James Tanton

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.

Today's Social/Pedagogical Question: People spend hours throwing orange balls through metal hoops, stroking horse hair along metal wires, creating passages of text 17 syllables long... When students ask "When will I ever use this?" what are they perhaps really asking/lamenting?

Can every NxN grid of squares (N≥3) be tiled with 1x3 and 3x1 trominoes with at most one cell left blank?

One cannot tile a 7x7 grid with 1x3 and 3x1 tiles. But one might be able to tile 48 of those squares, leaving one cell blank. What must be the color of the blank cell? Is every cell of that color a blank cell for some tiling of the remaining 48 cells?

At most ⌊N/a⌋ segments of length a fit along a line segment length N. (⌊ ⌋ = round down to the nearest integer.) I want to fit axb rectangles into an NxM rectangle, all oriented the same way, with sides parallel to the sides of the big rectangle. Formula for max number?

One can fit at most ⌊N/a⌋ segments of length a along a line segment of length N. (Here "⌊ ⌋" means round down to the nearest integer.) What's the formula for the maximal number of squares of side length a one can fit in a square of side length N? (Cubes?)

Two cups of marbles. Pour half of L into R, then half R into L. (Transfer any extra odd marble as well if there is one.) Repeat 16 marbles always enter 11|5<->5|11 oscillation no matter starting distribution. Unique final oscillation pattern for every starting count of marbles?

Two containers with marbles. Pour half contents of left container into right, then half the contents of right into left. (If an odd marble at any point, transfer the extra odd as well.) Repeat. This act must enter a cycle. Must that cycle be of period 2?

15 marbles distributed between two containers. Pour half contents of left container into right, then transfer half the contents of R into L. (If an odd marble at any point, transfer the extra odd as well.) Repeat over and over. What happens in long run?

1 cup of liquid distributed between A and B. A "move" consists of pouring 2/3 of the contents of one into the other. Choose any value x in the Cantor subset of [0,1]. Explain how you can end up x amount of liquid in cup A via a sequence of moves within 0.001 cup accuracy.

Two containers with 1 cup of liquid between them. n is a positive integer. A "move" pours all but 1/n th of the liquid of one container into the other. Express the amount of liquid in A as a base-n decimal: 0.abcd.... What do you see if repeatedly pour liquid back and forth?

1 cup of liquid between two containers. An A move: Pour half the contents of A into B. A B move: Pour half the contents of B and A. What sequence of A and B moves will land me 2/5 of a cup of liquid in cup A (or as close to 2/5 of a cup to any degree of accuracy I desire)?

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?

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.