Imagine that the earth shrunk and the oceans dried off. Would the earth be:
A) Smoother than a baseball but not a golf ball
B) Smoother than a golf ball but not a billiard ball
C) Smoother than a billiard ball
D) Rougher than all of the above
Does your intuition and calculations say the same thing?
Problem source: Martin Gardner via the Wordplay blog at New York Times.
- Last night I had the strangest dream.
- Please share!
- I dreamt of two numbers whose product and sum were both one.
- How unusual!
- Yes, indeed.
- Did you dream what the numbers were?
- No. But I found the difference in my sleep.
- What was the difference?
- Where have you been hiding lately? I haven’t seen you for days.
- I have been travelling.
- Really? That explains things.
- I read somewhere that travelling is good for your creativity.
- What will they think of next?! That one thinks with one’s knees?!
- I returned by train. The last leg was 500km long and the train’s average speed was exactly 50km/h.
- How on earth do you know?
- The journey lasted exactly ten hours.
- Well, that explains things.
- I was wondering, did we ever cover a distance along the way 50km long in exactly one hour?
- Of course, you had to. Didn’t you say the average speed was 50km/h?
Problem source: Martin Gardner , December 1979.
At my school the students will get report cards next week. For their effort in a subject they are assigned A to F. To explain what A to F stands for adjectives like ‘average’, ‘excellent’, ‘good’, ‘improvement needed’, ‘less than satisfactory’, and ’satisfactory’ are used.
My problem is that I don’t know if ‘good’ is better than ’satisfactory’. So the question is, can these adjectives be ordered?
- The other day I wanted to write f(x(1 + y)) on the board, but wrote something else instead.
- What did you write?
- f(x)(1 + f(y)).
- That is almost the same.
- Yes, I agree, and my students thought the slight change didn’t matter!
- Does it?
- What do you mean?
- Are there functions f for which the two give the same results for all x and y?
- I doubt it very much.
Problem source: Berkeley Math Circle Contest.
(Continued from yesterday)
- Come to think of it, I do see something.
- What?
- Well I can’t explain it and it may break down further out.
- Spit out! What do you see?
- Every number has at least three zeros in a row, like 000.
- What about 1023 or 1111111111?
- Sorry, it breaks down right there.
- No reason to be sorry.
- May I change my pattern slightly?
- Be my guest!
- Every number has either 111 or 000 in them somewhere.
- How old are you?
- 45. Why do you ask?
- Do you know how much 45×45 - 1 is?
- 2024, if I am not mistaken.
- You are not mistaken.
- Good!
- If you only had two fingers how would you express 2024?
- You mean in binary?
- Oh, sorry, I forgot you are educated!
- 2024 = 11111101000 in binary.
- That was fast!
- I just Googled it by typing “2024 in binary”.
- Do you notice anything about the result?
- It is much longer than 2024.
- Anything else?
- No.
- OK. Here is the challenge. Express n times n minus 1 in binary, discover a pattern, and prove it.
- What is n?
- Any number.
- That seems hard!
- Above you see some data created with Open Office Calc using the Base(x;2) function.
- I tried to solve the system above, but I could not find any solution!
- What value did you use for the parameter a?
- I don’t remember.
- When I solved it I got one, two, three, and four solutions.
- For different values of a?
- For different values of a.
- Do you remember the values?
- Do I remember my own name?
Problem source: Bamo problems.
- What a nice square!
- Thank you. I made it myself!
- How did you do it?
- I put 1 or -1 in all the cells in the upper-left 2×2 square, the blue cells.
- And then?
- Then I multiplied the cells in each row and put the result at the end of the row, the yellow cells
- And you did the same with the columns?
- Correct.
- But what is the number in the brown cell at the bottom-right?
- It is the sum of the row and column products, i.e. the sum of the yellow cells.
- You mean you added up the numbers above it and to its left?
- Yeah, man!
- I would like to make a square like that!
- Try to make a blue 1997×1997 square (which becomes 1998×1998 when you calculate all the products and the sum in the lower-right).
- That sounds too easy.
- OK then. Make the lower-right cell equal 0 in your 1998×1998 square.
Problem source: Bamo Problems.
