brain

Members of a species of bacteria occur in two different types: alpha and beta.

For every integer n1, the family of functions f{n,a,b} is defined by f{n,a,b}(x)equiv a x + b mod n for a,b,x integer an

This problem uses half open interval notation where [a,b) represents a le x < b.

Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.

The smallest number m such that 10 divides m! is m=5.

Consider a rectangle made up of W times H square cells each with area 1.

The quadtree encoding allows us to describe a 2^N times 2^N black and white image as a sequence of bits (0 and 1).

A Harshad or Niven number is a number that is divisible by the sum of its digits.

A composite number can be factored many different ways.

A number consisting entirely of ones is called a repunit.

The game Number Mind is a variant of the well known game Master Mind.

N disks are placed in a row, indexed 1 to N from left to right.

Consider the divisors of 30: 1,2,3,5,6,10,15,30.

The logical-OR of two bits is 0 if both bits are 0, otherwise it is 1.

For an integer n ge 4, we define the lower prime square root of n, denoted by operatorname{lps}(n), as the largest prime

Let displaystyle S(n)=sumlimits{k=0}^{n}binom{n}{k}k^n.

Amidakuji (Japanese: 阿弥陀籤) is a method for producing a random permutation of a set of objects.

An integer of the form p^q q^p with prime numbers p neq q is called a hybrid-integer.

Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept.

Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including 0 a

N trolls are in a hole that is DN cm deep.

(x,y) is called a nested radical pair if x and y are non-zero integers such that dfrac{x}{y} is not a cube of a rational

Two positive integers x and y (x y) can generate a sequence in the following manner: - ax = y is the first term, - a{z+1

An infinite sequence of real numbers a(n) is defined for all integers n as follows: For example, a(0) = dfrac{1}{1!} + d

Euler's totient function, phi(n) [sometimes called the phi function], is used to determine the number of positive number

In a room N chairs are placed around a round table.

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different orde

A positive integer n is powerful if p^2 is a divisor of n for every prime factor p in n.

Let n be a positive integer.

A standard 52-card deck comprises 13 ranks in four suits.

If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next te

Two friends A and B are great fans of Chess.

A snowflake of order n is formed by overlaying an equilateral triangle (rotated by 180 degrees) onto each equilateral tr

The positive integral solutions of the equation x^y=y^x are (2,4), (4,2) and (k,k) for all k 0.

In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbe

Define Q(n) to be the smallest number that occurs in exactly n Pythagorean triples (a,b,c) where a lt b lt c.

Given a character string s, we define L(k,s) to be the length of the longest substring of s which appears at least k tim

For a positive integer n, d(n) is defined to be the sum of the digits of n.

Let A be an affine plane over a radically integral local field F with residual characteristic p.

Let's call two numbers friend numbers if their representation in base 10 has at least one common digit.

Let S(A) represent the sum of elements in set A of size n.

Define g(n, m) to be the largest integer k such that 2^k divides binom{n}m.

Find the smallest x + y + z with integers x gt y gt z gt 0 such that x + y, x - y, x + z, x - z, y + z, y - z are all pe

Consider the following process that can be applied recursively to any positive integer n: - if n = 1 do nothing and the

The binomial coefficients displaystyle binom n k can be arranged in triangular form, Pascal's triangle, like this: | | |

Sam and Tom are trying a game of (partially) covering a given line segment of length L by taking turns in placing unit s

It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere.

Let Bbb R^2 be the set of pairs of real numbers (x, y).

By replacing the 1st digit of the 2-digit number 3, it turns out that six of the nine possible values: 13, 23, 43, 53, 7

For integers m, n (0 leq n lt m), let L(m, n) be an m times m grid with the top-right n times n grid removed.

Find the number of non-empty subsets of 1^1, 2^2, 3^3,dots, 250250^{250250}, the sum of whose elements is divisible by 2

All square roots are periodic when written as continued fractions and can be written in the form: For example, let us co

A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size.

The most naive way of computing n^{15} requires fourteen multiplications: But using a "binary" method you can compute it

The cube, 41063625 (345^3), can be permuted to produce two other cubes: 56623104 (384^3) and 66430125 (405^3).

Let T(r) be the number of integer quadruplets x, y, z, t such that x^2 + y^2 + z^2 + t^2 le r^2.

We use xoplus y for the bitwise XOR of x and y.

Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods an

Build a triangle from all positive integers in the following way: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Denote the average of k numbers x1, ..., xk by bar{x} = frac{1}{k} sumi xi.

Let f(n) = n^2 - 3n - 1.

The first 15 Fibonacci numbers are: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610.

A positive integer, n, is divided by d and the quotient and remainder are q and r respectively.

Starting with three numbers a, b, c, at each step do one of the three operations: - change a to 2(b + c) - a; - change b

Let an be the largest real root of a polynomial g(x) = x^3 - 2^n cdot x^2 + n.

A number consisting entirely of ones is called a repunit.

A bag contains one red disc and one blue disc.

N times N disks are placed on a square game board.

The well-known Rubik's Cube puzzle has many fascinating mathematical properties.

For any set A of numbers, let operatorname{sum}(A) be the sum of the elements of A.

The RSA encryption is based on the following procedure: Generate two distinct primes p and q.

Create a sequence of numbers using the "Blum Blum Shub" pseudo-random number generator: Concatenate these numbers s0s1s2

Let b(n) be the largest power of 2 that divides n.

Consider the term small sqrt{x+sqrt{y}+sqrt{z}} that is representing a nested square root.

An unbiased coin is tossed repeatedly until two consecutive heads are obtained.

Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform d

Starting from a positive integer n, at each step we subtract from n the largest perfect cube not exceeding n, until n be

A 4 times 4 grid is filled with digits d, 0 le d le 9.

A square piece of paper with integer dimensions N times N is placed with a corner at the origin and two of its sides alo

Let pi(x) be the prime counting function, i.e.

Solution to Project Euler Problem 710.

The blancmange curve is the set of points (x, y) such that 0 le x le 1 and y = sum limits{n = 0}^{infty} {dfrac{s(2^n x)

A laborious ant walks randomly on a 5 times 5 grid.

A mountain range consists of a line of mountains with slopes of exactly 45^circ, and heights governed by the prime numbe

A segment is uniquely defined by its two endpoints.

We define two sequences S = S(1), S(2), ..., S(n) and S2 = S2(1), S2(2), ..., S2(n): S(k) = (pk)^k bmod 10007 where pk i

For any positive integer k, a finite sequence ai of fractions xi/yi is defined by: a1 = 1/k and ai = (x{i - 1} + 1) / (y

Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns.

Consider the sequence of real numbers an defined by the starting value a0 and the recurrence displaystyle a{n+1}=an-frac

Consider a wire of length 1 unit between two posts.

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top

A train is used to transport four carriages in the order: ABCD.

The Möbius function, denoted mu(n), is defined as: - mu(n) = (-1)^{omega(n)} if n is squarefree (where omega(n) is the n

A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:

Each 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.

A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone t

It is easily proved that no equilateral triangle exists with integral length sides and integral area.

Consider a positive integer sequence S = (s1, s2, dots, sn).

Consider the following Diophantine equation: where x, y and z are positive integers.

Consider equations of the form: a^2 + b^2 = N, 0 le a le b, a, b and N integer.