Project Euler

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↴ Project Euler

Problem 431: Square Space Silo

Problem 447: Retractions C

Problem 448: Average least common multiple

Problem 85: Counting rectangles

Problem 132: Large repunit factors

Problem 172: Investigating numbers with few repeated digits

Problem 17: Number letter counts

Problem 322: Binomial coefficients divisible by 10

Problem 56: Powerful digit sum

Problem 62: Cubic permutations

Problem 63: Powerful digit counts

Problem 77: Prime summations

Problem 356: Largest roots of cubic polynomials

Problem 455: Powers With Trailing Digits

Problem 1: Multiples of 3 and 5

Problem 2: Even Fibonacci Numbers

Problem 7: 10001st prime

Problem 3: Largest prime factor

Problem 4: Largest palindrome product

Problem 5: Smallest multiple

Problem 6: Sum square difference

Problem 8: Largest product in a series

Problem 9: Special Pythagorean triplet

Problem 10: Summation of primes

Problem 11: Largest product in a grid

Problem 272: Modular Cubes, part 2

Problem 12: Highly divisible triangular number

Problem 13: Large sum

Problem 14: Longest Collatz sequence

Problem 15: Lattice paths

Problem 16: Power digit sum

Problem 18: Maximum path sum I

Problem 19: Counting Sundays

Problem 20: Factorial digit sum

Problem 21: Amicable numbers

Problem 78: Coin partitions

Problem 23: Non-abundant sums

Problem 24: Lexicographic permutations

Problem 25: 1000-digit Fibonacci number

Problem 26: Reciprocal cycles

Problem 27: Quadratic primes

Problem 28: Number spiral diagonals

Problem 29: Distinct powers

Problem 30: Digit n powers

Problem 31: Coin sums

Problem 32: Pandigital products

Problem 33: Digit cancelling fractions

Problem 34: Digit factorials

Problem 100: Arranged probability

Problem 35: Circular primes

Problem 36: Double-base palindromes

Problem 37: Truncatable primes

Problem 38: Pandigital multiples

Problem 39: Integer right triangles

Problem 40: Champernowne's constant

Problem 41: Pandigital prime

Problem 42: Coded triangle numbers

Problem 43: Sub-string divisibility

Problem 44: Pentagon numbers

Problem 45: Triangular, pentagonal, and hexagonal

Problem 46: Goldbach's other conjecture

Problem 47: Distinct primes factors

Problem 48: Self powers

Problem 49: Prime permutations

Problem 50: Consecutive prime sum

Problem 51: Prime digit replacements

Problem 52: Permuted multiples

Problem 53: Combinatoric selections

Problem 54: Poker hands

Problem 55: Lychrel numbers

Problem 57: Square root convergents

Problem 58: Spiral primes

Problem 59: XOR decryption

Problem 60: Prime pair sets

Problem 61: Cyclical figurate numbers

Problem 64: Odd period square roots

Problem 65: Convergents of e

Problem 66: Diophantine equation

Problem 67: Maximum path sum II

Problem 210: Obtuse Angled Triangles

Problem 68: Magic 5-gon ring

Problem 69: Totient maximum

Problem 70: Totient permutation

Problem 71: Ordered fractions

Problem 79: Passcode derivation

Problem 72: Counting fractions

Problem 73: Counting fractions in a range

Problem 74: Digit factorial chains

Problem 75: Singular integer right triangles

Problem 76: Counting summations

Problem 80: Square root digital expansion

Problem 81: Path sum: two ways

Problem 82: Path sum: three ways

Problem 83: Path sum: four ways

Problem 84: Monopoly odds

Problem 86: Cuboid route

Problem 88: Product-sum numbers

Problem 89: Roman numerals

Problem 87: Prime power triples

Problem 90: Cube digit pairs

Problem 91: Right triangles with integer coordinates

Problem 92: Square digit chains

Problem 93: Arithmetic expressions

Problem 94: Almost equilateral triangles

Problem 110: Diophantine Reciprocals II

Problem 95: Amicable chains

Problem 96: Su Doku

Problem 97: Large non-Mersenne prime

Problem 98: Anagramic squares

Problem 99: Largest exponential

Problem 101: Optimum polynomial

Problem 102: Triangle containment

Problem 104: Pandigital Fibonacci ends

Problem 103: Special subset sums: optimum

Problem 224: Almost right-angled triangles II

Problem 105: Special subset sums: testing

Problem 106: Special subset sums: meta-testing

Problem 107: Minimal network

Problem 108: Diophantine Reciprocals I

Problem 109: Darts

Problem 225: Tribonacci non-divisors

Problem 111: Primes with runs

Problem 112: Bouncy numbers

Problem 113: Non-bouncy numbers

Problem 115: Counting block combinations II

Problem 114: Counting block combinations I

Problem 116: Red, green or blue tiles

Problem 117: Red, green, and blue tiles

Problem 118: Pandigital prime sets

Problem 119: Digit power sum

Problem 120: Square remainders

Problem 126: Cuboid layers

Problem 121: Disc game prize fund

Problem 122: Efficient exponentiation

Problem 123: Prime square remainders

Problem 124: Ordered radicals

Problem 125: Palindromic sums

Problem 127: abc-hits

Problem 128: Hexagonal tile differences

Problem 130: Composites with prime repunit property

Problem 129: Repunit divisibility

Problem 131: Prime cube partnership

Problem 133: Repunit nonfactors

Problem 134: Prime pair connection

Problem 135: Same differences

Problem 136: Singleton difference

Problem 137: Fibonacci golden nuggets

Problem 138: Special isosceles triangles

Problem 139: Pythagorean tiles

Problem 141: Investigating progressive numbers, n, which are also square

Problem 140: Modified Fibonacci golden nuggets

Problem 142: Perfect Square Collection

Problem 143: Investigating the Torricelli point of a triangle

Problem 144: Investigating multiple reflections of a laser beam

Problem 145: How many reversible numbers are there below one-billion?

Problem 146: Investigating a Prime Pattern

Problem 156: Counting Digits

Problem 147: Rectangles in cross-hatched grids

Problem 149: Searching for a maximum-sum subsequence

Problem 148: Exploring Pascal's triangle

Problem 150: Searching a triangular array for a sub-triangle having minimum-sum

Problem 151: Paper sheets of standard sizes: an expected-value problem

Problem 152: Writing one half as a sum of inverse squares

Problem 153: Investigating Gaussian Integers

Problem 154: Exploring Pascal's pyramid

Problem 155: Counting Capacitor Circuits

Problem 157: Solving the diophantine equation

Problem 158: Exploring strings for which only one character comes lexicographically after its neighbour to the left

Problem 159: Digital root sums of factorisations

Problem 160: Factorial trailing digits

Problem 161: Triominoes

Problem 162: Hexadecimal numbers

Problem 163: Cross-hatched triangles

Problem 164: Numbers for which no three consecutive digits have a sum greater than a given value

Problem 165: Intersections

Problem 166: Criss Cross

Problem 167: Investigating Ulam sequences

Problem 168: Number Rotations

Problem 169: Exploring the number of different ways a number can be expressed as a sum of powers of 2

Problem 170: Find the largest 0 to 9 pandigital that can be formed by concatenating products

Problem 171: Finding numbers for which the sum of the squares of the digits is a square

Problem 173: Using up to one million tiles how many different "hollow" square laminae can be formed?

Problem 174: Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements

Problem 175: Fractions involving the number of different ways a number can be expressed as a sum of powers of 2

Problem 176: Right-angled triangles that share a cathetus

Problem 177: Integer angled Quadrilaterals

Problem 178: Step Numbers

Problem 179: Consecutive positive divisors

Problem 180: Rational zeros of a function of three variables

Problem 181: Investigating in how many ways objects of two different colours can be grouped

Problem 357: Prime generating integers

Problem 182: RSA encryption

Problem 183: Maximum product of parts

Problem 184: Triangles containing the origin

Problem 185: Number Mind

Problem 373: Circumscribed Circles

Problem 186: Connectedness of a network

Problem 187: Semiprimes

Problem 188: The hyperexponentiation of a number

Problem 189: Tri-colouring a triangular grid

Problem 190: Maximising a weighted product

Problem 191: Prize Strings

Problem 192: Best Approximations

Problem 194: Coloured Configurations

Problem 193: Squarefree Numbers

Problem 196: Prime triplets

Problem 195: Inscribed circles of triangles with one angle of 60 degrees

Problem 197: Investigating the behaviour of a recursively defined sequence

Problem 198: Ambiguous Numbers

Problem 199: Iterative Circle Packing

Problem 266: Pseudo Square Root

Problem 200: Find the 200th prime-proof sqube containing the contiguous sub-string "200"

Problem 201: Subsets with a unique sum

Problem 202: Laserbeam

Problem 203: Squarefree Binomial Coefficients

Problem 204: Generalised Hamming Numbers

Problem 205: Dice Game

Problem 206: Concealed Square

Problem 207: Integer partition equations

Problem 209: Circular Logic

Problem 208: Robot Walks

Problem 211: Divisor Square Sum

Problem 212: Combined Volume of Cuboids

Problem 213: Flea Circus

Problem 214: Totient Chains

Problem 215: Crack-free Walls

Problem 216: Investigating the primality of numbers of the form 2n2-1

Problem 217: Balanced Numbers

Problem 218: Perfect right-angled triangles

Problem 230: Fibonacci Words

Problem 219: Skew-cost coding

Problem 220: Heighway Dragon

Problem 221: Alexandrian Integers

Problem 222: Sphere Packing

Problem 223: Almost right-angled triangles I

Problem 227: The Chase

Problem 226: A Scoop of Blancmange

Problem 228: Minkowski Sums

Problem 229: Four Representations using Squares

Problem 231: The prime factorisation of binomial coefficients

Problem 232: The Race

Problem 233: Lattice points on a circle

Problem 234: Semidivisible numbers

Problem 235: An Arithmetic Geometric sequence

Problem 236: Luxury Hampers

Problem 237: Tours on a 4 x n playing board

Problem 238: Infinite string tour

Problem 239: Twenty-two Foolish Primes

Problem 240: Top Dice

Problem 241: Perfection Quotients

Problem 242: Odd Triplets

Problem 244: Sliders

Problem 243: Resilience

Problem 245: Coresilience

Problem 246: Tangents to an ellipse

Problem 247: Squares under a hyperbola

Problem 248: Numbers for which Euler’s totient function equals 13!

Problem 250: 250250

Problem 249: Prime Subset Sums

Problem 251: Cardano Triplets

Problem 252: Convex Holes

Problem 253: Tidying up

Problem 254: Sums of Digit Factorials

Problem 256: Tatami-Free Rooms

Problem 255: Rounded Square Roots

Problem 257: Angular Bisectors

Problem 258: A lagged Fibonacci sequence

Problem 259: Reachable Numbers

Problem 260: Stone Game

Problem 261: Pivotal Square Sums

Problem 262: Mountain Range

Problem 263: An engineers' dream come true

Problem 264: Triangle Centres

Problem 265: Binary Circles

Problem 267: Billionaire

Problem 268: Counting numbers with at least four distinct prime factors less than 100

Problem 269: Polynomials with at least one integer root

Problem 271: Modular Cubes, part 1

Problem 270: Cutting Squares

Problem 273: Sum of Squares

Problem 274: Divisibility Multipliers

Problem 275: Balanced Sculptures

Problem 276: Primitive Triangles

Problem 277: A Modified Collatz sequence

Problem 278: Linear Combinations of Semiprimes

Problem 279: Triangles with integral sides and an integral angle

Problem 280: Ant and seeds

Problem 281: Pizza Toppings

Problem 282: The Ackermann function

Problem 283: Integer sided triangles for which the area * perimeter ratio is integral

Problem 284: Steady Squares

Problem 285: Pythagorean odds

Problem 286: Scoring probabilities

Problem 287: Quadtree encoding (a simple compression algorithm)

Problem 288: An enormous factorial

Problem 289: Eulerian Cycles

Problem 290: Digital Signature

Problem 291: Panaitopol Primes

Problem 292: Pythagorean Polygons

Problem 375: Minimum of subsequences

Problem 293: Pseudo-Fortunate Numbers

Problem 294: Sum of digits - experience #23

Problem 295: Lenticular holes

Problem 296: Angular Bisector and Tangent

Problem 297: Zeckendorf Representation

Problem 298: Selective Amnesia

Problem 299: Three similar triangles

Problem 301: Nim

Problem 300: Protein folding

Problem 302: Strong Achilles Numbers

Problem 303: Multiples with small digits

Problem 304: Primonacci

Problem 305: Reflexive Position

Problem 306: Paper-strip Game

Problem 405: A rectangular tiling

Problem 307: Chip Defects

Problem 308: An amazing Prime-generating Automaton

Problem 309: Integer Ladders

Problem 310: Nim Square

Problem 343: Fractional Sequences

Problem 311: Biclinic Integral Quadrilaterals

Problem 312: Cyclic paths on Sierpiński graphs

Problem 313: Sliding game

Problem 314: The Mouse on the Moon

Problem 315: Digital root clocks

Problem 321: Swapping Counters

Problem 316: Numbers in decimal expansions

Problem 317: Firecracker

Problem 318: 2011 nines

Problem 319: Bounded Sequences

Problem 320: Factorials divisible by a huge integer

Problem 323: Bitwise-OR operations on random integers

Problem 324: Building a tower

Problem 325: Stone Game II

Problem 326: Modulo Summations

Problem 327: Rooms of Doom

Problem 328: Lowest-cost Search

Problem 329: Prime Frog

Problem 330: Euler's Number

Problem 331: Cross flips

Problem 332: Spherical triangles

Problem 333: Special partitions

Problem 334: Spilling the beans

Problem 335: Gathering the beans

Problem 336: Maximix Arrangements

Problem 337: Totient Stairstep Sequences

Problem 348: Sum of a square and a cube

Problem 338: Cutting Rectangular Grid Paper

Problem 339: Peredur fab Efrawg

Problem 341: Golomb's self-describing sequence

Problem 340: Crazy Function

Problem 342: The totient of a square is a cube

Problem 354: Distances in a bee's honeycomb

Problem 344: Silver dollar game

Problem 346: Strong Repunits

Problem 347: Largest integer divisible by two primes

Problem 345: Matrix Sum

Problem 355: Maximal coprime subset

Problem 349: Langton's ant

Problem 350: Constraining the least greatest and the greatest least

Problem 351: Hexagonal orchards

Problem 352: Blood tests

Problem 353: Risky moon

Problem 358: Cyclic numbers

Problem 359: Hilbert's New Hotel

Problem 360: Scary Sphere

Problem 361: Subsequence of Thue-Morse sequence

Problem 362: Squarefree factors

Problem 364: Comfortable distance

Problem 363: Bézier Curves

Problem 365: A huge binomial coefficient

Problem 366: Stone Game III

Problem 367: Bozo sort

Problem 368: A Kempner-like series

Problem 369: Badugi

Problem 370: Geometric triangles

Problem 371: Licence plates

Problem 372: Pencils of rays

Problem 374: Maximum Integer Partition Product

Problem 376: Nontransitive sets of dice

Problem 377: Sum of digits, experience 13

Problem 378: Triangle Triples

Problem 379: Least common multiple count

Problem 380: Amazing Mazes!

Problem 381: (prime-k) factorial

Problem 382: Generating polygons

Problem 384: Rudin-Shapiro sequence

Problem 383: Divisibility comparison between factorials

Problem 385: Ellipses inside triangles

Problem 386: Maximum length of an antichain

Problem 388: Distinct Lines

Problem 387: Harshad Numbers

Problem 389: Platonic Dice

Problem 433: Steps in Euclid's algorithm

Problem 390: Triangles with non rational sides and integral area

Problem 391: Hopping Game

Problem 392: Enmeshed unit circle

Problem 393: Migrating ants

Problem 394: Eating pie

Problem 395: Pythagorean tree

Problem 396: Weak Goodstein sequence

Problem 397: Triangle on parabola

Problem 398: Cutting rope

Problem 399: Squarefree Fibonacci Numbers

Problem 401: Sum of squares of divisors

Problem 400: Fibonacci tree game

Problem 402: Integer-valued polynomials

Problem 404: Crisscross Ellipses

Problem 403: Lattice points enclosed by parabola and line

Problem 406: Guessing Game

Problem 407: Idempotents

Problem 409: Nim Extreme

Problem 410: Circle and tangent line

Problem 411: Uphill paths

Problem 432: Totient sum

Problem 412: Gnomon numbering

Problem 408: Admissible paths through a grid

Problem 413: One-child Numbers

Problem 414: Kaprekar constant

Problem 415: Titanic sets

Problem 417: Reciprocal cycles II

Problem 416: A frog's trip

Problem 418: Factorisation triples

Problem 419: Look and say sequence

Problem 445: Retractions A

Problem 420: 2x2 positive integer matrix

Problem 421: Prime factors of n15+1

Problem 422: Sequence of points on a hyperbola

Problem 424: Kakuro

Problem 423: Consecutive die throws

Problem 430: Range flips

Problem 425: Prime connection

Problem 426: Box-ball system

Problem 427: n-sequences

Problem 428: Necklace of Circles

Problem 429: Sum of squares of unitary divisors

Problem 434: Rigid graphs

Problem 435: Polynomials of Fibonacci numbers

Problem 436: Unfair wager

Problem 438: Integer part of polynomial equation's solutions

Problem 439: Sum of sum of divisors

Problem 446: Retractions B

Problem 437: Fibonacci primitive roots

Problem 440: GCD and Tiling

Problem 441: The inverse summation of coprime couples

Problem 442: Eleven-free integers

Problem 443: GCD sequence

Problem 444: The Roundtable Lottery

Problem 449: Chocolate covered candy

Problem 450: Hypocycloid and Lattice points

Problem 451: Modular inverses

Problem 452: Long Products

Problem 453: Lattice Quadrilaterals

Problem 454: Diophantine reciprocals III

Problem 456: Triangles containing the origin II

Problem 457: A polynomial modulo the square of a prime

Problem 458: Permutations of Project

Problem 459: Flipping game

Problem 460: An ant on the move

Problem 461: Almost Pi

Problem 462: Permutation of 3-smooth numbers

Problem 463: A weird recurrence relation

Problem 464: Möbius function and intervals

Problem 465: Polar polygons

Problem 466: Distinct terms in a multiplication table

Problem 467: Superinteger

Problem 469: Empty chairs

Problem 468: Smooth divisors of binomial coefficients

Problem 470: Super Ramvok

Problem 476: Circle Packing II

Problem 471: Triangle inscribed in ellipse

Problem 472: Comfortable Distance II

Problem 473: Phigital number base

Problem 474: Last digits of divisors

Problem 475: Music festival

Problem 477: Number Sequence Game

Problem 479: Roots on the Rise

Problem 478: Mixtures

Problem 480: The Last Question

Problem 22: Names scores