think again!

- Congratulations on your new job!
- Oh, you mean the weather man thing?! That’s nothing! Anyone can do it.
- Really? I thought it was kind of hard.
- Not at all. Look at me!
- But what about all those averages you have to calculate?
- What do you mean? What averages?
- “The average temperature yesterday was 18°C.” How do you do that?
- We have a guy who adds the max and min temperature of the day and divides by two.
- How does he know the max and min temperatures? Does he stick around all day looking at the thermometer?
- He did that until recently. Now we have bought him a special thermometer so he doesn’t have to.
- I see.
- Do you know what kind of weather we will have tomorrow? The show starts in thirty minutes and I could need some help.
- One question. I have one question that just popped into my head. Do you have time?
- Shoot!
- How many people work on the set?
- Ten in all, I think.
- Do you know their heights?
- Their heights?
- Yes.
- No, but I could find out if you are interested.
- I want to calculate the average height.
- Oh, that?! You don’t need the height of all of them for that. I will find out the tallest guy and the shortest and just add their heights and divide by two.
- Will that give me the average height?
- You are talking to the weather man! If I don’t know, who does?

——

- Have you met our new neighbour?
- I saw him the other day, but I did not talk to him. He looks a bit strange.
- What do you mean?
- I don’t know. Just strange.
- Stranger than me?
- Sometimes you are strange too!
- Like when?
- You remember when you wrote down the temperatures outside every minute for an entire day?
- You find that strange? I just wanted to find the average temperature for the day.
- How would you do that?
- Add up all my readings and divide by 1440.
- Why 1440?
- That’s how many readings I would have had.
- “Would have had”! Sounds like the story of your life, doesn’t it?
- I had to stop at reading 1334 as my fingers cramped.
- Is that what you call it when you give up?
- By the way, I talked to the new neighbour yesterday!
- Really?! What did you talk about. Don’t tell me you talked about your average temperature project?
- We did, actually. He was very interested.
- I told you he was weird!
- He asked me if I could give him the temperatures I measured at 4, 8, 16, and 22 hours.
- Is he a military man?
- I don’t think so. Anyway, I told him the temperatures were 5, 12, 24, and 15°C. Do you know what he did then?
- Hit you on the fingers?
- Don’t be silly! He invited me inside and showed me this graph on his computer.

- And why one earth did he do that?
- Can’t you see what it is?
- It is obviously the graph of
f(x) = -65/6048*x^3+121/432*x^2-11/27*x+536/189,
but I don’t see how that relates to anything.
- Look carefully at the values for x = 4, 8, 16, and 22!
- I see, the graph shows the temperatures for the day you were measuring. But, how come?! You said you only gave him four values.
- I did! He told me that if one assumes the temperature follows the graph of a third degree polynomial, four points is enough to draw it.
- And does the rest of the graph match your data?
- I haven’t had time to check that part yet, but there is more!
- He showed you another graph?
- He told me how to find the average temperature exactly.
- Anyone can do that, just integrate the function from 0 to 24 and divide by 24. It comes to 14.582, if I am not very much mistaken.
- Your number crunching skills do not stop to amaze me! However, he did something simpler.
- Like what?
- He added f(5.071667) and f(18.928333) and divided by two.
- What did he get?
- 14.582.
- Get out of here!

Will the neighbour’s method always work? How did he pick his two points? Will other point pairs work? What is the neighbour’s name? Does the temperature during a day really follow a third degree polynomial?

Problem kindly suggested by John Shonder.

——

To research the problem I used MapleV Student Edition.

interp([4,8,16, 22], [5,12,24, 15], x) ;
gave me the third degree polynomial called f(x) in the story using Lagrange interpolation formula.

plot(-65/6048*x^3+121/432*x^2-11/27*x+536/189,x=0..24);
gave me the graph.

evalf((int(-65/6048*x^3+121/432*x^2-11/27*x+536/189, x=0..24))/24, 5);
gave me the definite integral divided by 24, i.e. 14.582.

By the way, I had to delete the file Opengl32.dll in the folder BIN.wnt to get Maple to read my key punches.

- Can we show you a new card trick, Abena?
- Go ahead!
- Pick five cards in this standard deck of cards.
- Any five?
- Let them be random. Shuffle the deck and pick five cards with eyes closed.
- OK. Then what?
- Email the five cards to Enu.
- How do I do that?
- Write down the value and suit of each card and email her.
- OK.
- Enu will email me four of the five cards, and me, brilliant as I am, will be able to deduce the fifth card!
- You are bluffing!
- Try me!

Problem creator: William Fitch Cheney.

- So how was the first day in kindergarden?
- Boring! said Pat.
- Booooring! joined Chris.
- How come? asked the concerned father.
- 12/17 of the other kids were girls! sighed Pat.
- Really! I found that 5/7 of the other kids were girls!

How many girls and boys were in the kindergarden, and what are the genders of Pat and Chris?

Problem source: dansmath@home.

2008 is just around the corner so the next question is just natural: In how many ways can 2008 be written as the sum of consecutive positive integers?

“The first known published crossword puzzle was created by a journalist named Arthur Wynne from Liverpool, and he is usually credited as the inventor of the popular word game. December 21, 1913 was the date and it appeared in a Sunday newspaper, the New York World. Wynne’s puzzle(see above) differed from today’s crosswords in that it was diamond shaped and contained no internal black squares. During the early 1920’s other newspapers picked up the newly discovered pastime and within a decade crossword puzzles were featured in almost all American newspapers. It was in this period crosswords began to assume their familiar form. Ten years after its rebirth in the States it crossed the Atlantic and re-conquered Europe.” More

Online crossword puzzles let you cheat, save, and print your solution. Here is one example.

- It was not my fault.
- I agree, but you were still late for the Go tournament!
- I know. That is why I could only play 3 instead of 5 games.
- You won 2 of the games. Sarah won 3 of her 5 games.
- Really! She has improved!
- So, what is best, 2 of 3 or 3 of 5?
- Why don’t you ask your readers?
- Good idea!
- You may like to add that a game of Go can not end in a draw.
- How come?
- We use a komi of 5.5.
- I don’t know what that means, but I will take your word for it.

Let a = games won and b = games played.

Is (2,3) better than (3,5)?

If the function used is a/b the answer is ‘yes’ since 2/3 = .67 and 3/5 is .6.

However if the function is a*a/b the answer is ‘no’ since 2*2/3 = 1.3 and 3*3/5 = 1.8.

To experiment with various functions there is a box below where you can enter the function before you press the Go button. Enter “a/b” (without the quotes) and press Go. All possible pairs of (a,b) with a less than six get their score calculated.

Now, edit the formula to a*a/b, and see what happens.

You can use more exciting formulas for a and b, like “Math.min(a,b)”, “a*Math.random()”. See Math functions in javascript for available functions. I have added the combin(a,b) function, so you could enter things like “combin(b,a)”.

So, the question is, is (2,3) better than (3,5)?

- I want to multiply 43 by 21.
- Why on earth do you want to do that?
- It’s kind of personal, if you don’t mind.
- OK. Do you have a calcualtor?
- No.
- Do you know the multiplication table?
- I don’t even know if I have a dining table.
- Can you count?
- Yes.
- Then you don’t have a problem. Draw 4 straight lines and 3 straight lines with a gap between them.
- Like this?

- Exactly!
- Then what?
- Draw 2 straight lines and 1 straight line with a gap between them at an angle to the first lines.
- Like this?

- Exactly!
- Then what?
- Count the intersections.
- Like this?

- Exactly!
- Add the numbers in each column, carry if you have to.
- I have a weak back.
- Lift with your knees.
- I got 903.
- That’s how you multiply if you don’t know how to multiply.

Good method? Challenge: devise a method for division using nothing but straight lines and counting.

Go to Google Images and search for any picture. In the browser’s address bar copy and paste this:

javascript:R=0; x1=.1; y1=.05; x2=.25; y2=.24; x3=1.6; y3=.24; x4=300; y4=200; x5=300; y5=200; DI= document.images; DIL=DI.length; function A(){for(i=0; i<DIL; i++){DIS=DI[ i ].style; DIS.position='absolute'; DIS.left=Math.sin(R*x1+i*x2+x3)*x4+x5; DIS.top=Math.cos(R*y1+i*y2+y3)*y4+y5}R++}setInterval('A()',5 ); void(0)

Press Enter.

Explain what happens. In other words, explain the code above.

When you have to explain the sin() function to the guy sitting next to you on the bus or in the dentist’s waiting room, try this as your starting point and tell me how it went.

Problem source: Crazy Cool Internet Trick! by Justin Benton.

Do you know others?

Erich Friedman is a math professor and puzzle enthusiast. Red points is his own creation. Colour five of the points in the image above red in such a way that the distance between any two of them them are all different. That is ten distances, by the way.

Here, is an example: