E.g., for algebra, "x^4+4 doesn't factor, does it?"

x^4 + 4 = (x^2 + 2i)(x^2 - 2i) = (x + i + 1)(x - i - 1)(x - i + 1)(x + i - 1) = (x^2 + 2x + 2)(x^2 - 2x + 2) So, yes, it does factor, and it only requires knowledge of the principal eighth roots of unity to do so.

Geometry: "How long are the two equal chords that trisect the area of the unit disk?"

Two dimensions are too easy for me, so I'll instead find the diameters of the two equal discs that trisect the volume of the unit ball. By using volumes of revolution, we have: y = sqrt(1 - x^2) Integrate(y^2, dx) = x - x^3/3 + constant By setting this as zero for x = -1, so the constant must be 2/3. The full (scaled) volume is 4/3, so we want a volume of 4/9 to the left of the disc. Hence, we have the cubic equation: x - x^3/3 + 2/3 = 4/9 or, more manageably, x^3 - 3x - 2/3 = 0. Now, I shall let a^3 + b^3 = -2/3, ab = 1, so we obtain: x^3 + a^3 + b^3 - 3abx = 0 ('difference of three cubes') a^3 and b^3 are the roots of the quadratic y^2 + 2y/3 + 1 = 0, which has the solutions -1/3 + sqrt(-8)/3 and -1/3 - sqrt(-8)/3. Now, there are three roots for x. The only appropriate solution is: x = omega^2 cbrt(1/3 - i sqrt(8)/3) + omega cbrt(1/3 + i sqrt(8)/3) This is approximately -0.226074, leading to a diameter of sqrt(1-x^2) = 1.94882.

Or, "To construct the tangent to a circle at a given point, why are you showing us this complicated compass procedure with perpendicular bisectors, etc., when all you need is to add four more points at random, join all five in a star, draw two more lines, and you're done, no compass needed?" (See http://www.tweedledum.com/rwg/tan.htm .)

Enabling students to construct tangents to any arbitrary conic would be giving them too much power... Sincerely, Adam P. Goucher

On Fri, May 11, 2012 at 9:26 PM, Bill Gosper <billgosper@gmail.com> wrote:

Long before XKCD named it, I was a big fan of teacher-trolling-by-proxy. All you need is an bright evil kid who can feign a look of innocent bewilderment. E.g., for algebra, "x^4+4 doesn't factor, does it?" Geometry: "How long are the two equal chords that trisect the area of the unit disk?" Or, "To construct the tangent to a circle at a given point, why are you showing us this complicated compass procedure with perpendicular bisectors, etc., when all you need is to add four more points at random, join all five in a star, draw two more lines, and you're done, no compass needed?" (See http://www.tweedledum.com/rwg/tan.htm .) Calculus: Which convergence test should I use for sum (csc n)/n^2 ?

[...] Here's a nice one from Neil: Gee, if the area of a sphere is d/dr of 4/3 π r^3 = 4 π r^2, then the area of a cube must be d/ds of s^3 = 3 s^2 ! This should take care of all sorts of polyhedra! --rwg Why don't you say sphere area = π d^2? Are you secretly one of those τ people?