thnik again!

- Give me three distinct primes?
- Will 5, 11, and 19 do?
- That’s fine.
- What is the sum of the sum and the product of these?
- Did you swallow a whale?
- What is 5 + 11 + 19 + 5 * 11 * 19?
-  1080.
- Prove that regardless of which three distinct primes you start with, p + q + r + p * q * r is never a prime.
- Not even once?

Problem  source: Berkeley Math Circle.

There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else - but persistent. - Raoul Bott

- Yesterday, I went to this crazy party!
- Why do I never get invited to parties like that?
- No, you wouldn’t have liked this one.
- Why not?
- We all had to answer 100 yes/no questions before we could eat!
- You are kidding?!
- Afterwards we compared our answers.
- What on earth for?
- I had 98 of the same answers as Belinda.
- Belinda was there?! I should have been invited!!!
- Belinda had 98 of the same answer as Carlos.
- What about you and Carlos?
- We did not compare. He said we didn’t have to.
- How come?
- He said he could calculate it from the data we already had.
- Really?! My feeling is that all you can calculate is the range of common answers.
- Could be. But I tell you something stranger.
- Shoot.
- You know Carl?
- Carlos brother? Of course, I know him. We play Go every Tuesday.
- He asked us what we would do if the theoretical minimum number of common answers were bigger than the number of common answers between Carlos and I!
- Is he on medication?
- No, but he reads New York Times a lot.

Underneath this flabby exterior is an enormous lack of character. - Oscar Levant

Cut out the five pieces above and reassemble them to make a square.

Problem source: wu riddles.

The other day I suggested an alternative way of arranging the World Cup matches in soccer. Please have a look at Reflections.

The surprise is that Round 1 has been played in the alternative World Cup thanks to a spreadsheet and a healthy random function. Greece beat Uruguay surprisingly 4-0 and have taken the lead. In Round 2 they will meet Switzerland who beat Australia convincingly 3-0. In order not to repeat matches Germany was moved down one position and so on.

Round 2 will be played as soon as the weather permits. Remember there are only four rounds before the knock-out part with 2, 4, or 8 teams takes place. Strong teams like Spain and Italy started with a loss and have to play extremely well to have a chance to reach the second stage.

- Do you have time?
- I feel time has me, but shoot.
- I want you to play a solitaire game several times in a row.
- What could me more fun?
- I will shuffle this deck of cards and put the pile face down on the table.
- I can dig that.
- You will turn one of the cards at the time starting from the top.
- OK.
- Before you do I want you to guess when you will draw the first red ace and when you will draw the second red ace.
- I have reason to believe that I will draw the first before the second.
- Brilliant! But can you be more precise? Where in the stack will the first red ace most likely be and where will the second be?
- I may have to peep?

Problem source: Martin Gardner, 1970, via Wordplay.

The mathematics is not there till we put it there. - Sir Arthur Eddington

How do you determine the best of 32 soccer teams? That’s the question.

One way is being tried out in South Africa right now. Seed some teams and make up 8 groups with 4 teams in each. Call it round 1. Let every team meet every other team in the group in round 1 and let the two best teams advance to the second round. In the second round the 16 teams play an elimination tournament, one loss and you are out.

In my opinion the system has several serious flaws.

1. Not every match counts. You may lose a match in round 1, but if you advance to round 2 it is of no importance.
2. Because of point 1 the early matches are less exciting as they may be of no importance.
3. The teams do not play the same number of matches. If you don’t go to round 2 you play only 3 matches, while the finalists play 7.
4. Some teams are seeded based on previous performance giving them an advantage. All teams should enter the tournament on the same footing.
5. The system only determines the best team. The losing finalists may not be the second best team and for the other teams they don’t get a final score showing how well they did.