 8, and 83 = 512. Another example of a number with this property is 614

Square remaindersLet r be the remainder when (a?1)n + (a+1)n is divided by a2.For example, if a = 7 and n = 3, then r = 42: 63 + 83 = 728 ≡ 42 mod 49. And as n varies, so too will r, but f

Disc game prize fundA bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its colour is noted. After each turn the disc is returned to the bag, an

Problem 90Cube digit pairsEach of the six faces on a cube has a different digit (0 to 9) written on it; the same is done to a second cube. By placing the two cubes side-by-side in differen

Problem 91Right triangles with integer coordinatesThe points P (x1, y1) and Q (x2, y2) are plotted at integer co-ordinates and are joined to the origin, O(0,0), to form ΔOPQ.Ther

Problem 93Arithmetic expressionsBy using each of the digits from the set, {1, 2, 3, 4}, exactly once, and making use of the four arithmetic operations (+, −, *, /) and brackets/parentheses

Problem 43Sub-string divisibilityThe number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interestin

Problem 49Prime permutationsThe arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii

Problem 92Square digit chainsA 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

Sum square differenceThe sum of the squares of the first ten natural numbers is,12 + 22 + … + 102 = 385The square of the sum of the first ten natural numbers is,(1 + 2 + … + 10)2 = 5

Large sumWork out the first ten digits of the sum of the following one-hundred 50-digit numbers.37107287533902102798797998220837590246510135740250463769376774900097126481248969700780504170