Three checkers labeled A, B, C in that order L to R on a line.
A move = leapfrogging one checker over another to an empty spot. E.g. Moving A over C gives checkers in order BCA.
There are 6 orderings of three letters. Can we see all 6?
Four checkers A,B,C,D and all 24 orderings?

I've had requests to make my COLLEGE ALGEBRA FOR HUMANS notes available as hard-copy print versions. Each chapter turns out to be book length (!), so here are eight chapters each available on lulu.com (search for the title). Of course, free PDFs are here too:

Two fixed points A & B and fixed angle x.
Rotate a third point counterclockwise x degrees around A, then B, then A, then B, then A, then B, and so on.
Prove that if x is of the form x = a/b* 180 degrees for integers 0<a<b, then you will return to start (in 2b moves).

n fixed points in the plane A, B, C, ....
Choose another point and rotate it 90-degrees counterclockwise about each point in turn and repeat.
For n = 2, return to start in 4 quarter turns.
For n = 3, return to start in 12 quarter turns.
For n = 4? n = 5?
General claim?

Three points A, B, C fixed in a plane.
Choose another point and rotate it 90 degrees counterclockwise around A, then B, then C, then A, then B, then C, and so on.
Prove that you will return to start after twelve such quarter turns.

Okay ... here is tomorrow's X today. I accidentally posted it this morning for 10 seconds, but people noticed it and are already into it! (As I am ... I thought this is cool from some incidental math play I was doing yesterday.)

Two fixed points A and B in the plane.
Choose any

Two fixed points A and B.
Choose another point P and rotate is 90 degrees counterclockwise around A and then around B and then around A again and then around B again.
What do you notice?

The (infinitely long) red line is the set of points equidistant from a point A and a second geometric object B.
Must B be a point?

3^2 + 4^2 = 5^2.
The lilac triangle is a right triangle.
What can you say then about the area of a tilted little square compared to the area of a non-tilted little square?

3^2 + 4^2 = 5^2.
Do you trust this picture (modulo my lack of talent on MicroSoft Paint!)?
Is the lilac triangle a right triangle?

Backwards ...

Circle C & point P inside the circle.
For each point on C draw the circle that passes through P. What is the convex hull of all these circles?

[I can get parametric equns for this enveloping curve. But I suspect an explicit equation for it is quite unpleasant!]

Sunil Singh James Tanton describes Exploding Dots as a way to visualize long division, not a replacement. Once you understand it, you realize it's the same thing as long division and can use the simpler, standard algorithm with confidence.

gdaymath.com/lessons/explod…

Two intersecting circles.
Draw circles tangent to each of these two given circles. What 'curve' do the centers of these circle trace?

Following David Marain Fred G. Harwood
Call N 'sandwiched' if N-1 and N+1 are both prime numbers.
a) If N>4 is sandwiched, then N is a multiple of 2x3=6. Why?
b) If N and 2N are each sandwiched, then N is a multiple of 2x3x5=30. Why?
c) If N, 2N, and 3N each each sandwiched, then N is a

If a pile represents +1 and a pit represents -1, what number does this sign board represent ?
Exploding Dots is a great way to introduce negative numbers to students.
James Tanton Dhimath

Two circle, one inside the other.
Plot circles tangent to each of the these two circles. What curve do the centers of these circles outline?

What is the locus of points equidistant from two disjoint circles, one outside of the other, of the same radius? Different radii?

The set of points equidistant from ...
** a point and a line: a parabola
** two points: a line
** a point and a segment: a parabolic arc and two rays
** two lines segments?
That no doubt depends on the placement of the segments. What can you determine?

A fixed circle and a line, not touching.
Packing in circles that are tangent to each object, what curve is traced by their centers? Is it again a parabola (as per yesterday's puzzle where the circle and the line did touch)?

Classic:
Pack circles that just touch a fixed line and a fixed given circle tangent to that line. What curve is traced out by the centers of those circles? Is it a parabola--as seems to be the natural guess?