## Henry Heaton - Problem setter and solver

Henry Heaton was both a proposer of problems and an enthusiastic solver of problems. Although he was interested in a broad range of mathematical problems, he specialised in problems concerning probability, particularly geometric probability problems. We give below a selection of problems that Heaton proposed and those he solved. Some are taken from textbooks (and the reference is given) but many are original. Some of the problems appear more than once. For example a problem will appear when Heaton proposes it and then again when solutions are presented. We give each reference. The problems we list below are all taken from

*The Analyst*or from the*American Mathematical Monthly*. We note, however, that Heaton also published and solved problems in several other journals.**Problem:**A ball rolls down the convex surface of a fixed sphere, the friction being just sufficient to prevent sliding; find the point where it leaves the sphere.

**Solution by:**Henry Heaton.

**Journal Reference:***The Analyst***1**(4) (1874), 69.**Problem:**Find the real value of (√-1)^{(√-1)}.

**Solution by:**Henry Heaton, Des Moines, Iowa.

**Journal Reference:***The Analyst***2**(2) (1875), 59.**Problem:**Through two given points draw a circle bisecting the circumference of a given circle.

**Solution by:**Henry Heaton, Des Moines, Iowa.

**Journal Reference:***The Analyst***3**(3) (1876), 85.**Problem:**A circle, radius*r*, is placed at random on another equal circle. Prove that the average area of the greatest ellipse that can be inscribed in the area common to the two circles, is π*r*^{2}(π/2 - 4/3).

**Solution by:**Henry Heaton, Des Moines, Iowa.

**Journal Reference:***The Analyst***3**(3) (1876), 89-90.**Problem:**If four bricks are placed on each other at random, with their longest axes horizontal and in the same vertical plane, determine the probability that the pile will stand.

**Solution by:**Henry Heaton.

**Journal Reference:***The Analyst***3**(3) (1876), 152-154.

[**Note.**The solution given in the reference is due to the Editor of*The Analyst*, J E Hendricks, but he states in a footnote that the problem has been solved by Henry Heaton.]

**Problem:**To inscribe a square in a given quadrilateral.

**Solution by:**Henry Heaton, B.S., Des Moines, Iowa.

**Journal Reference:***The Analyst***3**(5) (1876), 160.**Problem:**A radius is drawn in the circle*x*^{2}+*y*^{2}=*a*^{2}and from its extremity an ordinate. From the foot of the ordinate a line is drawn perpendicular to the radius. Find and discuss the envelope of this last line.

**Solution by:**Henry Heaton, B.S., Des Moines, Iowa.

**Journal Reference:***The Analyst***3**(5) (1876), 161-162.**Problem:**Proposed by Henry Heaton.

When*x*= 0 find the value of (*x*- sin*x*)/(tan*x*-*x*).

**Journal Reference:***The Analyst***3**(5) (1876), 163.**Problem:**If a body be impelled by the force of a fluid having the velocity*v*_{1}and if the force of the impelling fluid to move the body be as*mv*+*nv*^{2}, where*v*is the velocity of the fluid relative to the body, what must be the velocity of the body in order that the work performed on the body by the fluid, in a unit of time, may be a maximum?

**Solution by:**Henry Heaton.

**Journal Reference:***The Analyst***3**(6) (1876), 190-191.**Problem:**A point is taken at random in the surface of a given circle, and from it a line equal in length to the radius is drawn, so as to lie wholly in the surface of the circle. Find the chance that the line intersects a given diameter.

**Solution by:**Henry Heaton, B.S., Des Moines, Iowa.

**Journal Reference:***The Analyst***4**(1) (1877), 23-24.**Problem:**An underwriter insures three vessels, the first an iron steamer, the second a steamer not of iron and the third a sailing vessel, at $20,000, $15,000, and $10,000, respectively. One of them is known to have been burned at sea; and three persons, A, B, C, whose respective veracities are^{3}/_{4},^{4}/_{5}, and^{5}/_{6}, report as follows: A, that the lost vessel was an iron steamer; B, that it was not a sailing vessel; and C, that it was a sailing vessel. Required the expectation of loss to the underwriter, the a priori probability of destruction by fire being twice as great in case of a steamer as of a sailing vessel.

**Solution by:**Henry Heaton, B.S., Des Moines, Iowa.

**Journal Reference:***The Analyst***4**(2) (1877), 62-63.**Problem:**Two equal circles, radii*r*, are drawn on the surface of a circle, radius 2*r*; find the average of the area common to the two circles.

**Solution by:**Henry Heaton.

**Journal Reference:***The Analyst***4**(3) (1877), 93.**Problem:**Two equal spheres placed in a paraboloid with its axis vertical touch one another at the focus. If*W*be the weight of a sphere,*R*,*R*' the pressures upon it, prove that*W*^{2}:*R*.*R*' :: 3 : 2.

**Solution by:**Henry Heaton, Sabula, Iowa.

**Journal Reference:***The Analyst***4**(6) (1877), 189.**Problem:**Prove that the locus of points, which have the sum of the squares of their distances from fixed points constant, is the surface of a sphere.

**Solution by:**Henry Heaton, Sabula, Iowa.

**Journal Reference:***The Analyst***5**(3) (1878), 91-92.

**Problem:**If*M*,*N*,*P*,*Q*are four random points in the surface of a circle, find the chance that*E*, the intersection of the straight lines through*M*,*N*and*P*,*Q*, lies between*M*,*N*and between*P*,*Q*.

**Solution by:**Henry Heaton, Sabula, Iowa.

**Journal Reference:***The Analyst***5**(4) (1878), 123.**Problem:**A harbour*A*is so situated with reference to two headlands*B*and*C*, that the angle*BAC*is a right angle. A ship sails in a course making an angle of 55° with*AB*, 45 ms. to*D*, when*DB*=*DC*: she then sails forward on the same course 15 ms. to*E*, when*BEC*is a straight line. Required*AB*,*AC*,*DB*,*EB*and*EC*.

**Solution by:**Henry Heaton, Sabula, Iowa.

**Journal Reference:***The Analyst***5**(5) (1878), 153.**Problem:**Find the algebraic equation whose roots are the real quantities found by giving integral values to*k*in

*x*= cos(2*k*π/*n*),

*n*being a given integer. Also the equation whose solution is

*x*= sin(2*k*π/*n*).

**Solution by:**Henry Heaton, Sabula, Iowa.

**Journal Reference:***The Analyst***5**(6) (1878), 186-187.**Problem:**If three points be taken at random in the circumference of a circle, required the probability that the triangle formed by joining them will be acute.

**Solution by:**Henry Heaton, Atlantic, Iowa.

**Journal Reference:***The Analyst***6**(5) (1879), 155-156.**Problem:**Solve, algebraically, the equation*x*^{17}- 1 = 0.

**Solution by:**Henry Heaton, Atlantic, Iowa.

**Journal Reference:***The Analyst***6**(6) (1879), 186-187.**Query:**By Henry Heaton.

Is there any known general method of elimination when we have two or more equations containing two or more unknown quantities, the equations being of the third degree or higher?

**Journal Reference:***The Analyst***8**(2) (1881), 63.**Problem:**Show that the definite integral from*t*= 0 to*t*= π/2 of

√(1 -*c*)*dt*/(1 -*c*cos^{n}*t*) = π/√(2*n*)

where*c*is indefinitely nearly equal to unity,*n*being a positive quantity.

**Solution by:**Henry Heaton, Lewis, Iowa.

**Journal Reference:***The Analyst***8**(6) (1881), 197-198.**Problem:**Given two points*A*and*B*, and a circle*K*having its centre at 0. Let any circle*L*be drawn through*A*and*B*so as to cut the circumference of the circle*K*in two variable points*m*and*n*. Show that the circle through 0,*A*and*B*is cut by the variable circle through 0,*m*and*n*, in a fixed point*P*.

**Solution by:**Henry Heaton.

**Journal Reference:***The Analyst***9**(4) (1882), 125.**Problem:**Proposed by Henry Heaton, Lewis, Iowa.

Evaluate the integral of √(1 + cos^{4}*t*)*dt*from*t*= 0 to*t*= π/2.

**Journal Reference:***The Analyst***9**(4) (1882), 128.**Problem:**Transform the definite integral of*f*(*x*) from*b*to*a*so that the limits of integration be*m*and*n*.

**Solution by:**Henry Heaton, Atlantic, Iowa.

**Journal Reference:***The Analyst***10**(1) (1883), 29.**Problem:**Proposed by Henry Heaton, M.S., Atlantic, Iowa.

Through two given points to pass four spherical surfaces tangent to two given spheres.

**Journal Reference:***Amer. Math. Monthly***1**(1) (1894), 24.**Problem:**Proposed by Henry Heaton, M.S., Atlantic City, Iowa.

Through a given point to draw four circles tangent to two given circles.

**Journal Reference:***Amer. Math. Monthly***1**(1) (1894), 24.

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***1**(7) (1894), 232-233.

**Journal Reference:***Amer. Math. Monthly***1**(8) (1894), 268.**Problem:**Proposed by Professor Henry Heaton, Atlantic, Iowa.

Through two given points to draw two circles tangent to a given circle.

**Journal Reference:***Amer. Math. Monthly***1**(2) (1894), 51.

**Journal Reference:***Amer. Math. Monthly***1**(8) (1894), 271.**Problem:**Proposed by Professor Henry Heaton, M.S., Atlantic, Iowa.

Through three given points to pass two spherical surfaces tangent to a given sphere.

**Journal Reference:***Amer. Math. Monthly***1**(4) (1894), 132.

**Journal Reference:***Amer. Math. Monthly***1**(11) (1894), 394.**Problem:**Proposed by Henry Heaton, M.S., Atlantic, Iowa.

Through two given points to pass four spherical surfaces tangent to two given spheres.

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***1**(6) (1894), 199-200.**Problem:**Draw a circle bisecting the circumference of three given circles.

**Solution by:**Henry Heaton, M.S., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***1**(8) (1894), 270-271.**Problem:**Suppose that in a meadow the grass is of uniform quality and growth and that 6 oxen or 10 colts could eat up 3 acres of the pasture in^{18}/_{26}of the time in which 10 oxen and 6 colts could eat up 8 acres; or that 600 sheep would require 2^{6}/_{7}weeks longer than 660 sheep to eat up 9 acres. In what time would an ox, a colt and a sheep together eat up an acre of the pasture on the supposition that 589 sheep eat as much in a week as 6 oxen and 11 colts? By Arithmetic, if possible.

**Solution by:**Henry Heaton, M.S., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***3**(5) (1896), 139.

**Journal Reference:***Amer. Math. Monthly***3**(12) (1896), 323-324.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

Required the average area of all triangles two of whose sides are*a*and*b*.

**Journal Reference:***Amer. Math. Monthly***3**(5) (1896), 154.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

If every point of an ellipse be joined with every other point, what is the average length of the chords thus drawn?

**Journal Reference:***Amer. Math. Monthly***3**(6/7) (1896), 192.**Problem:**Prove that every algebraic equation can be transformed into another equation of the same degree, but which wants its*n*th term.

**Solution by:**Henry Heaton, M.Sc., County Surveyor, Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***3**(8/9) (1896), 210-211.**Problem:**Given*x*^{2}+*x*√(*xy*) = 10, and*y*^{2}+*y*√(*xy*) = 20 to find*x*and*y*by quadratics.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***3**(8/9) (1896), 211-212.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

In a circle whose radius is a, chords are draw through a point distant b from the centre. What is the average length of such chords, (1), if a chord is drawn from every point of the circumference, and (2), if they are drawn through the point at equal angular intervals?

**Journal Reference:***Amer. Math. Monthly***3**(10) (1896), 259.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

What is the average length of all the chords that may be drawn from one extremity of the major axis of an ellipse if they are drawn at equal angular intervals?

**Journal Reference:***Amer. Math. Monthly***3**(11) (1896), 288.**Problem:**Solve the simultaneeous equations:

*a*^{2}*x*= (2*x*^{2}-*a*^{2})√(*x*^{2}+*y*^{2})

*b*^{2}*y*= (2*y*^{2}-*b*^{2})√(*x*^{2}+*y*^{2})

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***3**(12) (1896), 303-304.**Problem:**Solve the differential equation,^{dy}/_{dx}=*y*(*x*-*y*)/*x*(*x*+*y*), and show that Heaton_problems.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***3**(12) (1896), 313-314.**Problem:**A certain solid has a square, side =*a*, for its base, and all parallel sections are squares, the two sections through the middle points of the opposite side of the square are semi-circles, however. Find surface, volume, and centre of gravity of each.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***3**(12) (1896), 314-315.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

What is the average length of the chords that may be drawn from one extremity of the major axis of an ellipse to every point of the curve?

**Journal Reference:***Amer. Math. Monthly***3**(12) (1896), 330.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

If every point of an ellipse he joined with every other point, what is the average length of the chords thus drawn?

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***4**(1) (1897), 29-30.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

In a circle whose radius is*a*, chords are drawn through a point distant*b*from the centre. What is the average length of such chords,

(1), if a chord is drawn from every point of the circumference, and

(2), if they are drawn through the point at equal angular intervals?

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***4**(3) (1897), 93.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

What is the average length of all the chords that may be drawn from one extremity of the major axis of an ellipse if they are drawn at equal angular intervals?

**Journal Reference:***Amer. Math. Monthly***4**(3) (1897), 93-94.**Problem:**There is a triangle whose sides repulse a centre of force within the triangle with an intensity that varies inversely as the distance of the centre of force from each point of the sides of the triangle. What the position of equilibrium of the centre?

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***4**(4) (1897), 112-113.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

What is the average length of the chords that may be drawn from one extremity of the major axis of an ellipse to every point of the curve?

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***4**(4) (1897), 119.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

A man is at the centre of a circle whose diameter is equal to three of his steps. If each step is taken in a perfectly random direction, what is the probability, (1), that he will step outside the circle at the second step, and, (2), that he will step outside at the third step?

**Journal Reference:***Amer. Math. Monthly***4**(4) (1897), 122.**Problem:**A square whose side is 2*a*and an equilateral triangle whose altitude is 3*a*are fastened together at their centres, but otherwise free to move. If they are thrown on a floor at random, what is the average area common to both?

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***4**(6/7) (1897), 191.**Problem:**A straight line of length*a*is divided into three parts by two points taken at random; find the chance that no part is greater than*b*. [From Hall and Knight's Higher Algebra.]

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***4**(8/9) (1897), 221-222.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

From a point on the surface of a circle two lines are drawn to the circumference. Required the average area that may be cut from the circle in this way if the lines are supposed to be drawn at equal angular intervals.

Query I. How does this differ from problem 32?

Query II. Is sector the proper word to use for the surface, thus cut off?

Query III. Is it absolutely correct to use the word random in average problems?

**Journal Reference:***Amer. Math. Monthly***4**(8/9) (1897), 229.**Problem:**A chain 16 feet long is hung over a smooth pin with one end 2 feet higher than the other end and then let go. Show that the chain will run off the pin in about 7/5 second. [Wright's Mechanics, page 92.]

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***4**(11) (1897), 279-280.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

A man is at the centre of a circle whose diameter is equal to three of his steps. If each step is taken in a perfectly random direction, what is the probability,

(1), that he will step outside the circle at the second step, and,

(2), that he will step outside at the third step?

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***4**(11) (1897), 283.**Problem:**One circle touches another internally, and a third circle whose radius is a mean proportional between their radii, passes through the point of contact. Prove that the other intersections of the third circle with the first two are in a line parallel to the common tangent of the first two. [From Phillips and Fisher's Geometry.]

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(1) (1898), 18.**Problem:**If the extremities of the base of a triangle be joined by straight lines to the exterior angles of squares constructed upon its two sides, the superior pair of lines thus drawn intersect at right angles; the inferior pair intersect at a point in a line drawn from the vertical angle perpendicular to the base.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(1) (1898), 19.**Problem:**Find the volume and surface generated by revolving about the*Y*axis, the catenary y =^{1}/_{2}a(e^{x/a}+ e^{-x/a}), from*x*= 0 to*x*=*a*. [Osborne's Calculus, page 255, example 8]

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(1) (1898), 22-23.**Problem:**Find the chance that the centre of gravity of a triangle lies inside the triangle formed by three points taken at random within the triangle. [From Williamson's Integral Calculus.]

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(2) (1898), 54-55.**Problem:**A chord is drawn through two points taken at random in the surface of a circle. If a second chord be drawn through two other points taken at random in the, surface, find the chance that the quadrilateral formed by joining the extremities of the two chords will contain the centre of the circle.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(2) (1898), 56-57.**Problem:**An endless uniform chain is hung over two small smooth pegs in the same horizontal line. Show that, when it is in a position of equilibrium, the ratio of the distance between the vertices of the two catenaries to half the length of the chain is the tangent of half the angle of inclination of the portions near the pegs.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(3) (1898), 92-93.**Problem:**Find the radius of at sphere of given specific gravity which will rest just immersed in a fluid whose density varies as its depth.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(4) (1898), 112.**Problem:**What must be the ratio of the two legs of a uniform and heavy right triangle suspended from the centre of the inscribed circle, if this triangle will rest with the shorter leg in a horizontal position?

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***5**(4) (1898), 113**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

From a point on the surface of a circle two lines are drawn to the circumference. Required the average area that may be cut from the circle in this way if the lines are supposed to be drawn at equal angular intervals.

Query I. How does this differ from problem 32?

Query II. Is sector the proper word to use for the surface thus cut off?

Query III. Is it absolutely correct to use the word random in average problems?

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***5**(4) (1898), 114-115.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

A witness in court who undertook to recognize the signature of an individual failed four times in succession. What is the probability that he was correct the fifth time? An actual occurrence.

**Journal Reference:***Amer. Math. Monthly***5**(6/7) (1898), 187.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

A witness in court who undertook to recognize the signature of an individual failed four times in succession. What is the probability that he was correct the fifth time? An actual occurrence.

**Journal Reference:***Amer. Math. Monthly***6**(3) (1899), 89.**Problem:**When a watch is wound up, the mainspring is closely coiled around a cylindrical piece called the hub of the barrel-arbor. When entirely run down the spring forms an annulus against the inner circumference of the barrel. Show that if the width of the annulus is a little more than one-fourth of the radius of the barrel, the spring will run the watch the greatest number of hours at one winding, the diameter of the hub being one-third the inside diameter of the barrel.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***6**(6/7) (1899), 177-178.**Problem:**Proposed by Henry Heaton, M.Sc., Atlantic, Iowa.

A witness in court who undertook to recognize the signature of an individual failed four times in succession. What is the probability that he was correct the fifth time? An actual occurrence.

**Journal Reference:***Amer. Math. Monthly***6**(8/9) (1899), 207.**Problem:**On an average, 1 vessel out of every n is wrecked. Find the chance that out of m vessels expected p at least will arrive safely.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***6**(8/9) (1899), 210-211.**Problem:**Draw a circle tangent to a given circle and tangent to a given chord at a given point.

**Solution by:**Henry Heaton, A.M., Atlantic Iowa.

**Journal Reference:***Amer. Math. Monthly***6**(11) (1899), 275.**Problem:**If perpendiculars are dropped from the vertices of a regular polygon upon any diameter of the circumscribed circle, the sum of the perpendiculars which fall on one side of this diameter is equal to the sum of those which fall on the opposite side. [From Chauvenet's Treatise on Elementary Geometry.]

**Solution by:**Henry Heaton, B.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***7**(2) (1900), 40.**Problem:**Show that

log[*x*-*a*-*b*√(-1)] =^{1}/_{2}log[(*x*-*a*)^{2}+*b*^{2}]- √(-1)tan^{-1}(*b*/(*x*-*a*)).

Naperian logarithms being used.

**Solution by:**Henry Heaton, M.Sc., Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***7**(4) (1900), 118-119.**Problem:**Find the sum of the first n+1 terms of the series

1 +*m*/1! +*m*(*m*+1)/2! +*m*(*m*+1)(*m*+2)/3! + ...

**Solution by:**Henry Heaton, Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***12**(3) (1905), 70.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

What is the average length of all straight lines that can be drawn within a given square parallel to one of the diagonals?

**Journal Reference:***Amer. Math. Monthly***12**(6/7) (1905), 146.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

What is the average length of all straight lines that can be drawn within a given square?

**Journal Reference:***Amer. Math. Monthly***12**(8/9) (1905), 161.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

What is the average length of all straight lines that can be drawn within a given square parallel to one of the diagonals?

**Journal Reference:***Amer. Math. Monthly***12**(10) (1905), 181.**Problem:**Proposed by Henry Heaton, Atlantic. Iowa.

What is the average length of all straight lines that can be drawn within a given square?

**Journal Reference:***Amer. Math. Monthly***12**(11) (1905), 198.**Problem:**Proposed by Henry Heaton, Atlantic. Iowa.

What is the average length of all straight lines that can be drawn within

a given square in every possible direction and every possible length from every point of the square; if all the lines are equally distributed about the starting point and equally distributed as to length.

[The problem as restated above is somewhat different from the one solved in our columns last month. As the above conveys the original meaning of the Proposer it is published as a third solution.]

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***12**(12) (1905), 227-228.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

Chords are drawn through every point of the surface of a given circle in

every possible direction. What is their average length?

**Journal Reference:***Amer. Math. Monthly***12**(12) (1905), 236.**Problem:**Find the sum, to*n*terms, of

1 +*n*/2 +*n*(*n*+ 2)/2.4 +*n*(*n*+ 2)(*n*+ 4)/2.4.6 + ... .

**Solution by:**Henry Heaton, Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***13**(1) (1906), 13-14.**Problem:**What is the average length of all straight lines that can be drawn within a given triangle?

**Solution by:**Henry Heaton, Atlantic, Iowa.

**Journal Reference:***Amer. Math. Monthly***13**(1) (1906), 15-16.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

Chords are drawn through every point of the surface of a given circle in every possible direction. What is their average length?

**Journal Reference:***Amer. Math. Monthly***13**(2) (1906), 39.**Problem:**Solve the simultaneous equations*x*+*y*= 10, 3*x*= log_{10}(*y*).

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***13**(4) (1906), 81-82.**Problem:**Sum the infinite series*n*^{2}/(4*n*^{2}- 1)^{2}beginning with*n*= 1,*n*being always odd.

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***13**(4) (1906), 83.**Problem:**Evaluate the indefinite integral of (√(1 +*y*))/(1 +*y*^{2})*dy*.

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***13**(4) (1906), 83-84.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

Through every point of the circumference of a given circle, chords are drawn in every possible direction. What is their average length?

**Journal Reference:***Amer. Math. Monthly***13**(8/9) (1906), 169.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

Through every point of the sides of a given square, straight lines are drawn across the square in every possible direction. What is their average length.

**Journal Reference:***Amer. Math. Monthly***13**(11) (1906), 220.**Problem:**A circular arc, with centre at one corner of a given square, is drawn through a point taken at random in the square. What is the average length of the arc within the square?

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***13**(12) (1906), 233.**Problem:**Proposed by Henry Heaton, Belfield, N.D.

Through every point of the circumference of a given circle, chords are drawn in every possible direction. What is their average length?

**Journal Reference:***Amer. Math. Monthly***13**(12) (1906), 234.**Problem:**Find the number of real roots of the equation 100sin*x*=*x*, and show the largest root is approximately 96.10. Find tan 39° to three places of decimals. How many real roots of tan*x*= 1/*x*^{2}lie between 0 and 27π?

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***14**(1) (1907), 16-17.**Problem:**Two points are taken at random in a triangle, the line joining them dividing the triangle into two portions. Find the mean value of that portion containing the centre of gravity.

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***14**(2) (1907), 33-34.- Through every point of a given square straight lines are drawn in every possible direction, terminating in the sides of the square. What is the average length of such lines?

**Solution by:**the Proposer.

**Journal Reference:***Amer. Math. Monthly***14**(2) (1907), 38.

**Journal Reference:***Amer. Math. Monthly***14**(6/7) (1907), 139.

**Journal Reference:***Amer. Math. Monthly***14**(8/9) (1907), 159-160.

**Journal Reference:***Amer. Math. Monthly***14**(10) (1907), 181. **Problem:**Two random planes cut a given sphere. What is the chance that they

intersect within the sphere?

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***14**(3) (1907), 61-62.**Problem:**Proposed by Henry Heaton, Belfield, N.D.

Through every point of the circumference of a given circle, chords are drawn in every possible direction. What is their average length?

No solution has been received.

**Journal Reference:***Amer. Math. Monthly***14**(5) (1907), 108.**Problem:**A point within a given triangle is joined to each of the corners. What is the average of the sum of the lengths of these three lines?

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***14**(6/7) (1907), 137-139.**Problem:**If a line l is divided into*n*parts by*n*-1 points taken at random on it, what is the mean value of the pth power of one of the parts taken at random?

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***14**(8/9) (1907), 157-158.**Problem:**Two random lines cut a given circle. What is the chance that they intersect within the circle?

**Solution by:**Henry Heaton, Belfield, N.D.

**Journal Reference:***Amer. Math. Monthly***14**(12) (1907), 231.**Problem:**Proposed by Henry Heaton, Atlantic, Iowa.

Show that sin 3° = 1/16 (√30 + √10 - √6 - √2) + 1/8 (√(5 + √5) - √(15 + 3√5)).

**Journal Reference:***Amer. Math. Monthly***24**(9) (1917), 426.

**Journal Reference:***Amer. Math. Monthly***25**(4) (1918), 171.