brain

Consider a wire of length 1 unit between two posts.

Let sk be the number of 1’s when writing the numbers from 0 to k in binary.

Let T(n, m) be the number of m-tuples of positive integers such that the sum of any two neighbouring elements of the tup

On a chess board, a centaur moves like a king or a knight.

The Carmichael function lambda(n) is defined as the smallest positive integer m such that a^m = 1 modulo n for all integ

Julie proposes the following wager to her sister Louise.

Two positive integers a and b are 2-friendly when gcd(a,b) = 2^t, t gt 0.

We shall call a positive integer A an "Alexandrian integer", if there exist integers p, q, r such that: and For example,

Let S = 2, 3, 5, dots, 4999 be the set of prime numbers less than 5000.

Given an n-tuple of numbers another n-tuple is created where each element of the new n-tuple is chosen randomly from the

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

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

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

It is possible to write five as a sum in exactly six different ways: How many different ways can one hundred be written

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

The triangle triangle ABC is inscribed in an ellipse with equation frac {x^2} {a^2} + frac {y^2} {b^2} = 1, 0 lt 2b lt a

In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way: - Hi

In the classic "Crossing Ladders" problem, we are given the lengths x and y of two ladders resting on the opposite walls

A certain type of chandelier contains a circular ring of n evenly spaced candleholders.

Consider the numbers 15, 16 and 18: 15=3times 5 and 3+5=8.

Let x1, x2, dots, xn be a sequence of length n such that: - x1 = 2 - for all 1 lt i le n: x{i - 1} lt xi - for all i and

In a sliding game a counter may slide horizontally or vertically into an empty space.

A row of five grey square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (le

Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try managing extremely large finite hotels, inste

Let Sm = (x1, x2, dots , xm) be the m-tuple of positive real numbers with x1 + x2 + cdots + xm = m for which Pm = x1 cdo

For an integer M, we define R(M) as the sum of 1/(p cdot q) for all the integer pairs p and q which satisfy all of these

Let's call S the (infinite) string that is made by concatenating the consecutive positive integers (starting from 1) wri

Euler's totient function, phi(n) [sometimes called the phi function], is defined as the number of positive integers not

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

There is a method that is used by Bell ringers to generate all variations of the order that bells are rung.

Two players play a game with a single pile of stones of initial size n.

A positive integer N is stealthy, if there exist positive integers a, b, c, d such that ab = cd = N and a+b = c+d+1.

A set of lattice points S is called a titanic set if there exists a line passing through exactly two points in S.

Let r be the remainder when (a - 1)^n + (a + 1)^n is divided by a^2.

We stack n plates into k non-empty piles where each pile is a different size.

The following undirected network consists of seven vertices and twelve edges with a total weight of 243.

We say that a d-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divi

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

Define two functions on lattice points: r(x,y) = (x+1,2y) s(x,y) = (2x,y+1) A path to equality of length n for a pair (a

Consider the following set of dice with nonstandard pips: Die A: 1 4 4 4 4 4 Die B: 2 2 2 5 5 5 Die C: 3 3 3 3 3 6 A gam

A symmetrical convex grid polygon is a polygon such that: - All its vertices have integer coordinates.

The primes 3, 7, 109, and 673, are quite remarkable.

Alice walks on a lattice grid.

It can be verified that there are 23 positive integers less than 1000 that are divisible by at least four distinct prime

For a positive integer n gt 1, let p(n) be the smallest prime dividing n, and let alpha(n) be its p-adic order, i.e.

If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken

Let us call an integer sided triangle with sides a le b le c barely acute if the sides satisfy a^2 + b^2 = c^2 + 1.

We define the rounded-square-root of a positive integer n as the square root of n rounded to the nearest integer.

A particular school offers cash rewards to children with good attendance and punctuality.

A list of size n is a sequence of n natural numbers.

Let E(x0, y0) be the number of steps it takes to determine the greatest common divisor of x0 and y0 with Euclid's algori

Let N(i) be the smallest integer n such that n! is divisible by (i!)^{1234567890} Let S(u)=sum N(i) for 10 le i le u.

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

By using each of the digits from the set, 1, 2, 3, 4, exactly once, and making use of the four arithmetic operations (+,

Consider an n-player game played in consecutive pairs: Round 1 takes place between players 1 and 2, round 2 takes place

Each one of the 25 sheep in a flock must be tested for a rare virus, known to affect 2 of the sheep population.

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

Let T(n) be the number of tours over a 4 times n playing board such that: - The tour starts in the top left corner.

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

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

A k-bounded partition of a positive integer N is a way of writing N as a sum of positive integers not exceeding k.

A positive integer, n, is factorised into prime factors.

The four right-angled triangles with sides (9,12,15), (12,16,20), (5,12,13) and (12,35,37) all have one of the shorter s

Let x be a real number.

The Torpids are rowing races held annually in Oxford, following some curious rules: - A division consists of n boats (ty

A non-decreasing sequence of integers an can be generated from any positive real value theta by the following procedure:

A pack of cards contains 4n cards with four identical cards of each value.

The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 2^{

The number of divisors of 120 is 16.

Consider the following algorithm for sorting a list: - 1.

The proper divisors of a number are all the divisors excluding the number itself.

This problem combines the game of Nim with the Towers of Hanoi.

Let P(n) be the set of the first n positive integers 1, 2, dots, n.

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

Given an integer sided triangle ABC: Let I be the incenter of ABC.

The 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 tri

Writing down the numbers which have a digit sum of 10 in ascending order, we get: 19, 28, 37, 46,55,64,73,82,91,109, 118

For a number written in Roman numerals to be considered valid there are basic rules which must be followed.

Consider positive integer solutions to a^2+b^2+c^2 = 3abc For example, (1,5,13) is a solution.

For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a,b)1 = a, U(a,b)2 = b and for k gt 2, U(a,b

Alice plays the following game, she starts with a list of integers L and on each step she can either: - remove two eleme

A secret integer t is selected at random within the range 1 le t le n.

Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring d

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

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

Alice is playing a game with n cards numbered 1 to n.

Alice and Bob are taking turns playing a game consisting of c different coins on a chessboard of size n by n.

We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any e

We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and hor

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

Let n be a natural number and p1^{alpha1}cdot p2^{alpha2}cdots pk^{alphak} its prime factorisation.

Using a combination of grey square tiles and oblong tiles chosen from: red tiles (measuring two units), green tiles (mea

A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.

Two players play a game.

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

For a positive number n, define S(n) as the sum of the integers x, for which 1 lt x lt n and x^3 equiv 1 bmod n.

Let f(N) be the number of points with integer coordinates that are on a circle passing through (0,0), (N,0),(0,N), and (

Given a fixed real number c, define a random sequence (Xn){nge 0} by the following random process: - X0 = c (with probab

For every positive number n we define the function mathop{streak}(n)=k as the smallest positive integer k such that n+k

In the following equation x, y, and n are positive integers.