Euler in Babylon

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter

Divisibility of factorials

Problem 549

Gozinta Chains

Problem 548

Distance of random points within hollow square laminae

Problem 547

The Floor's Revenge

Problem 546

Faulhaber's Formulas

Problem 545

Chromatic Conundrum

Problem 544

Prime-Sum Numbers

Problem 543

Geometric Progression with Maximum Sum

Problem 542

Divisibility of Harmonic Number Denominators

Problem 541

Counting primitive Pythagorean triples

Problem 540

Odd elimination

Problem 539

Maximum quadrilaterals

Problem 538

Counting tuples

Problem 537

Modulo power identity

Problem 536

Fractal Sequence

Problem 535

Weak Queens

Problem 534

Minimum values of the Carmichael function

Problem 533

Nanobots on Geodesics

Problem 532

Chinese leftovers

Problem 531

GCD of Divisors

Problem 530

10-substrings

Problem 529

Constrained Sums

Problem 528

Randomized Binary Search

Problem 527

Largest prime factors of consecutive numbers

Problem 526

Rolling Ellipse

Problem 525

First Sort I

Problem 523

First Sort II

Problem 524

Hilbert's Blackout

Problem 522

Smallest prime factor

Problem 521

Simbers

Problem 520

Tricoloured Coin Fountains

Problem 519

Prime triples and geometric sequences

Problem 518

A real recursion

Problem 517

5-smooth totients

Problem 516

Dissonant Numbers

Problem 515

Geoboard Shapes

Problem 514

Integral median

Problem 513

Sums of totients of powers

Problem 512

Sequences with nice divisibility properties

Problem 511

Tangent Circles

Problem 510

Divisor Nim

Problem 509

Integers in base i-1

Problem 508

Shortest Lattice Vector

Problem 507

Clock sequence

Problem 506

Bidirectional Recurrence

Problem 505

Square on the Inside

Problem 504

Compromise or persist

Problem 503

Counting Castles

Problem 502

Eight Divisors

Problem 501

Problem 500!!!

Problem 500

St. Petersburg Lottery

Problem 499

Remainder of polynomial division

Problem 498

Drunken Tower of Hanoi

Problem 497

Incenter and circumcenter of triangle

Problem 496

Writing n as the product of k distinct positive integers

Problem 495

Collatz prefix families

Problem 494

Under The Rainbow

Problem 493

Exploding sequence

Problem 492

Double pandigital number divisible by 11

Problem 491

Jumping frog

Problem 490

Common factors between two sequences

Problem 489

Unbalanced Nim

Problem 488

Sums of power sums

Problem 487

Palindrome-containing strings

Problem 486

Maximum number of divisors

Problem 485

Arithmetic Derivative

Problem 484

Repeated permutation

Problem 483

The incenter of a triangle

Problem 482

Chef Showdown

Problem 481

The Last Question

Problem 480

Roots on the Rise

Problem 479

Mixtures

Problem 478

Number Sequence Game

Problem 477

Circle Packing II

Problem 476

Music festival

Problem 475

Last digits of divisors

Problem 474

Phigital number base

Problem 473

Comfortable Distance II

Problem 472

Triangle inscribed in ellipse

Problem 471

Super Ramvok

Problem 470

Empty chairs

Problem 469

Smooth divisors of binomial coefficients

Problem 468

Superinteger

Problem 467

Distinct terms in a multiplication table

Problem 466

Polar polygons

Problem 465

Möbius function and intervals

Problem 464

A weird recurrence relation

Problem 463

Permutation of 3-smooth numbers

Problem 462

Almost Pi

Problem 461

An ant on the move

Problem 460

Flipping game

Problem 459

Permutations of Project

Problem 458

A polynomial modulo the square of a prime

Problem 457

Triangles containing the origin II

Problem 456

Powers With Trailing Digits

Problem 455

Diophantine reciprocals III

Problem 454

Lattice Quadrilaterals

Problem 453

Long Products

Problem 452

Modular inverses

Problem 451

Hypocycloid and Lattice points

Problem 450

Chocolate covered candy

Problem 449

Average least common multiple

Problem 448

Retractions A

Problem 445

Retractions B

Problem 446

Retractions C

Problem 447

The Roundtable Lottery

Problem 444

GCD sequence

Problem 443

Eleven-free integers

Problem 442

The inverse summation of coprime couples

Problem 441

GCD and Tiling

Problem 440

Sum of sum of divisors

Problem 439

Integer part of polynomial equation's solutions

Problem 438

Fibonacci primitive roots

Problem 437

Unfair wager

Problem 436

Polynomials of Fibonacci numbers

Problem 435

Rigid graphs

Problem 434

Steps in Euclid's algorithm

Problem 433

Totient sum

Problem 432

Square Space Silo

Problem 431

Range flips

Problem 430

Sum of squares of unitary divisors

Problem 429

Necklace of circles

Problem 428

n-sequences

Problem 427

Box-ball system

Problem 426

Prime connection

Problem 425

Kakuro

Problem 424

Consecutive die throws

Problem 423

Sequence of points on a hyperbola

Problem 422

Prime factors of n15+1

Problem 421

2x2 positive integer matrix

Problem 420

Look and say sequence

Problem 419

Factorisation triples

Problem 418

Reciprocal cycles II

Problem 417

A frog's trip

Problem 416

Titanic sets

Problem 415

Kaprekar constant

Problem 414

One-child Numbers

Problem 413

Gnomon numbering

Problem 412

Uphill paths

Problem 411

Circle and tangent line

Problem 410

Nim Extreme

Problem 409

Admissible paths through a grid

Problem 408

Idempotents

Problem 407

Guessing Game

Problem 406

A rectangular tiling

Problem 405

Crisscross Ellipses

Problem 404

Lattice points enclosed by parabola and line

Problem 403

Integer-valued polynomials

Problem 402

Sum of squares of divisors

Problem 401

Fibonacci tree game

Problem 400

Squarefree Fibonacci Numbers

Problem 399

Cutting rope

Problem 398

Triangle on parabola

Problem 397

Weak Goodstein sequence

Problem 396

Pythagorean tree

Problem 395

Eating pie

Problem 394

Migrating ants

Problem 393

Enmeshed unit circle

Problem 392

Hopping Game

Problem 391

Triangles with non rational sides and integral area

Problem 390

Platonic Dice

Problem 389

Distinct Lines

Problem 388

Harshad Numbers

Problem 387

Maximum length of an antichain

Problem 386

Ellipses inside triangles

Problem 385

Rudin-Shapiro sequence

Problem 384

Divisibility comparison between factorials

Problem 383

Generating polygons

Problem 382

(prime-k) factorial

Problem 381

Amazing Mazes!

Problem 380

Least common multiple count

Problem 379

Triangle Triples

Problem 378

Sum of digits, experience 13

Problem 377

Nontransitive sets of dice

Problem 376

Minimum of subsequences

Problem 375

Maximum Integer Partition Product

Problem 374

Circumscribed Circles

Problem 373

Pencils of rays

Problem 372

Licence plates

Problem 371

Geometric triangles

Problem 370

Badugi

Problem 369

A Kempner-like series

Problem 368

Bozo sort

Problem 367

Stone Game III

Problem 366

A huge binomial coefficient

Problem 365

Comfortable distance

Problem 364

Bézier Curves

Problem 363

Squarefree factors

Problem 362

Subsequence of Thue-Morse sequence

Problem 361

Scary Sphere

Problem 360

Hilbert's New Hotel

Problem 359

Cyclic numbers

Problem 358

Prime generating integers

Problem 357

Largest roots of cubic polynomials

Problem 356

Maximal coprime subset

Problem 355

Distances in a bee's honeycomb

Problem 354

Risky moon

Problem 353

Blood tests

Problem 352

Hexagonal orchards

Problem 351

Constraining the least greatest and the greatest least

Problem 350

Langton's ant

Problem 349

Largest integer divisible by two primes

Problem 347

Matrix Sum

Problem 345

Strong Repunits

Problem 346

Sum of a square and a cube

Problem 348

Silver dollar game

Problem 344

Fractional Sequences

Problem 343

The totient of a square is a cube

Problem 342

Golomb's self-describing sequence

Problem 341

Crazy Function

Problem 340

Peredur fab Efrawg

Problem 339

Cutting Rectangular Grid Paper

Problem 338

Totient Stairstep Sequences

Problem 337

Maximix Arrangements

Problem 336

Gathering the beans

Problem 335

Spilling the beans

Problem 334

Special partitions

Problem 333

Spherical triangles

Problem 332

Cross flips

Problem 331

Euler's Number

Problem 330

Prime Frog

Problem 329

Lowest-cost Search

Problem 328

Rooms of Doom

Problem 327

Modulo Summations

Problem 326

Stone Game II

Problem 325

Building a tower

Problem 324

Bitwise-OR operations on random integers

Problem 323

Binomial coefficients divisible by 10

Problem 322

Swapping Counters

Problem 321

Factorials divisible by a huge integer

Problem 320

Bounded Sequences

Problem 319

2011 nines

Problem 318

Firecracker

Problem 317

Numbers in decimal expansions

Problem 316

Digital root clocks

Problem 315

The Mouse on the Moon

Problem 314

Sliding game

Problem 313

Cyclic paths on Sierpiński graphs

Problem 312

Biclinic Integral Quadrilaterals

Problem 311

Nim Square

Problem 310

Integer Ladders

Problem 309

An amazing Prime-generating Automaton

Problem 308

Chip Defects

Problem 307

Paper-strip Game

Problem 306

Reflexive Position

Problem 305

Primonacci

Problem 304

Multiples with small digits

Problem 303

Strong Achilles Numbers

Problem 302

Nim

Problem 301

Protein folding

Problem 300

Three similar triangles

Problem 299

Selective Amnesia

Problem 298

Zeckendorf Representation

Problem 297

Angular Bisector and Tangent

Problem 296

Lenticular holes

Problem 295

Sum of digits - experience #23

Problem 294

Pseudo-Fortunate Numbers

Problem 293

Pythagorean Polygons

Problem 292

Panaitopol Primes

Problem 291

Digital Signature

Problem 290

Eulerian Cycles

Problem 289

An enormous factorial

Problem 288

Quadtree encoding (a simple compression algorithm)

Problem 287

Pythagorean odds

Problem 285

Scoring probabilities

Problem 286

Steady Squares

Problem 284

Integer sided triangles for which the area/perimeter ratio is integral

Problem 283

The Ackermann function

Problem 282

Pizza Toppings

Problem 281

Ant and seeds

Problem 280

Triangles with integral sides and an integral angle

Problem 279

Linear Combinations of Semiprimes

Problem 278

A Modified Collatz sequence

Problem 277

Primitive Triangles

Problem 276

Balanced Sculptures

Problem 275

Divisibility Multipliers

Problem 274

Sum of Squares

Problem 273

Modular Cubes, part 1

Problem 271

Modular Cubes, part 2

Problem 272

Cutting Squares

Problem 270

Polynomials with at least one integer root

Problem 269

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

Problem 268

Billionaire

Problem 267

Pseudo Square Root

Problem 266

Binary Circles

Problem 265

Triangle Centres

Problem 264

An engineers' dream come true

Problem 263

Mountain Range

Problem 262

Pivotal Square Sums

Problem 261

Stone Game

Problem 260

Reachable Numbers

Problem 259

A lagged Fibonacci sequence

Problem 258

Angular Bisectors

Problem 257

Tatami-Free Rooms

Problem 256

Rounded Square Roots

Problem 255

Sums of Digit Factorials

Problem 254

Tidying up

Problem 253

Convex Holes

Problem 252

Cardano Triplets

Problem 251

250250

Problem 250

Prime Subset Sums

Problem 249

Numbers for which Euler’s totient function equals 13!

Problem 248

Squares under a hyperbola

Problem 247

Tangents to an ellipse

Problem 246

Coresilience

Problem 245

Sliders

Problem 244

Resilience

Problem 243

Odd Triplets

Problem 242

Perfection Quotients

Problem 241

Top Dice

Problem 240

Twenty-two Foolish Primes

Problem 239

Infinite string tour

Problem 238

Tours on a 4 x n playing board

Problem 237

Luxury Hampers

Problem 236

An Arithmetic Geometric sequence

Problem 235

Semidivisible numbers

Problem 234

Lattice points on a circle

Problem 233

The Race

Problem 232

The prime factorisation of binomial coefficients

Problem 231

Fibonacci Words

Problem 230

Four Representations using Squares

Problem 229

Minkowski Sums

Problem 228

The Chase

Problem 227

A Scoop of Blancmange

Problem 226

Almost right-angled triangles I

Problem 223

Almost right-angled triangles II

Problem 224

Tribonacci non-divisors

Problem 225

Sphere Packing

Problem 222

Alexandrian Integers

Problem 221

Heighway Dragon

Problem 220

Skew-cost coding

Problem 219

Perfect right-angled triangles

Problem 218

Balanced Numbers

Problem 217

Investigating the primality of numbers of the form 2n2-1

Problem 216

Crack-free Walls

Problem 215

Totient Chains

Problem 214

Flea Circus

Problem 213

Combined Volume of Cuboids

Problem 212

Divisor Square Sum

Problem 211

Obtuse Angled Triangles

Problem 210

Circular Logic

Problem 209

Robot Walks

Problem 208

Concealed Square

Problem 206

Dice Game

Problem 205

Generalised Hamming Numbers

Problem 204

Integer partition equations

Problem 207

Squarefree Binomial Coefficients

Problem 203

Laserbeam

Problem 202

Subsets with a unique sum

Problem 201

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

Problem 200

Iterative Circle Packing

Problem 199

Ambiguous Numbers

Problem 198

Investigating the behaviour of a recursively defined sequence

Problem 197

Prime triplets

Problem 196

Inscribed circles of triangles with one angle of 60 degrees

Problem 195

Coloured Configurations

Problem 194

Squarefree Numbers

Problem 193

Best Approximations

Problem 192

Prize Strings

Problem 191

Maximising a weighted product

Problem 190

Tri-colouring a triangular grid

Problem 189

The hyperexponentiation of a number

Problem 188

Semiprimes

Problem 187

Connectedness of a network

Problem 186

Number Mind

Problem 185

Triangles containing the origin

Problem 184

Maximum product of parts

Problem 183

RSA encryption

Problem 182

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

Problem 181

Rational zeros of a function of three variables

Problem 180

Consecutive positive divisors

Problem 179

Step Numbers

Problem 178

Integer angled Quadrilaterals

Problem 177

Right-angled triangles that share a cathetus

Problem 176

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

Problem 175

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

Problem 174

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

Problem 173

Investigating numbers with few repeated digits

Problem 172

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

Problem 171

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

Problem 170

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

Problem 169

Number Rotations

Problem 168

Investigating Ulam sequences

Problem 167

Criss Cross

Problem 166

Intersections

Problem 165

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

Problem 164

Cross-hatched triangles

Problem 163

Hexadecimal numbers

Problem 162

Triominoes

Problem 161

Factorial trailing digits

Problem 160

Digital root sums of factorisations

Problem 159

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

Problem 158

Solving the diophantine equation 1/a+1/b= p/10n

Problem 157

Counting Digits

Problem 156

Counting Capacitor Circuits

Problem 155

Exploring Pascal's pyramid

Problem 154

Investigating Gaussian Integers

Problem 153

Writing 1/2 as a sum of inverse squares

Problem 152

Paper sheets of standard sizes: an expected-value problem

Problem 151

Searching a triangular array for a sub-triangle having minimum-sum

Problem 150

Searching for a maximum-sum subsequence

Problem 149

Exploring Pascal's triangle

Problem 148

Rectangles in cross-hatched grids

Problem 147

Investigating a Prime Pattern

Problem 146

How many reversible numbers are there below one-billion?

Problem 145

Investigating multiple reflections of a laser beam

Problem 144

Investigating the Torricelli point of a triangle

Problem 143

Perfect Square Collection

Problem 142

Investigating progressive numbers, n, which are also square

Problem 141

Modified Fibonacci golden nuggets

Problem 140

Pythagorean tiles

Problem 139

Special isosceles triangles

Problem 138

Fibonacci golden nuggets

Problem 137

Same differences

Problem 135

Singleton difference

Problem 136

Prime pair connection

Problem 134

Large repunit factors

Problem 132

Repunit nonfactors

Problem 133

Prime cube partnership

Problem 131

Composites with prime repunit property

Problem 130

Repunit divisibility

Problem 129

Hexagonal tile differences

Problem 128

abc-hits

Problem 127

Cuboid layers

Problem 126

Palindromic sums

Problem 125

Ordered radicals

Problem 124

Prime square remainders

Problem 123

Efficient exponentiation

Problem 122

Disc game prize fund

Problem 121

Square remainders

Problem 120

Digit power sum

Problem 119

Pandigital prime sets

Problem 118

Red, green, and blue tiles

Problem 117

Red, green or blue tiles

Problem 116

Counting block combinations II

Problem 115

Counting block combinations I

Problem 114

Non-bouncy numbers

Problem 113

Bouncy numbers

Problem 112

Primes with runs

Problem 111

Diophantine reciprocals II

Problem 110

Darts

Problem 109

Diophantine reciprocals I

Problem 108

Minimal network

Problem 107

Special subset sums: meta-testing

Problem 106

Special subset sums: testing

Problem 105

Pandigital Fibonacci ends

Problem 104

Special subset sums: optimum

Problem 103

Triangle containment

Problem 102

Optimum polynomial

Problem 101

Arranged probability

Problem 100

Largest exponential

Problem 99

Anagramic squares

Problem 98

Large non-Mersenne prime

Problem 97

Su Doku

Problem 96

Amicable chains

Problem 95

Almost equilateral triangles

Problem 94

Arithmetic expressions

Problem 93

Square digit chains

Problem 92

Right triangles with integer coordinates

Problem 91

Cube digit pairs

Problem 90

Roman numerals

Problem 89

Product-sum numbers

Problem 88

Prime power triples

Problem 87

Cuboid route

Problem 86

Counting rectangles

Problem 85

Monopoly odds

Problem 84

Path sum: four ways

Problem 83

Path sum: three ways

Problem 82

Path sum: two ways

Problem 81

Square root digital expansion

Problem 80

Passcode derivation

Problem 79

Coin partitions

Problem 78

Prime summations

Problem 77

Counting summations

Problem 76

Singular integer right triangles

Problem 75

Digit factorial chains

Problem 74

Counting fractions in a range

Problem 73

Counting fractions

Problem 72

Ordered fractions

Problem 71

Totient permutation

Problem 70

Totient maximum

Problem 69

Magic 5-gon ring

Problem 68

Maximum path sum II

Problem 67

Diophantine equation

Problem 66

Convergents of e

Problem 65

Odd period square roots

Problem 64

Powerful digit counts

Problem 63

Cubic permutations

Problem 62

Cyclical figurate numbers

Problem 61

Prime pair sets

Problem 60

XOR decryption

Problem 59

Spiral primes

Problem 58

Square root convergents

Problem 57

Powerful digit sum

Problem 56

Lychrel numbers

Problem 55

Poker hands

Problem 54

Combinatoric selections

Problem 53

Permuted multiples

Problem 52

Prime digit replacements

Problem 51

Consecutive prime sum

Problem 50

Prime permutations

Problem 49

Self powers

Problem 48

Distinct primes factors

Problem 47

Goldbach's other conjecture

Problem 46

Triangular, pentagonal, and hexagonal

Problem 45

Pentagon numbers

Problem 44

Sub-string divisibility

Problem 43

Coded triangle numbers

Problem 42

Pandigital prime

Problem 41

Champernowne's constant

Problem 40

Integer right triangles

Problem 39

Pandigital multiples

Problem 38

Truncatable primes

Problem 37

Double-base palindromes

Problem 36

Circular primes

Problem 35

Digit factorials

Problem 34

Digit cancelling fractions

Problem 33

Pandigital products

Problem 32

Coin sums

Problem 31

Digit fifth powers

Problem 30

Distinct powers

Problem 29

Number spiral diagonals

Problem 28

Quadratic primes

Problem 27

Reciprocal cycles

Problem 26

1000-digit Fibonacci number

Problem 25

Lexicographic permutations

Problem 24

Non-abundant sums

Problem 23

Names scores

Problem 22

Amicable numbers

Problem 21

Factorial digit sum

Problem 20

Counting Sundays

Problem 19

Maximum path sum I

Problem 18

Number letter counts

Problem 17

Power digit sum

Problem 16

Lattice paths

Problem 15

Longest Collatz sequence

Problem 14

Large sum

Problem 13

Highly divisible triangular number

Problem 12

Largest product in a grid

Problem 11

Summation of primes

Problem 10

Special Pythagorean triplet

Problem 9

Largest product in a series

Problem 8

10001st prime

Problem 7

Sum square difference

Problem 6

Smallest multiple

Problem 5

Largest palindrome product

Problem 4

Largest prime factor

Problem 3

Even Fibonacci numbers

Problem 2

Multiples of 3 and 5

Problem 1