thnik again!

Lewis Carroll found that ‘ape’ can be turn into ‘man’ in six steps. A step consists of replacing one letter to give a word in the dictionary. By the way, he claimed to have invented the world ladder puzzle game.

Mathematica can be used for many things. One of them is to beat Lewis Carroll. It can turn ‘ape’ into ‘man’ in only five steps.

Here is a list of definitions. Can you fill in the lest column by changing just one letter in the word above?

Click here for a longer list.

Problem source: Wolfram Blog. You are advised to try to crack the problems above before you look at the blog.

- What is this?
- Oh, I am just playing with triangles.
- Can I play?
- Sure!
- What is the aim of the game?
- I am trying to find out in how many regions I can divide a table top with two triangles.
- Any triangles?
- It would be nice if they fit on the table top.
- In the image above you have created 6 regions.
- Right. I am mighty proud of it!!
- Why?
- It is the maximum number of regions possible!
- Really?!
- Now I am wondering what I can do with two quadrilaterals.
- You mean 4-gons?
- Yes, and what about n-gons?

Problem source:  Uncover a Few blog by Joshua Zucker.

The other day I got this email from David Brooks:

998001 is an interesting number.  Its inverse (1/998001) is a repeating decimal with a period of 2997.

It starts out 0.000 001 002 003 004 005 … and continues counting until it gets to 997.  Then it skips 998, does 999, and starts to repeat itself.

I was wondering if they have a name this this kind of integer - an integer whose reciprocal or inverse produces a decimal that shows a known or familiar sequence of numbers.  This one counts natural numbers.  But I have found others that count by even numbers, powers of 2, and Fibonacci numbers.

I would like to know if anyone has already researched these, or if I might have started on something new in mathematics.

Any info you have would be helpful.

Can you help?

- What are you reading?
- Did you know that 3685 = ( 3⁶ + 8 ) * 5?
- My wife told me the other day.
- Some kind of wife!
- She is imaginary!
- Oh, mine is complex.
- Aren’t these numbers called Friedman numbers?
- Yes, but why?
- All it says in the book is this:

A Friedman number is a positive integer which can be written in some non-trivial way using its own digits, together with the symbols + – x / ^ ( ) and concatenation. For example, 25 = 5² and 126 = 21 * 6.

- Is 1285 a Friedman number?
- Yes.
- How do you know?
- I am not allowed to tell.

- The radius of the big circle is 10.
- You mean the circle which has the four yellow circle and five green circles within it?
- Is that green? When did you have your eyes checked the last time?
- What is the radius of circle 1?
- Let me find out.
- What is the radius of circle 2?
- Don’t talk. I am busy.

The diagram is taken from Math Magic Packing Archive.

- Sometimes I wonder.
- Me too!
- Just wonder. Are there intelligent lives on other planets?
- Are there intelligent lives on this planet?
- Why do we live?
- Why do people wonder why we live?
- Are you mocking me?
- I wonder what gave you that idea.
- Cut the crap!
- OK. Here is what I am really wondering about.
- Let’s hear it.
- Do squares with one million digits of which half are even exist?
- Now, that is interesting!
- Because it has a precise answer?
- Someone told me once that questions without answers are more frequent.
- I wonder what the ratio is.

Problems source:  The Emissary Newsletter’s Puzzle section by Elwyn Berlekamp and Joe P Buhler.

- How was the movie?
- It was OK. But I had seen it before.
- ‘Stand and Deliver’ wasn’t it?
- Yes, about Jaime Escalante’s unbelievable success with his maths students.
- I must see it.
- Yes, you must. But something that happened before the movie began was equally interesting.
- How come?
- I went with my girlfriend Tara and we were supposed to meet her friend Hugo at the cinema.
- Hugo, the plumber.
- He is an apprentice plumber, yes.
- I have always wanted to be a plumber.
- We didn’t find him, so we were searching the rows and files for him. I searched the rows with my eyes from left to right starting at the first row and then the second from left to right, and so on.
- I could never be that systematic.
- Tara did the same thing, but with the files.
- Did any of you find him.
- Yes.
- I am wondering which method is best. Searching by rows or by files?
- We were wondering about the very same thing after the movie!
- Were there more files than rows?
- Yes.
- What did you and Tara find out?
- Not a lot, but Hugo solved the puzzle in no time.
- I have always wanted to work with water.

Problems source:  The Emissary Newsletter’s Puzzle section by Elwyn Berlekamp and Joe P Buhler.

Magnus Carlsen, the no 1 ranked chess player, challenges you here. Solve three puzzles in record time and be no 1 on his list.

By the way, the world’s no 1 is now playing in the Tata Steel tournament. Read about it in his blog, or follow it on the official web site or watch the games commented at chessbomb.

Oh, I almost forgot, how did you do in the challenge? Me?! Number 354 on the list with no improvement in sight.

- Nice to see you home again!
- Thank you.
- It must have been quite an ordeal!
- The food was good.
- OK. But you were arrested unlawfully?
- It made me think.
- I am glad you see the positive in every situation.
- I mean, I had to think hard to be released.
- How come?
- Just after my arrest I was given 100 coins and I was asked to divide them into two piles with 25 magic coins in each.
- Magic coins?
- I was told 50 of the coins were magic, but I could not see which.
- So how could you do the task?
- I found a way that guaranteed success in 50 or less tries.
- Amazing!
- I agree.
- Now that you are a free woman again, what are your plans?
- I am building prisons.
- What?
- My plan is to arrest politicians and release them when they have demonstrated some thinking skills.
- Interesting! Which skills do you have in mind?
- Any.

Problems source:  The Emissary Newsletter’s Puzzle section by Elwyn Berlekamp and Joe P Buhler.

- How old are you?
- 67.
- That is a lot!
- Thank you!
- What is 67 in base 2?
- Let me get out my wallet and see.
- Why your wallet?
- My memory is fading.
- So what?
- I have it written down on a piece of paper that I keep in my wallet.
- Sorry for asking!
- 67 is 1000011 in base 2.
- Now put some + signs between the digits.
- How many?
- As many as you like, but not more than one + sign between two digits.
- Will 100 + 0 + 1 + 1 do?
- Excellent! How much is the sum?
- 6.
- What is 6 in base 2?
- 110.
- Now repeat the process. Add + signs at will.
- 1 + 1 + 0.
- How much is that?
- 2.
- Repeat the process.
- 10 is 2 in base 2. 1 + 0 = 1.
- That’s it?
- What’s what?
- You can stop.
- My time is up?
- You reached 1.
- That is the aim of this game?
- Oh, I forgot to say so?
- No harm done.
- Well, yes. The question is if any positive number can be reduced to 1 in this way?
- Who want’s to know?
- The tomato farmers of Italy.
- Really?
- Yesterday I got an email from the World Processing Tomato Council (WPTC) asking for help.
- What have you found out?
- I believe it can always be done. Even more, I think it always can be done in n steps or less.
- What is n?
- It is whispered in high circles that n ”is a real shocker.”

Problems source:  The Emissary Newsletter’s Puzzle section by Elwyn Berlekamp and Joe P Buhler.