James Tanton

A power of three can be the side-length of an integer right triangle. (Why?) Can it be the hypotenuse of one?

An integer right triangle has a power of two as one of its side lengths. Except for the 3-4-5 triangle, must that power of two be the shortest side length of the triangle?

How many integer right triangles have a side of length 2^100?

Today's Puzzle: How many integer different right triangles have a side-length of 8?

Can the hypotenuse of an integer right triangle be a power of two?

Is the 3-4-5 triangle the only integer right triangle with a side length of 4 ?

Quickie: Is there an integer right triangle with a side length of 2?

There are eight ways to arrange 4 pink and green dots in a row so that now two consecutive dots are pink. How many way are there to arrange 50 pink and green dots this way?

**MATH DECODED** **Explaining the WHY behind the Numbers and How they Operate!** It's a book launch tomorrow! Friday 5:30 pm Los Angeles time. Saturday 8:30 am Hong Kong time. jamestantonbooklaunch.lovable.app Come along! It shall be fun!

The number 3 can be written as a sum of integers: * 1 way if the number 1 cannot be used. * 4 ways if the number 1 can be used * 13 ways if two different versions of the number 1 can be used. How many ways if three different versions of the number one can be used?

Today's Puzzle: Place N dots in a row. Touching dots must be green. Single dots can be pink or green. Is the count of possibilities always a Hemachandra/Fibonacci number?

21 happens to be a Hemachandra/Fibonacci number. Coincidence?

There are 13 ways to underline some or all or none of the three dots in a row. How many ways are there to underline some or all or none of ten dots in a row?

Quickie: There are 4 ways to arrange three dots in a row so that some might touch. How many such ways are there to arrange 11 dots in a row with some touching, or not?

Today's Puzzle: Arrange red, yellow, or green dots in a row with no red dot immediately to the right of a green dot. 21 such arrangements with 3 dots. 8 of those begin with red. 21 and 8 are both Hemachandra/Fibonacci numbers. Coincidence?

Today's Puzzle: Arrange red, yellow, or green dots in a row with no red dot immediately to the right of a green dot. 21 such arrangements with 3 dots. How many with 4 dots in a row? 5? 6?

A bag contains 1 red, 1 blue, and 1 white ball. Pull one out at random: If red, you win! If blue, you lose. If white, put back and add 1 more red and 1 more blue to the bag and repeat. What are the chances of winning a) via just thinking? b) via writing an infinite sum?

It's a book launch! jamestantonbooklaunch.lovable.app The incredibly gracious and generous Dr. Jennifer Chang Wathall is hosting a zoom event a week from tomorrow and will ask me all about my whys and what-fors and secrets behind MATH DECODED (Corwin). It shall be mighty fun! Come join!

A bag contains N coins: all are normal H|T coins, except one, it's double-headed, H|H. A friend pulls out a coin at random, tosses it once, and reports seeing heads. Given this, what are the chances she pulled out the H|H coin? For which N is this probability 50%?

Another classic: A bag contains 99 regular coins (H|T) and 1 coin that has heads on both sides (H|H). Without looking, I pick a coin at random and have a friend to toss it ten times. She reports getting 10 heads in a row. What are the chances I picked the H|H coin for her?