The triangle shown with one side in red and its third corner on a dot has an even number of dots (6) on its boundary. Pick any other dot and use the red side and that dot to make a triangle. It too will have an even number of dots on its boundary! Why must this be so?

Entertaining Mathematical Puzzles by Martin Gardner PDF: abrarhashmi.wordpress.com/wp-content/upl…

Looked at some old writings. Wondering how I came up with the numbers 4, 6, 12 to make this puzzle solvable. Hmm! What other values for speeds work for this puzzle? (3,5,15, yes!, 2,5,10, no!) [Assume the path is composed of clearly-defined uphill, downhill, and flat sections.]

David Marain Fred G. Harwood Rémi Eismann G_Tulloch Yacob Goitom Ramin Khosravi John Allen Paulos A visual proof 😉

It is known that the three altitudes x,y,z of a triangle satisfy 1/x +1/y > 1/z, 1/y + 1/z > 1/x, and 1/z + 1/x > 1/y. If three numbers x, y, z meet these conditions, must there be a triangle with these lengths for its altitudes?

Quickie: One can smoothly transform a 3-4-5 triangle into a 4-4-4 triangle with intermediate (3+k)-4-(5-k) triangles preserving perimeter along the way. (k transitions from 0 to 1.) The top vertex of the triangle moves along a section of a curve. What curve?

Is the transition below smoothly transforming an a-b-c triangle into a d-e-f triangle too naive? Could an intermediate phase not actually be a triangle?

A meta-analysis of elite Olympic athletes (top 16 in the world) found they: • Started their sport at 10 years old • Focused on their sport at 15 • Skiing, soccer, basketball, and hockey players sampled other sports for 7 years The authors concluded: “Only after the age of 12

I desperately need to know if Mike ad-libbed all that or if he’s using the prompter. Either way, it’s one of the all-time great closes to a sporting event I’ve ever seen. It resonated deeply with the audience. If he did that off the top of his head, crown him now. Wow.

There is only 1 way to "split" a pile of N coins into one pile. There are N/2, rounded down to the nearest integer, ways to split them into 2 non-empty piles. (Order of the piles irrelevant.) Prove there are (N^2)/12, rounded to the nearest integer, ways to split into 3 piles!

Mathematics. "The Zero's Journey." By Grant Snider, Grant Snider, incidentalcomics.com, Used with permission.

12 toothpicks: There are 12 ways to split into three non-empty piles and exactly one-quarter of those examples make the sides of a triangle. What's the next number N for which exactly 25% of the piles of three you can make form a triangle?

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?

A Favourite: Can you make a three-strand braid with no free ends? Give it a try! (Notice how each strand is relatively flat.)

A Challenge: Can you make a four-strand braid with now free ends? Give it a try! (Notice how each strand is relatively flat.)

During one of the worst losing streaks of my career, our team president walked into my office. Keli McGregor. One of the best men I've ever known. He could have come to vent. To question my decisions. To ask hard questions. Instead, he said: "Cut to the chase, Clint. What's

Draw a loop of V and H steps in a 6x6 grid of squares visiting each cell exactly once. Pick's Theorem says the area of the polygon you create is sure to be 17. But can you see why without it the big theorem? Think about the L and R turns you make as you walk around the polygon.

One can walk a series of unit steps in a 6x6 square grid visiting each cell exactly once with as few as 5 rightward steps (and no less). What is the maximal number of rightward steps possible?

Some red and blue line segments on are drawn on a page in pen. Must there be a line that splits the amount of red ink each side of it exactly in half and, simultaneously, splits the amount of blue ink either side of it exactly in half too?

Math news: A powerful new invariant offers "a combination of strength and speed… that... can probe knots that were previously far out of reach." Reported by Erica Klarreich in @quantamagazine: quantamagazine.org/a-powerful-new…