Two containers contain 1 cup of liquid between them. Initially, an unequal distribution of liquid. Do I have any hope of seeing an equal distribution of liquid by pouring half the contents of one container into the other as many times as I want?

1000 candies on a table each considered a pile of size 1. A "move" consists of taking three piles of the same size, eating one of them and combining the other two to make a bigger pile. What is the largest number of moves you could possibly conduct?

1000 pennies on a table each considered a pile of size 1. A "move" consists of combining any three piles of the same size into a single pile. What is the maximum number of moves you could make?.[How does this answer change if you must combine four piles of the same size? Ten?]

1000 pennies on a table, each considered a pile of size 1. A "move" consists of combining two piles of the same size into one pile. What is the maximal number of moves you could make?

Some Red/Black dumbbells are placed in a circle. If you count the number of RR adjacencies and the number of BB adjacencies, these counts are sure to be the same. Why?. Can you come up with more than one way to explain why this is so?

In this picture, 8 "pins" are arranged in a circle. There are 2 head/tail mix meeting points. Arrange 9 pins in a circle also with 2 head/tail meeting points.

The inf. long "number" made of repeating blocks of 052631578947368421 doubles to give itself with the last entry removed. Is this number really the result of putting all the powers of two in each place and performing all the base-10 carries? (After all, this doubles same way.)

Create an infinitely long "number" which, when multiplied by 19, gives the infinitely long answer . 99999999999. (Philosophically: Why must your "number" fall into a repeating pattern?)

428571 has the property that moving its last digit up front divides the number by three. Find a number with the property that moving its last digit up front halves the number instead -- or prove that not such (non-zero) number exists.

There is an 18-digit number with the property that moving its last digit up front doubles it value. (Origin: Presh Talwalker?) . Find a number such that if you move its last two digits up front (as a pair, keep their order the same) produces double the number.

It is well known that multiplying 052631578947368421 by two has the effect of moving the last digit up front. Most people don't like this example because of the leading zero. Find five more numbers with this curious property, none with a leading zero. (How many egs are there?)

Here are two numbers for which dividing by 3 has the effect of moving final digits up front. Is my list of such numbers complete?

The number 428571 has the property that dividing it by 3 has the same effect as removing its last digit and placing it up front. Is there another number that behaves this way with respect to division by 3?

There are 11 ways to subdivide a pentagon into polygons, and 11 ways to insert parentheses into a sum of four terms. Coincidence? .Same true for subdivision of a hexagon and parentheses in a sum of five term? .An (N+1)-gon and a sum of N terms?

There a 5 ways to place parentheses in a sum of four terms so that one is only ever adding two terms at a time and 5 ways to subdivide a pentagon with a distinguished edge into triangles. What exactly is the correspondence between the two sets? Does your mapping generalize?

Most people make sense of the dots-on-a-circle sequence using the classic formula V - E + F = C established yesterday. (We made sense of it without it a few days ago.) .Can you reconstruct the classic approach using the classic formula?

For picture with V dots (vertices), E lines (edges) connecting two dots or a dot to itself, making F finite regions (faces), and C components, we must have V - E + F = C. Can you see why?

A "line" starts and ends on the rim of a circle. Only two lines are allowed to cross to make an intersection point. Prove, in all such diagrams:."circles + lines + intersection points = pieces"

Five lines drawn in a circle. Minimum count of pieces you could see is 6, maiximum 16. Possible to see every count in between? (That is, for N between 6 and 16, is there an arrangement of five lines with producing N regions?)

Prove that in the famous dots-on-a-circle puzzle, with maximal number of regions produced by lines, we are sure to have: .circles + lines + intersections = pieces.(And this is C(n,0)+C(n,2)+C(n,4)!)