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?

Here are some of the Thorny Math Questions I attend to in these COLLEGE ALGEBRA FOR HUMANS notes.
gdaymath.com/lessons/gmp/9-…
I still can't tell if I've written a College Algebra book or more of a book for pre-service teachers and general PD. What do folk think?
Either way ... here…

The set of points equidistant from an infinitely long line and a point is a parabola.
Describe the set of points equidistant from a square and a point inside the square.
(What if the point is on the rim of the square?)

The set of points equidistant from an infinitely long line and a point is a parabola. (What if the point is on the line?)
Describe the set of points equidistant from a square a a point outside the square.

The set all points equidistant from a fixed, infinitely long line and a point is a parabola.
How would you describe the set of all points equidistant from a finite line segment and a point?
Is it a smooth curve?

Daus sense zeros 🎲

👉🏼 Llença quatre daus de l'1 al 6. Perds si hi ha dos o més valors que, si es combinen amb els signes ➕ o ➖, el resultat és zero. Es pot guanyar? 🔍

Quina és la probabilitat de guanyar amb 3 daus? Explica com ho saps 🤔

🙌🏽 Proposta de James Tanton

Base b is 'robust' if each fraction 1/2,1/3,..,1/(b-1) is a ratio of two integers expressed in base b that use each digit 1, 2, 3,.... (up to some value) exactly once. (1/b cannot be so expressed. Why?)
🄸🄹🄺 shows base 10 is robust.
Base 3 and base 4 are robust. Base 5? 6?

It is possible to express each of 1/2, 1/3, and 1/4 as a ratio of two integers where each of the digits 1, 2, 3, .... (up to some value) are used exactly once.
Is there an easier way to express 1/3?

All solutions to all chapters are now done and included. Phew!

gdaymath.com/lessons/gmp/9-…

Question: Have I written a College Algebra tome, or is it really a pre-service teacher's piece?
ARITHMETIC & ALGEBRA FOR TEACHERS, TEACHING, AND RADICAL COMPREHENSION or some such?

Permutation & combination questions are annoying!
I roll 10 identical dice simultaneously.
a) How many of those outcomes have the form
xxx aa bb p q r
--six distinct values for a triple, two pairs of doubles, three singles?
b) How many different 'forms' are there?

NO ZEROS DICE GAME:
I roll four dice and get four values. I 'lose' if there are two or three or four values that combine with + & - signs to make zero. (e.g. 4, 2, 3, 2 loses since 2-2=0, and 2, 4, 5, 6 loses since 2+4-6 = 0)
Can I win?
If so, what are chances of winning?

Here are the Propp base-1.5 codes for the numbers 0 thru 40. (Each number is the sum of powers of 3/2 using the digits 0,1,2.)
Every fifth code, and only these codes, has the property that the alternating sum of its digits is a multiple of 5. Why so?
James Propp

Here are the Propp base-1.5 codes of the numbers 0 thru 40. (Each number is a sum of powers of 3/2 using digits 0,1,2.)
Just looking at the codes, is there a way to tell if a number is even? Does every 2nd code have a recognizable property? I personally don't see one!
James Propp

Here are 0 thru 40 in Propp Base 1.5. (Each number is a sum of powers of 3/2 with coeffs 0, 1, 2).
I don't know how to perform division in base 1.5.
I don't even know how to divide by two!
eg Dividing 2120010 (forty two) by 2 should give 21220 (twenty one). Thoughts?
James Propp

Here are the numbers 0 thru 40 in Propp base 1.5. (Each number is the sum of powers of 3/2 with coeffs 0,1,2.)

Can one perform long multiplication in base 1.5?
For example, is '44510' really 2120010, the code to forty two? (What are the 'carry' rules?)

James Propp

Here are the numbers 0 thru 40 in Propp Base 1.5. (Each number is a sum of powers of 3/2 using coeffs 0,1,2.)

Can one do long addition in base 1.5?
What are the rules for 'carrying'?
James Propp

What's the area of the parallelogram--in terms of x and y--inside this unit square?
[If x=y, then the area is (1-x)^2/(1+x^2).]

What is the area of the orange square--in terms of x--sitting inside the unit square?
(In particular, what do we get for x=1/2?)