brain

In the game, Monopoly, the standard board is set up in the following way: !0084monopolyboard.png A player starts on the

Given two points (x1, y1, z1) and (x2, y2, z2) in three dimensional space, the Manhattan distance between those points i

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

A firecracker explodes at a height of pu{100 m} above level ground.

We shall define a pythagorean polygon to be a convex polygon with the following properties: - there are at least three v

Consider graphs built with the units A: and B: , where the units are glued along the vertical edges as in the graph .

How many 20 digit numbers n (without any leading zero) exist such that no three consecutive digits of n have a sum great

Consider numbers t(n) of the form t(n) = 2n^2 - 1 with n gt 1.

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

For some positive integers k, there exists an integer partition of the form 4^t = 2^t + k, where 4^t, 2^t, and k are all

For an odd prime p, define f(p) = leftlfloorfrac{2^{(2^p)}}{p}rightrfloorbmod{2^p} For example, when p=3, lfloor 2^8/3rf

We define a pseudo-geometric sequence to be a finite sequence a0, a1, dotsc, an of positive integers, satisfying the fol

Define: xn = (1248^n bmod 32323) - 16161 yn = (8421^n bmod 30103) - 15051 Pn = (x1, y1), (x2, y2), dots, (xn, yn) For ex

A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the

The eight divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

Consider the alphabet A made out of the letters of the word "text{project}": A=text c,text e,text j,text o,text p,text r

For positive integers n and m, we define two polynomials Fn(x) = x^n and Gm(x) = (x-1)^m.

Consider the equation 17^pa+19^pb+23^pc = n where a, b, c and p are positive integers, i.e.

An ant moves on a regular grid of squares that are coloured either black or white.

A list initially contains the numbers 2, 3, dots, n.

The smallest positive integer n for which the numbers n^2 + 1, n^2 + 3, n^2 + 7, n^2 + 9, n^2 + 13, and n^2 + 27 are con

Let Sk be the set containing 2 and 5 and the first k primes that end in 7.

There is a grid of length and width 50515093 points.

For a positive integer n we define q(n) to be the number of solutions to: where 0 leq ai, bi lt n.

Given an irrational number alpha, let Salpha(n) be the sequence Salpha(n)=lfloor {alpha cdot n} rfloor - lfloor {alpha c

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).

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

Three mirrors are arranged in the shape of an equilateral triangle, with their reflective surfaces pointing inwards.

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

We are trying to find a hidden number selected from the set of integers 1, 2, dots, n by asking questions.

The palindromic number 595 is interesting because it can be written as the sum of consecutive squares: 6^2 + 7^2 + 8^2 +

Recall the blancmange function from Problem 226: T(x) = sumlimits{n = 0}^inftydfrac{s(2^nx)}{2^n}, where s(x) is the dis

Turan has the electrical water heating system outside his house in a shed.

How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter tha

A positive integer n is called squarefree, if no square of a prime divides n, thus 1, 2, 3, 5, 6, 7, 10, 11 are squarefr

An integer partition of a number n is a way of writing n as a sum of positive integers.

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

A Hamming number is a positive number which has no prime factor larger than 5.

Let f be a function from a finite set S to itself.

Given the set 1,2,dots,n, we define f(n, k) as the number of its k-element subsets with an odd sum of elements.

NOTE: This problem is a more challenging version of Problem 81.

Some positive integers n have the property that the sum [n + operatorname{reverse}(n)] consists entirely of odd (decimal

Triangle numbers Tk are integers of the form frac{k(k+1)} 2.

You are given a unique investment opportunity.

A certain type of tile comes in three different sizes - 1 times 1, 1 times 2, and 1 times 3 - and in four different colo

Consider the diophantine equation frac 1 a + frac 1 b = frac p {10^n} with a, b, p, n positive integers and a le b.

Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).

We define an S-number to be a natural number, n, that is a perfect square and its square root can be obtained by splitti

Each new term in the Fibonacci sequence is generated by adding the previous two terms.

Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots

In the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and

Let f(n) be the number of divisors of 2n^2 that are no greater than n.

Consider the number 3600.

A robot moves in a series of one-fifth circular arcs (72^circ), with a free choice of a clockwise or an anticlockwise ar

Frodo and Sam need to travel 100 leagues due East from point A to point B.

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

When (1+sqrt 7) is raised to an integral power, n, we always get a number of the form (a+bsqrt 7).

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

Let ABC be a triangle with all interior angles being less than 120 degrees.

Let a1, a2, dots, an be an integer sequence of length n such that: - a1 = 6 - for all 1 le i lt n: phi(ai) lt phi(a{i +

A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in

Let f(N) be the smallest positive integer that is not coprime to any positive integer n le N whose least significant dig

The Gauss Factorial of a number n is defined as the product of all positive numbers leq n that are relatively prime to n

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

A standard deck of 52 playing cards, which consists of thirteen ranks (Ace, Two, ..., Ten, King, Queen and Jack) each in

A game is played with two piles of stones and two players.

Let D0 be the two-letter string "Fa".

Given a circle C and an integer n 1, we perform the following operations.

Define m = M(n, d) to be the smallest positive integer such that when m^2 is written in base n it includes the base n di

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

Two positive numbers A and B are said to be connected (denoted by "A leftrightarrow B") if one of these conditions holds

Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4.

For any N, let f(N) be the last twelve hexadecimal digits before the trailing zeroes in N!.

A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next low

Let G(a, b) be the smallest non-negative integer n for which operatorname{mathbf{gcd}}Greatest common divisor(n^3 + b, (

A sequence of integers S = si is called an n-sequence if it has n elements and each element si satisfies 1 leq si leq n.

For a positive integer, n, define g(n) to be the maximum perfect square that divides n.

For fixed integers a, b, c, define the crazy function F(n) as follows: F(n) = n - c for all n gt b F(n) = F(a + F(a + F(

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) eac

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

Let S(n,m) = sumphi(n times i) for 1 leq i leq m.

"What? Where? When?" is a TV game show in which a team of experts attempt to answer questions.

How many integers 0 le n lt 10^{18} have the property that the sum of the digits of n equals the sum of digits of 137n?

Both 169 and 961 are the square of a prime.

Let a0, a1, dots be an integer sequence defined by: - a0 = 1; - for n ge 1, an is the sum of the digits of all preceding

A Pythagorean triangle is called supernatural if two of its three sides are consecutive integers.

A permutation of 2,3,ldots,n is a rearrangement of these numbers.

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 program written in the programming language Fractran consists of a list of fractions.

For positive real numbers a,b, an atimes b torus is a rectangle of width a and height b, with left and right sides ident

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

The McCarthy 91 function is defined as follows: We can generalize this definition by abstracting away the constants into

There are several ways to write the number dfrac{1}{2} as a sum of square reciprocals using distinct integers.

Inside a rope of length n, n - 1 points are placed with distance 1 from each other and from the endpoints.

The SET® card game is played with a pack of 81 distinct cards.

Consider a two dimensional grid of squares.

Two players play a game with two piles of stones, alternating turns.

Consider the set S of all possible products of n positive integers not exceeding m, that is S= x1x2cdots xn mid 1 le x1,

Let f(n) be the number of couples (x, y) with x and y positive integers, x le y and the least common multiple of x and y

The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime