Xavier, Yves and Zoé play Ping Pong a whole afternoon. 1 against 1, the third sitting and replacing the looser. Xavier played 10 times. Yves 15 times. Zoé 17 times.
Who lost the second match?
Kindly submitted by Paul Schwander.
- You look bored!
- Yeah. I don’t know what to do. Nothing inspires me.
- You are as sent from heaven!
- Yeah?
- Make a circle and colour every point on its circumference red or blue.
- Won’t that take a long time?
- And? You said you had nothing to do.
- OK. But what’s the point?
- Pun intended?
- What pun?
- The old lady in the bakery asserts that regardless of how you colour the points one can always find an isosceles triangle with equally coloured vertices on the circle.
- I didn’t know you shopped at the bakery.
Problem source: Mathematical problems by Andrew Adler.
- 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.
- 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.
- 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 problem yesterday was unfair.
- I am sorry. In what way?
- You said Euclid had proven something he hadn’t!
- Did I? Oh well, something I get things mixed up.
- Is that your only excuse?!
- Have you heard of Imre Lakatos?
- I am not quite sure. Did he write ‘How to effect children’s affections in rural Albania’?
- No, he wrote ‘Proofs and refutations: the logic of mathematical discovery’.
- I was not even close. Was he Albanian by any chance?
- He was Hungarian. More importantly, he suggested textbooks should be written in heuristic style to show how problems are solved and mathematics created.
- So the textbooks should be less authoritarian?
- Exactly, if something is true or not is for the student to ponder, not for the textbook to impose.
- Sounds like an impractical approach.
- Anyway, here is a problem Ravi a reader in India posted in a comment in New York Times the other day.
- Is it inspired by Lakatos?
- Give me a prime number.
- How about 11?
- What is 11 in base 3?
- You want me to write 11 as the sum of some 9s, 3s, and 1s?
- OK.
- 11 = 1*9 + 0*3 + 2*1. So 11 is 102 in base 3.
- Add 11 and 102.
- 113.
- Which is prime!
- So it is.
- Find the biggest prime that when added to its base 3 representation is a prime.
- What if there are infinitely many of them?
- Hadn’t thought of that!
- What are the numbers above?
- Some prime numbers I found at Wikipedia.
- That reminds me. I got a letter from a student of mine.
- A maths student?
- No she was more into art. Anyway, she has found a proof that amazes me.
- I am intrigued!
- She studied p1 * p2 * p3 * … * pn + 1 where pi is the ith prime number.
- I am lost!
- Let me give you an example. For n = 3 she multiplied the first three prime numbers and added one.
- What on earth for?
- 2 * 3 * 5 + 1 = 31.
- 31 is on the Wikipedia list.
- That is what she noticed! She proved that p1 * p2 * p3 * … * pn + 1 always gives a prime.
- I have some bad news. Euclid, who was not born yesterday, used that fact to show that there are an infinite number of primes.
- Really? I guess I should get out more often.
Start at 1 and go clockwise around the star till you end up at 1 again. Your task is to select as many numbers as you can in rising order. The best you can do with the star above is 1, 3, 4 with a score of 3.
Now do the same counter clockwise. Start at 1 and select numbers in rising order. The best you can do is 1, 2, 3, 5 with a score of 4.
The highest of the clockwise and counter clockwise score is called the star’s rising score. In this case it is four.
Today’s little non-fattening food for thought: Rearrange the numbers on the star so the score is three. Can you reduce it to two, too?
Problem source: Wordplay.
- Do you play Poker?
- Can’t say that I do.
- I find that strange.
- I have only 2Kb of memory.
- That is not a lot!
- Right! I forget what cards have been played.
- Then I have the game for you.
- As long as it is not a card game I am game.
- It is a card game, but of a different sort.
- Please explain.
- We will put all the cards on the table face up.
- I am beginning to like it already.
- You draw five cards. Any cards you like.
- To build a strong Poker hand?
- Then I will draw any five cards.
- And the best hand wins? I seem to have an advantage!
- Not so fast! Suits have equal value.
- So a flush of spade ties a flush of clubs?
- Correct.
- That is a bit sad.
- After both players have drawn five cards the first player, you, can discard any of his cards and draw from the table to again get five cards on his hand. The second player, me, can then do the same.
- I don’t see how I can win!
- But you can!
Problem source: Martin Gardner via WordPlay.
