1 - Multiples of 3 and 5
2 - Even Fibonacci numbers
3 - Largest prime factor
4 - Largest palindrome product
5 - Smallest multiple
6 - Sum square difference
7 - 10001st prime
8 - Largest product in a series
9 - Special Pythagorean triplet
10 - Summation of primes
11 - Largest product in a grid
12 - Highly divisible triangular number
13 - Large sum
14 - Longest Collatz sequence
15 - Lattice paths
16 - Power digit sum
17 - Number letter counts
18 - Maximum path sum I
19 - Counting Sundays
20 - Factorial digit sum
21 - Amicable numbers
22 - Names scores
23 - Non-abundant sums
24 - Lexicographic permutations
25 - 1000-digit Fibonacci number
26 - Reciprocal cycles
27 - Quadratic primes
28 - Number spiral diagonals
29 - Distinct powers
30 - Digit fifth powers
31 - Coin sums
32 - Pandigital products
33 - Digit cancelling fractions
34 - Digit factorials
35 - Circular primes
36 - Double-base palindromes
37 - Truncatable primes
38 - Pandigital multiples
39 - Integer right triangles
40 - Champernowne's constant
41 - Pandigital prime
42 - Coded triangle numbers
43 - Sub-string divisibility
44 - Pentagon numbers
45 - Triangular, pentagonal, and hexagonal
46 - Goldbach's other conjecture
47 - Distinct primes factors
48 - Self powers
49 - Prime permutations
50 - Consecutive prime sum
51 - Prime digit replacements
52 - Permuted multiples
53 - Combinatoric selections
54 - Poker hands
55 - Lychrel numbers
56 - Powerful digit sum
57 - Square root convergents
58 - Spiral primes
59 - XOR decryption
60 - Prime pair sets
61 - Cyclical figurate numbers
62 - Cubic permutations
63 - Powerful digit counts
64 - Odd period square roots
65 - Convergents of e
66 - Diophantine equation
67 - Maximum path sum II
68 - Magic 5-gon ring
69 - Totient maximum
70 - Totient permutation
71 - Ordered fractions
72 - Counting fractions
73 - Counting fractions in a range
74 - Digit factorial chains
75 - Singular integer right triangles
76 - Counting summations
77 - Prime summations
78 - Coin partitions
79 - Passcode derivation
80 - Square root digital expansion
81 - Path sum: two ways
82 - Path sum: three ways
83 - Path sum: four ways
84 - Monopoly odds
85 - Counting rectangles
86 - Cuboid route
87 - Prime power triples
88 - Product-sum numbers
89 - Roman numerals
90 - Cube digit pairs
91 - Right triangles with integer coordinates
92 - Square digit chains
93 - Arithmetic expressions
94 - Almost equilateral triangles
95 - Amicable chains
96 - Su Doku
97 - Large non-Mersenne prime
98 - Anagramic squares
99 - Largest exponential
100 - Arranged probability
101 - Optimum polynomial
102 - Triangle containment
103 - Special subset sums: optimum
104 - Pandigital Fibonacci ends
105 - Special subset sums: testing
106 - Special subset sums: meta-testing
107 - Minimal network
108 - Diophantine reciprocals I
109 - Darts
110 - Diophantine reciprocals II
111 - Primes with runs
112 - Bouncy numbers
113 - Non-bouncy numbers
114 - Counting block combinations I
115 - Counting block combinations II
116 - Red, green or blue tiles
117 - Red, green, and blue tiles
118 - Pandigital prime sets
119 - Digit power sum
120 - Square remainders
121 - Disc game prize fund
122 - Efficient exponentiation
123 - Prime square remainders
124 - Ordered radicals
125 - Palindromic sums
126 - Cuboid layers
127 - abc-hits
128 - Hexagonal tile differences
129 - Repunit divisibility
130 - Composites with prime repunit property
131 - Prime cube partnership
132 - Large repunit factors
133 - Repunit nonfactors
134 - Prime pair connection
135 - Same differences
136 - Singleton difference
137 - Fibonacci golden nuggets
138 - Special isosceles triangles
139 - Pythagorean tiles
140 - Modified Fibonacci golden nuggets
141 - Investigating progressive numbers, n, which are also square
142 - Perfect Square Collection
143 - Investigating the Torricelli point of a triangle
144 - Investigating multiple reflections of a laser beam
145 - How many reversible numbers are there below one-billion?
146 - Investigating a Prime Pattern
147 - Rectangles in cross-hatched grids
148 - Exploring Pascal's triangle
149 - Searching for a maximum-sum subsequence
150 - Searching a triangular array for a sub-triangle having minimum-sum
151 - Paper sheets of standard sizes: an expected-value problem
152 - Writing 1/2 as a sum of inverse squares
153 - Investigating Gaussian Integers
154 - Exploring Pascal's pyramid
155 - Counting Capacitor Circuits
156 - Counting Digits
157 - Solving the diophantine equation 1/a+1/b= p/10n
158 - Exploring strings for which only one character comes lexicographically after its neighbour to the left
159 - Digital root sums of factorisations
160 - Factorial trailing digits
161 - Triominoes
162 - Hexadecimal numbers
163 - Cross-hatched triangles
164 - Numbers for which no three consecutive digits have a sum greater than a given value
165 - Intersections
166 - Criss Cross
167 - Investigating Ulam sequences
168 - Number Rotations
169 - Exploring the number of different ways a number can be expressed as a sum of powers of 2
170 - Find the largest 0 to 9 pandigital that can be formed by concatenating products
171 - Finding numbers for which the sum of the squares of the digits is a square
172 - Investigating numbers with few repeated digits
173 - Using up to one million tiles how many different "hollow" square laminae can be formed?
174 - Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements
175 - Fractions involving the number of different ways a number can be expressed as a sum of powers of 2
176 - Right-angled triangles that share a cathetus
177 - Integer angled Quadrilaterals
178 - Step Numbers
179 - Consecutive positive divisors
180 - Rational zeros of a function of three variables
181 - Investigating in how many ways objects of two different colours can be grouped
182 - RSA encryption
183 - Maximum product of parts
184 - Triangles containing the origin
185 - Number Mind
186 - Connectedness of a network
187 - Semiprimes
188 - The hyperexponentiation of a number
189 - Tri-colouring a triangular grid
190 - Maximising a weighted product
191 - Prize Strings
192 - Best Approximations
193 - Squarefree Numbers
194 - Coloured Configurations
195 - Inscribed circles of triangles with one angle of 60 degrees
196 - Prime triplets
197 - Investigating the behaviour of a recursively defined sequence
198 - Ambiguous Numbers
199 - Iterative Circle Packing
200 - Find the 200th prime-proof sqube containing the contiguous sub-string "200"
201 - Subsets with a unique sum
202 - Laserbeam
203 - Squarefree Binomial Coefficients
204 - Generalised Hamming Numbers
205 - Dice Game
206 - Concealed Square
207 - Integer partition equations
208 - Robot Walks
209 - Circular Logic
210 - Obtuse Angled Triangles
211 - Divisor Square Sum
212 - Combined Volume of Cuboids
213 - Flea Circus
214 - Totient Chains
215 - Crack-free Walls
216 - Investigating the primality of numbers of the form 2n2-1
217 - Balanced Numbers
218 - Perfect right-angled triangles
219 - Skew-cost coding
220 - Heighway Dragon
221 - Alexandrian Integers
222 - Sphere Packing
223 - Almost right-angled triangles I
224 - Almost right-angled triangles II
225 - Tribonacci non-divisors
226 - A Scoop of Blancmange
227 - The Chase
228 - Minkowski Sums
229 - Four Representations using Squares
230 - Fibonacci Words
231 - The prime factorisation of binomial coefficients
232 - The Race
233 - Lattice points on a circle
234 - Semidivisible numbers
235 - An Arithmetic Geometric sequence
236 - Luxury Hampers
237 - Tours on a 4 x n playing board
238 - Infinite string tour
239 - Twenty-two Foolish Primes
240 - Top Dice
241 - Perfection Quotients
242 - Odd Triplets
243 - Resilience
244 - Sliders
245 - Coresilience
246 - Tangents to an ellipse
247 - Squares under a hyperbola
248 - Numbers for which Euler’s totient function equals 13!
249 - Prime Subset Sums
250 - 250250
251 - Cardano Triplets
252 - Convex Holes
253 - Tidying up
254 - Sums of Digit Factorials
255 - Rounded Square Roots
256 - Tatami-Free Rooms
257 - Angular Bisectors
258 - A lagged Fibonacci sequence
259 - Reachable Numbers
260 - Stone Game
261 - Pivotal Square Sums
262 - Mountain Range
263 - An engineers' dream come true
264 - Triangle Centres
265 - Binary Circles
266 - Pseudo Square Root
267 - Billionaire
268 - Counting numbers with at least four distinct prime factors less than 100
269 - Polynomials with at least one integer root
270 - Cutting Squares
271 - Modular Cubes, part 1
272 - Modular Cubes, part 2
273 - Sum of Squares
274 - Divisibility Multipliers
275 - Balanced Sculptures
276 - Primitive Triangles
277 - A Modified Collatz sequence
278 - Linear Combinations of Semiprimes
279 - Triangles with integral sides and an integral angle
280 - Ant and seeds
281 - Pizza Toppings
282 - The Ackermann function
283 - Integer sided triangles for which the area/perimeter ratio is integral
284 - Steady Squares
285 - Pythagorean odds
286 - Scoring probabilities
287 - Quadtree encoding (a simple compression algorithm)
288 - An enormous factorial
289 - Eulerian Cycles
290 - Digital Signature
291 - Panaitopol Primes
292 - Pythagorean Polygons
293 - Pseudo-Fortunate Numbers
294 - Sum of digits - experience #23
295 - Lenticular holes
296 - Angular Bisector and Tangent
297 - Zeckendorf Representation
298 - Selective Amnesia
299 - Three similar triangles
300 - Protein folding
301 - Nim
302 - Strong Achilles Numbers
303 - Multiples with small digits
304 - Primonacci
305 - Reflexive Position
306 - Paper-strip Game
307 - Chip Defects
308 - An amazing Prime-generating Automaton
309 - Integer Ladders
310 - Nim Square
311 - Biclinic Integral Quadrilaterals
312 - Cyclic paths on Sierpiński graphs
313 - Sliding game
314 - The Mouse on the Moon
315 - Digital root clocks
316 - Numbers in decimal expansions
317 - Firecracker
318 - 2011 nines
319 - Bounded Sequences
320 - Factorials divisible by a huge integer
321 - Swapping Counters
322 - Binomial coefficients divisible by 10
323 - Bitwise-OR operations on random integers
324 - Building a tower
325 - Stone Game II
326 - Modulo Summations
327 - Rooms of Doom
328 - Lowest-cost Search
329 - Prime Frog
330 - Euler's Number
331 - Cross flips
332 - Spherical triangles
333 - Special partitions
334 - Spilling the beans
335 - Gathering the beans
336 - Maximix Arrangements
337 - Totient Stairstep Sequences
338 - Cutting Rectangular Grid Paper
339 - Peredur fab Efrawg
340 - Crazy Function
341 - Golomb's self-describing sequence
342 - The totient of a square is a cube
343 - Fractional Sequences
344 - Silver dollar game
345 - Matrix Sum
346 - Strong Repunits
347 - Largest integer divisible by two primes
348 - Sum of a square and a cube
349 - Langton's ant
350 - Constraining the least greatest and the greatest least
351 - Hexagonal orchards
352 - Blood tests
353 - Risky moon
354 - Distances in a bee's honeycomb
355 - Maximal coprime subset
356 - Largest roots of cubic polynomials
357 - Prime generating integers
358 - Cyclic numbers
359 - Hilbert's New Hotel
360 - Scary Sphere
361 - Subsequence of Thue-Morse sequence
362 - Squarefree factors
363 - Bézier Curves
364 - Comfortable distance
365 - A huge binomial coefficient
366 - Stone Game III
367 - Bozo sort
368 - A Kempner-like series
369 - Badugi
370 - Geometric triangles
371 - Licence plates
372 - Pencils of rays
373 - Circumscribed Circles
374 - Maximum Integer Partition Product
375 - Minimum of subsequences
376 - Nontransitive sets of dice
377 - Sum of digits, experience 13
378 - Triangle Triples
379 - Least common multiple count
380 - Amazing Mazes!
381 - (prime-k) factorial
382 - Generating polygons
383 - Divisibility comparison between factorials
384 - Rudin-Shapiro sequence
385 - Ellipses inside triangles
386 - Maximum length of an antichain
387 - Harshad Numbers
388 - Distinct Lines
389 - Platonic Dice
390 - Triangles with non rational sides and integral area
391 - Hopping Game
392 - Enmeshed unit circle
393 - Migrating ants
394 - Eating pie
395 - Pythagorean tree
396 - Weak Goodstein sequence
397 - Triangle on parabola
398 - Cutting rope
399 - Squarefree Fibonacci Numbers
400 - Fibonacci tree game
401 - Sum of squares of divisors
402 - Integer-valued polynomials
403 - Lattice points enclosed by parabola and line
404 - Crisscross Ellipses
405 - A rectangular tiling
406 - Guessing Game
407 - Idempotents
408 - Admissible paths through a grid
409 - Nim Extreme
410 - Circle and tangent line
411 - Uphill paths
412 - Gnomon numbering
413 - One-child Numbers
414 - Kaprekar constant
415 - Titanic sets
416 - A frog's trip
417 - Reciprocal cycles II
418 - Factorisation triples
419 - Look and say sequence
420 - 2x2 positive integer matrix
421 - Prime factors of n15+1
422 - Sequence of points on a hyperbola
423 - Consecutive die throws
424 - Kakuro
425 - Prime connection
426 - Box-ball system
427 - n-sequences
428 - Necklace of circles
429 - Sum of squares of unitary divisors
430 - Range flips
431 - Square Space Silo
432 - Totient sum
433 - Steps in Euclid's algorithm
434 - Rigid graphs
435 - Polynomials of Fibonacci numbers
436 - Unfair wager
437 - Fibonacci primitive roots
438 - Integer part of polynomial equation's solutions
439 - Sum of sum of divisors
440 - GCD and Tiling
441 - The inverse summation of coprime couples
442 - Eleven-free integers
443 - GCD sequence
444 - The Roundtable Lottery
445 - Retractions A
446 - Retractions B
447 - Retractions C
448 - Average least common multiple
449 - Chocolate covered candy
450 - Hypocycloid and Lattice points
451 - Modular inverses
452 - Long Products
453 - Lattice Quadrilaterals
454 - Diophantine reciprocals III
455 - Powers With Trailing Digits
456 - Triangles containing the origin II
457 - A polynomial modulo the square of a prime
458 - Permutations of Project
459 - Flipping game
460 - An ant on the move
461 - Almost Pi
462 - Permutation of 3-smooth numbers
463 - A weird recurrence relation
464 - Möbius function and intervals
465 - Polar polygons
466 - Distinct terms in a multiplication table
467 - Superinteger
468 - Smooth divisors of binomial coefficients
469 - Empty chairs
470 - Super Ramvok
471 - Triangle inscribed in ellipse
472 - Comfortable Distance II
473 - Phigital number base
474 - Last digits of divisors
475 - Music festival
476 - Circle Packing II
477 - Number Sequence Game
478 - Mixtures
479 - Roots on the Rise
480 - The Last Question
481 - Chef Showdown
482 - The incenter of a triangle
483 - Repeated permutation
484 - Arithmetic Derivative
485 - Maximum number of divisors
486 - Palindrome-containing strings
487 - Sums of power sums
488 - Unbalanced Nim
489 - Common factors between two sequences
490 - Jumping frog
491 - Double pandigital number divisible by 11
492 - Exploding sequence
493 - Under The Rainbow
494 - Collatz prefix families
495 - Writing n as the product of k distinct positive integers
496 - Incenter and circumcenter of triangle
497 - Drunken Tower of Hanoi
498 - Remainder of polynomial division
499 - St. Petersburg Lottery
500 - Problem 500!!!
501 - Eight Divisors
502 - Counting Castles
503 - Compromise or persist
504 - Square on the Inside
505 - Bidirectional Recurrence
506 - Clock sequence
507 - Shortest Lattice Vector
508 - Integers in base i-1
509 - Divisor Nim
510 - Tangent Circles
511 - Sequences with nice divisibility properties
512 - Sums of totients of powers
513 - Integral median
514 - Geoboard Shapes
515 - Dissonant Numbers
516 - 5-smooth totients
517 - A real recursion
518 - Prime triples and geometric sequences
519 - Tricoloured Coin Fountains
520 - Simbers
521 - Smallest prime factor
522 - Hilbert's Blackout
523 - First Sort I
524 - First Sort II
525 - Rolling Ellipse
526 - Largest prime factors of consecutive numbers
527 - Randomized Binary Search
528 - Constrained Sums
529 - 10-substrings
530 - GCD of Divisors
531 - Chinese leftovers
532 - Nanobots on Geodesics
533 - Minimum values of the Carmichael function
534 - Weak Queens
535 - Fractal Sequence
536 - Modulo power identity
537 - Counting tuples
538 - Maximum quadrilaterals
539 - Odd elimination
540 - Counting primitive Pythagorean triples
541 - Divisibility of Harmonic Number Denominators
542 - Geometric Progression with Maximum Sum
543 - Prime-Sum Numbers
544 - Chromatic Conundrum
545 - Faulhaber's Formulas
546 - The Floor's Revenge
547 - Distance of random points within hollow square laminae
548 - Gozinta Chains
549 - Divisibility of factorials
550 - Divisor game
551 - Sum of digits sequence
552 - Chinese leftovers II
553 - Power sets of power sets
554 - Centaurs on a chess board
555 - McCarthy 91 function
556 - Squarefree Gaussian Integers
557 - Cutting triangles
558 - Irrational base
559 - Permuted Matrices
560 - Coprime Nim
561 - Divisor Pairs
562 - Maximal perimeter
563 - Robot Welders
564 - Maximal polygons
565 - Divisibility of sum of divisors
566 - Cake Icing Puzzle
567 - Reciprocal games I
568 - Reciprocal games II
569 - Prime Mountain Range
570 - Snowflakes
571 - Super Pandigital Numbers
572 - Idempotent matrices
573 - Unfair race
574 - Verifying Primes
575 - Wandering Robots
576 - Irrational jumps
577 - Counting hexagons
578 - Integers with decreasing prime powers
579 - Lattice points in lattice cubes
580 - Squarefree Hilbert numbers
581 - 47-smooth triangular numbers
582 - Nearly isosceles 120 degree triangles
583 - Heron Envelopes
584 - Birthday Problem Revisited
585 - Nested square roots
586 - Binary Quadratic Form
587 - Concave triangle
588 - Quintinomial coefficients
589 - Poohsticks Marathon
590 - Sets with a given Least Common Multiple
591 - Best Approximations by Quadratic Integers
592 - Factorial trailing digits 2
593 - Fleeting Medians
594 - Rhombus Tilings
595 - Incremental Random Sort
596 - Number of lattice points in a hyperball
597 - Torpids
598 - Split Divisibilities
599 - Distinct Colourings of a Rubik's Cube
600 - Integer sided equiangular hexagons
601 - Divisibility streaks
602 - Product of Head Counts
603 - Substring sums of prime concatenations
604 - Convex path in square
605 - Pairwise Coin-Tossing Game
606 - Gozinta Chains II
607 - Marsh Crossing
608 - Divisor Sums
609 - π sequences
610 - Roman Numerals II
611 - Hallway of square steps
612 - Friend numbers
613 - Pythagorean Ant
614 - Special partitions 2
615 - The millionth number with at least one million prime factors
616 - Creative numbers
617 - Mirror Power Sequence
618 - Numbers with a given prime factor sum
619 - Square subsets
620 - Planetary Gears
621 - Expressing an integer as the sum of triangular numbers
622 - Riffle Shuffles
623 - Lambda Count
624 - Two heads are better than one
625 - Gcd sum
626 - Counting Binary Matrices
627 - Counting products
628 - Open chess positions
629 - Scatterstone Nim
630 - Crossed lines
631 - Constrained Permutations
632 - Square prime factors
633 - Square prime factors II
634 - Numbers of the form $a^2b^3$
635 - Subset sums
636 - Restricted Factorisations
637 - Flexible digit sum
638 - Weighted lattice paths
639 - Summing a multiplicative function
640 - Shut the Box
641 - A Long Row of Dice
642 - Sum of largest prime factors
643 - 2-Friendly
644 - Squares on the line
645 - Every Day is a Holiday
646 - Bounded Divisors
647 - Linear Transformations of Polygonal Numbers
648 - Skipping Squares
649 - Low-Prime Chessboard Nim
650 - Divisors of Binomial Product
651 - Patterned Cylinders
652 - Distinct values of a proto-logarithmic function
653 - Frictionless Tube
654 - Neighbourly Constraints
655 - Divisible Palindromes
656 - Palindromic sequences
657 - Incomplete words
658 - Incomplete words II
659 - Largest prime
660 - Pandigital Triangles
661 - A Long Chess Match
662 - Fibonacci paths
663 - Sums of subarrays
664 - An infinite game
665 - Proportionate Nim
666 - Polymorphic Bacteria
667 - Moving Pentagon
668 - Square root smooth Numbers
669 - The King's Banquet
670 - Colouring a Strip
671 - Colouring a Loop
672 - One more one
673 - Beds and Desks
674 - Solving $\mathcal{I}$-equations
675 - $2^{\omega(n)}$
676 - Matching Digit Sums
677 - Coloured Graphs
678 - Fermat-like Equations
679 - Freefarea
680 - Yarra Gnisrever
681 - Maximal Area
682 - 5-Smooth Pairs
683 - The Chase II
684 - Inverse Digit Sum
685 - Inverse Digit Sum II
686 - Powers of Two
687 - Shuffling Cards
688 - Piles of Plates
689 - Binary Series
690 - Tom and Jerry
691 - Long substring with many repetitions
692 - Siegbert and Jo
693 - Finite Sequence Generator
694 - Cube-full Divisors
← Problem 221
Sphere Packing
Problem 223 →
Sphere Packing
Problem 222
What is the length of the shortest pipe, of internal radius 50mm, that can fully contain 21 balls of radii 30mm, 31mm, ..., 50mm?
Give your answer in micrometres (10
-6
m) rounded to the nearest integer.