Problems from Another Time

Individual problems from throughout mathematics history, as well as articles that include problem sets for students.

In a certain lake, swarming with red geese, the tip of a lotus bud was seen to extend a span above the surface of the water.

Nuts Over Counting

A fellow said that when he counted his nuts by twos, threes, fours, fives and sixes, there was still one left over; but when he counted them by sevens they came out even. What is the smallest number of nuts he could have?

Two Balls, No Strikes!

Imagine an urn with two balls, each of which may be either white or black. One of these balls is drawn and is put back before a new one is drawn.

A Circular Problem

Three circles of varying radius are mutually tangent. The area of the triangle connecting their centers is given. Find the radius of the third circle.

On the Run

A certain slave fled from Milan to Naples going 1/10 of the whole journey each day. At the beginning of the third day, his master sent a slave after him and this slave went 1/7 of the whole journey each day.

Prove the Straight Lines

Given two circles tangent at the point P with parallel diameters AB and CD, prove that APD and BPC are straight lines.

Area of an Equilateral Triangle

Suppose the area of an equilateral triangle be 600. The sides are required.

A Triangular Problem

In a right triangle, having been given the perimeter, a, and the length of the perpendicular from the right-angled vertex to the hypotenuse, b, it is required to find the length of the hypotenuse.

A Grinding Stone

Seven men held equal shares in a grinding stone 5 feet in diameter. What part of the diameter should each grind away?

Mutually Tangent Circles

Three congruent circles of radius 6 inches are mutually tangent to one another. Compute the area enclosed between them.