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Various handshaking problems are in circulation, the most common one being the following. In a room of n people, how many different handshakes are possible? The answer is (n; ...

How many times can a shape be completely surrounded by copies of itself without being able to tile the entire plane, i.e., what is the maximum (finite) Heesch number?

Consider a horse rider who wishes to feed his horse at a field, gather water from a river, and then return to his tent, all in the smallest overall distance possible. The ...

Find the maximum number of bishops B(n) that can be placed on an n×n chessboard such that no two attack each other. The answer is 2n-2 (Dudeney 1970, Madachy 1979), giving ...

Solve the Pell equation x^2-92y^2=1 in integers. The smallest solution is x=1151, y=120.

Given the functional (1) find in a class of arcs satisfying p differential and q finite equations phi_alpha(y_1,...,y_n;y_1^',...,y_n^')=0 for alpha=1,...,p ...

Given a map f from a space X to a space Y and another map g from a space Z to a space Y, does there exist a map h from X to Z such that gh=f? If such a map h exists, then h ...

Given the center of a circle, divide the circle into four equal arcs using a compass alone (a Mascheroni construction).

In his Meditationes algebraicae, Waring (1770, 1782) proposed a generalization of Lagrange's four-square theorem, stating that every rational integer is the sum of a fixed ...

For what value of x is f(x)=x^(1/x) a maximum? The maximum occurs at x=e, where f^'(x)=x^(-2+1/x)(1-lnx)=0, (1) which is zero at x=e and gives a maximum of ...