N students in a circle. Starting with student 1, following an IN-IN-OUT pattern the teacher circles clockwise multiple times, sending each student called OUT out, until one remains. For N=6, that's Student 1. (Ditto for N=4.) What's the next N for which student 1 remains?

Thrilled to see an fresh, kind, human, and delightful Algebra 1 curriculum come to life! (And yep, I'll stop being coy: It is based on my thinking and work.) See you at NCTM! Look for @SmartWithIt.

Global Math Week Day #5 #math #puzzle James Tanton The Global Math Project Dhimath

Students 1, 2, 3,..,N clockwise in a circle. Starting with student 1, teacher walks round multiple times saying IN-OUT-IN-IN-OUT-IN-IN-IN-OUT.. increasing the number of consecutive INs as she goes. Does this until one student remains. Student 1 remains for N=1,2,3. Any other N?

Global Math Week Day #6 #math #puzzle James Tanton The Global Math Project Dhimath

I am smitten with Sperner's Lemma: Label each of the four interior dots either A, B, C. Explain why you are forced to create at least three triangle fully labeled ABC.

Global Math Week Day #7 #math #puzzle James Tanton The Global Math Project Dhimath

I am smitten with Sperner's Lemma: Please label the interior dots A, B, C so that only one fully labeled ABC triangle appears.

Still smitten with Sperner's Lemma: I want to fill in each unlabeled circle with either A, B, or C so that no "fully labeled" ABC triangle appears anywhere. What letter must the red circle have?

Today's Puzzle: Every odd number is the difference of two squares. Which even numbers are the difference of two squares?

Every number is the difference of two triangular numbers: E.g. 5 = 15 - 10; 8 = 36 - 28; 52 = 1378 - 1326 But neither 5, 8, nor 52 is the *sum* of two triangular numbers. What is the first (positive) three-digit number that is not the sum of two triangular numbers?

Today's Puzzle: The square numbers: 0, 1, 4, 9, 16, ... (zero included). There are two ways to express 9 as a difference of two squares: 9 - 0 and 25 - 16. How many ways are there to so express 125? 135? Any given odd number?

Today's Puzzle: The square numbers: 0, 1, 4, 9, 16, .... (include zero). There is just one way to write 8 as a difference of two squares. How many ways are there to so write 360? (A general formula for the number of ways to write N as a difference of two squares?)

Today's Puzzle: The numbers 6 and 16 can each be written as the sum of two triangular numbers two different ways: 6 = 6+0 = 3+3; 16 = 15+1 = 10+6. What's the first three-digit number that can be expressed as the sum of two triangular numbers in at least two different ways?

It is known that if N is a sum of two square numbers then, in the prime factorization of N, any prime that appears that is one less than a multiple of 4 does so an even number of times. (Tricky!) Given that: Which numbers N can be expressed as the sum of two triangular numbers?

Today's Pedagogy Question: Here's a standardized test question. What sort of answer do you expect. What is "the" answer?

Are 0 and 3 the only triangular numbers whose doubles are also triangular?

There are infinitely many Pythagorean triples: square numbers that equal the sum of two non-zero squares. Are there infinitely many triangular numbers that equal the sum of two non-zero triangular numbers? (6 = 3 + 3 and 21 = 15 + 6 are two examples.)

Today's Puzzle: Double a triangular number can be triangular. E.g. 3 x 2 = 6. Ditto for triple, five-fold, six-fold, seven-fold: E.g. 15 x 3 = 45, 21 x 5 = 105, 6 x 6 = 36, 3 x 7 = 21 Can quadruple a triangular number again be triangular?

4 is not the quotient of two triangular numbers. Is 9?