Wild About Math! Making Math fun and accessible

A challenging 11111… puzzle

I received this email from Mr. Brian Silverman who gave me permission to publish it.

I noticed that you were asking for interesting factoids about the number 11.

This isn't quite a factoid about 11 but here's a math puzzle a Russian friend of mine said her son worked out with 7th graders. Seems to me to be beyond most undergrads here. I'm not including the answer, to give you a chance to work on it if you want. but will send it if you ask.

"The question is if it's true that among the numbers consisting of only "1"s (1; 11; 111; 1,111; etc.) there is a number (maybe many) that is divisible by 572,003?

Actually, 572003 is taken arbitrarily. 57 is the number of the school (schools here are mostly numbered instead of having names) and 2003 you possibly know what is recently used for (yes, yes, here in Moscow, too). "

Brian

This puzzle doesn't seem at all obvious to me, especially if one needs to solve it for arbitrary numbers other than 572003. I thought of putting it into my pile of problems to someday solve but then thought it'd be fun to post here.

Can Russian 7th graders solve this? How much help did they get?

Can you solve it?

A deeper understanding of Rubik’s cube

From MITnews:

The math of the Rubik’s cube

New research establishes the relationship between the number of squares in a Rubik’s-cube-type puzzle and the maximum number of moves required to solve it.

Erik Demaine, an associate professor of computer science and engineering at MIT; his father, Martin Demaine, a visiting scientist at MIT’s Computer Science and Artificial Intelligence Laboratory; graduate student Sarah Eisenstat; Anna Lubiw, who was Demaine’s PhD thesis adviser at the University of Waterloo; and Tufts graduate student Andrew Winslow showed that the maximum number of moves required to solve a Rubik’s cube with N squares per row is proportional to N^2/log N. “That that’s the answer, and not N^2, is a surprising thing,” Demaine says.

Hat tip to John Cook.

Mental Floss brain game

Mental Floss is one of my very favorite magazines. Here's a simple (if you see it) puzzle from their web-site:

The answer is here.

Can you name … ?

Can You Name All 16 Non-Negative Integers Whose English Names Spell With No Repeating Letters In 10 Minutes?

While there are an infinite amount of numbers, there are, surprisingly, only 16 different non-negative integers (in other words, whole numbers greater than -1) whose names in English, when spelled out, have no repeating letters.

For example, 7 doesn't fit, because when spelled out as S-E-V-E-N, the word has two Es. -2 might seem to work, as it is spelled M-I-N-U-S-T-W-O, and has no repeating letters, but it doesn't qualify because it's negative, and I specified non-negative integers.

Enter as many of these numbers, but you must enter your answers as digits, not spelled out. This keeps spelling from being an issue. If you enter a number greater than 999, you can enter the number with or without commas.

Take the quiz and see how you do.

Pickover’s picks

Clifford Pickover recently picked ten of his favorite Math puzzles. I found images from Wikipedia to represent them.

How many of the puzzles can you identify from the pictures? (See the bottom of this post for a link to Pickover's article with the answers.)

The puzzles are from here. Hat tip to Shecky at Math-Frolic!

An information theory puzzle

My brother shared this puzzle with me this morning. He heard it on the radio but no solution was offered. Neither of us know what the answer is so I'm looking forward to one of you posting the answer in the comments. Here's the puzzle:

Bob and Alice are both millionaires. They're both curious to know who is richer but they don't want to tell the other one how much money they have. Without engaging a trusted third party, how can they both know who is richer?

I wonder if the solution has something to do with public and private keys and/or authentication.

So, what's the answer?