brain

By replacing each of the letters in the word CARE with 1, 2, 9, and 6 respectively, we form a square number: 1296 = 36^2

A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when conn

The flipping game is a two player game played on an N by N square board.

Let g(n) be the number of undirected graphs with n nodes satisfying the following properties: - The graph is connected a

Let n be a positive integer.

Tom has built a random generator that is connected to a row of n light bulbs.

Let n be a positive integer.

We shall define a sqube to be a number of the form, p^2 q^3, where p and q are distinct primes.

A horizontal row comprising of 2n + 1 squares has n red counters placed at one end and n blue counters at the other end,

If a,b are two nonnegative integers with decimal representations a=(dots a2a1a0) and b=(dots b2b1b0) respectively, then

Let S(n) be the sum of all contiguous integer-substrings that can be formed from the integer n.

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

A deck of cards numbered from 1 to n is shuffled randomly such that each permutation is equally likely.

We can easily verify that none of the entries in the first seven rows of Pascal's triangle are divisible by 7: | | | | |

Let An be the smallest positive integer satisfying An bmod pi = i for all 1 le i le n, where pi is the i-th prime.

An ellipse E(a, b) is given at its initial position by equation: frac {x^2} {a^2} + frac {(y - b)^2} {b^2} = 1 The ellip

Consider the problem of building a wall out of 2 times 1 and 3 times 1 bricks (text{horizontal} times text{vertical} dim

Consider all lattice points (a,b,c) with 0 le a,b,c le N.

An equilateral triangle with integer side length n ge 3 is divided into n^2 equilateral triangles with side length 1 as

A series of three rooms are connected to each other by automatic doors.

Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length 1.

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

For a polygon P, let t(P) be the number of ways in which P can be tiled using rhombi and squares with edge length 1.

After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a matching cocktail table from the same compa

Let ra, rb and rc be the radii of three circles that are mutually and externally tangent to each other.

Consider the region constrained by 1 le x and 0 le y le 1/x.

For a positive integer n, let f(n) be the sum of the squares of the digits (in base 10) of n, e.g.

A and B play a game. A has originally 1 gram of gold and B has an unlimited amount. Each round goes as follows: - A choo

On planet J, a year lasts for D days.

Solution to Project Euler Problem 710.

A prime is drawn uniformly from all primes not exceeding N.

Given any positive integer n, we can construct a new integer by inserting plus signs between some of the digits of the b

Consider a single game of Ramvok: Let t represent the maximum number of turns the game lasts.

A company specialises in producing large rectangular metal sheets, starting from unit square metal plates.

Given positive integers a le b le c le d, it may be possible to form quadrilaterals with edge lengths a,b,c,d (in any or

Take the number 6 and multiply it by each of 1273 and 9854: By concatenating these products we get the 1 to 9 pandigital

The sequence Sn is defined by S0 = 290797 and Sn = S{n - 1}^2 bmod 50515093 for n 0.

Let fn(k) = e^{k/n} - 1, for all non-negative integers k.

The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the se

A binary matrix is a matrix consisting entirely of 0s and 1s.

Tom (the cat) and Jerry (the mouse) are playing on a simple graph G.

A circle C of circumference c centimetres has a smaller circle S of circumference s centimetres lying off-centre within

The lambda-calculus is a universal model of computation at the core of functional programming languages.

A googol (10^{100}) is a massive number: one followed by one-hundred zeros; 100^{100} is almost unimaginably large: one

By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles: Although the

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

Consider the triangles with integer sides a, b and c with a le b le c.

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

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

A game is played with many identical, round coins on a flat table.

The nth harmonic number Hn is defined as the sum of the multiplicative inverses of the first n positive integers, and ca

Considering 4-digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by

12n musicians participate at a music festival.

An infinite number of people (numbered 1, 2, 3, etc.) are lined up to get a room at Hilbert's newest infinite hotel.

Consider the number 54.

Let H be the hyperbola defined by the equation 12x^2 + 7xy - 12y^2 = 625.

Consider the natural numbers having at least 5 prime factors, which don't have to be distinct.

An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher ro

We use xoplus y to be the bitwise XOR of x and y.

Let's call an integer sided triangle with exactly one angle of 60 degrees a 60-degree triangle.

Peter is playing a solitaire game on an infinite checkerboard, each square of which can hold an unlimited number of toke

Given is an integer sided triangle ABC with sides a le b le c.

A small child has a “number caterpillar” consisting of N jigsaw pieces, each with one number on it, which, when connecte

The coefficients in the expansion of (x+1)^k are called binomial coefficients.

A group of chefs (numbered 1, 2, etc) participate in a turn-based strategic cooking competition.

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

A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, a1, a2,

Dr. One and Dr. Zero are playing the following partisan game. The game begins with one 1, two 2's, three 3's, ..., n n's

Using only a six-sided fair dice and a five-sided fair dice, we would like to emulate an n-sided fair dice.

A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.

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

A polygon is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit.

Consider the following configuration of 64 triangles: We wish to colour the interior of each triangle with one of three

You are given a pizza (perfect circle) that has been cut into m cdot n equal pieces and you want to have exactly one top

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

For an integer n, we define the square prime factors of n to be the primes whose square divides n.

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

k defects are randomly distributed amongst n integrated-circuit chips produced by a factory (any number of defects may b

6174 is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get

A positive fraction whose numerator is less than its denominator is called a proper fraction.

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

A gambler decides to participate in a special lottery.

The square root of 2 can be written as an infinite continued fraction.

4n people stand in a circle with their heads down.

Consider an infinite row of boxes.

Anton and Bertrand love to play three pile Nim.

Define f(0)=1 and f(n) to be the number of different ways n can be expressed as a sum of integer powers of 2 using each

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

If we calculate a^2 bmod 6 for 0 leq a leq 5 we get: 0,1,4,3,4,1.

A hexagonal orchard of order n is a triangular lattice made up of points within a regular hexagon with side n.

Let g(m) be the integer defined by the following double sum of products of binomial coefficients: You are given that g(1

A matrix M is called idempotent if M^2 = M.

Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any hea

The numbers 545, 5995 and 15151 are the three smallest palindromes divisible by 109.

Let us call a positive integer k a square-pivot, if there is a pair of integers m gt 0 and n ge k, such that the sum of

A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of si

A standard 52 card deck comprises thirteen ranks in four suits.

The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its squar

Let g(n) be a sequence defined as follows: g(4) = 13, g(n) = g(n-1) + gcd(n, g(n-1)) for n gt 4.

The game of Mahjong is played with tiles belonging to s suits.