thnik again!

Sometimes the planes don’t arrive on time and one has to find things to do while waiting. One popular sport is to race with someone running on the moving walkways. It gets especially exciting if one runs at the same speed as the one on the walkway and challenges him to run down the walkway and then back again. On the first leg he has an advantage as the walkway carries him along, but going back the advantage is yours and you end up arriving at the finish line at the same time.

Prove that the race is fair when v1 is the running speed of both competitors, v2 is the speed of the walk way, s is the length of the walk way, and the time is eight in the morning.

Problem source: The blog Division by Zero.

- Is that your new calculator!?
- It is more than a calculator!
- Can it divide by 0?
- No, what I mean is that it is a phone, a camera, a notebook, a netbook, and a coffe machine!
- I am amazed.
- When will you amaze me, I wonder!
- How much is 2618 * 11?
- Do you want fries with the answer?
- A café au lait will do.
- The answer is 28798. Careful, it may be a bit hot.
- Turn 2618 around to get 8162 and multiply it by 11. What do you get?
- 89782. But wait a minute, that is the first answer turned around. I am amazed!
- So am I.
- Are there more numbers like that? Why do they behave like that?
- How does your calculator make coffee? That is what I would like to know.

Problem source: Mattenøtter, Teknisk Ukeblad.

- Do you know what a semiprime is?
- If I concentrate I may remember what a prime is.
- 21 is a semiprime since it is the product of two prime numbers.
- You mean 3 and 7?
- 22 is also a semiprime.
- Get out of here!
- 2 and 11 gives 22.
- Are there more pairs of consecutive numbers that are semiprimes?
- There is one pair before 21 and 22.
- And after?

Problem source: math hombre.

- I am nine, she smiled.
- That’s great, I replied, turning to the sports page. I am 91.
- Do you know the cube of 2? she persisted.
- 8, I answered sub-consciously.
- Won’t do then, she said disappointed.
- Why not?
- The remainder when I divide it by 9 has to be 1.
- I see, I said, reading about the local basketball team. They had lost again.
- Do you know the cube of three, she asked innocently.
- Yes, but you won’t like it. Its remainder with 9 is not 1.
- I can never do this, she said.
- What? I asked, putting down the paper.
- I have to add up all numbers between 2 and 8 whose cube gives 1 when divided by 9.
- Who would give such a stupid task, I asked irritated.
- You did, she sighed. You wanted to read the paper in peace I guess.
- So much for that plan, I half smiled.

Being 91 and the proud owner of Calc spreadsheet I tried to solve her problem as if I was her. In other words I tried to find the sum of all numbers from 2 to 90 whose cube when divided by 91 gave 1 as the remainder.

It was a no-brainer. Here are my sheets.

When I showed it to her she refused to put down her comic book to give it a glance.

My second disappointment followed shortly thereafter. Project Euler asks for the sum for people who are 13082761331670030 years old.

My spreadsheet approach required no thinking, my area of expertise. Euler’s question left me feeling 99% inadequate and 1% curious on how on earth the problem can be solved.

- Four digits are added and the same four are multiplied.
- I bet the multiplication gave a much bigger result!
- No, that is what was strange, the result was the same.
- What were the four digits.
- I am afraid I don’t remember.
- Were they four zeroes?
- No. That I would have remembered.

The other day I sold two books I bought a few years back. One of them gave me a 20% profit, while the other gave me a 20% loss. Together, however, the two books gave me a 5% profit.

I can’t seem to remember how much I sold each bok for, although I do remember that the total sale price was 1260 baht.

Problem source: Rudi Kessel, Mattenøtter.

By M C Escher

A recursive acronym … is an abbreviation that refers to itself in the expression for which it stands. The term was first used in print in April 1986. – Wikipedia

Examples:

In Gödel, Escher, Bach: an Eternal Golden Braid, published 1979, Douglas Hofstadter uses the acronym GOD, ‘GOD Over Djinn’. As a genie explains to Achilles, GOD stands for GOD Over Djinn, remarking that “GOD can never be fully expanded.”

In The Simpsons episode That 90′s Show, Homer defines “GRUNGE” as Guitar Rock Utilizing Nihilist Grunge Energy.

Can you create an original recursive acronym? For example one for this blog: ‘think again’?

To factorise numbers can be done amazingly fast with a dumb algorithm on a computer. Run over to http://www.cryptographic.co.uk/factjava2.html by Andrew Hodges and see what patterns you discover with your favourite number.

http://www.cryptographic.co.uk/factorise.html gives a bit more background.

I was born in 1952. If you turn it partly upside down and read it from right to left you get 2⁵x61 = 1952.