I'm looking forward to thinking about these puzzles Denise posted:

- 24 can be written as the sum of three square numbers. How?

- Can 24 be written as the sum of two consecutive integers? Can 24 be written as the sum of three or more consecutive integers? How many ways?

[Which reminds me: Did you figure out the consecutive-integer puzzle from MTaP #22?]

- How many ways can the letters M-A-T-H be arranged to form a 4-letter “word”? Okay, since there’s no 24 in the question, you’ve probably already guessed the answer — but can you prove it?

- 24 is the largest number divisible by all numbers less than its square root. Can you find all of the other numbers for which this is true?

- 24 is an abundant number, which means that if you add up all the numbers that divide evenly into 24 (except for 24 itself), the sum will be greater than 24 itself. How many other abundant numbers can you find which are less than 100?

- What is the ones digit in the number 24^24?

[That means 24 raised to the 24th power.]

I can't tell you my favorites, because I haven't had time to look through more than a few entries yet. (And if I don't post this link now, I might just forget.)

I think I'll savor it all on Sunday morning. Enjoy!

For the second question, if you removed the condition, ie:

ReplyDeleteIn how many ways can 24 be written as the sum of consecutive integers?

would you include 24 by itself as one way?

Jonathan

I guess it depends on whether you can build up some grand unifying theory one way or the other.

ReplyDeleteLooks to me like the only way you get get 24 as the sum of more than one consecutive number is with 3 of them.

It also seems like an odd number (n) of consecutive numbers will only add up to a multiple of n. In fact, if the middle number is m, then the sum will be mxn.

And an even number (2n of them) of consecutive numbers will always add up to a number which is equivalent to n mod 2n. (With 4 numbers, your sum divided by 4 would always have remainder 2. With 6 numbers, your sum divided by 6 would always have remainder 3.)

Ok. Having just one number fits with my other ideas, so I say yes. What were your thoughts about it?

I like saying yes... but I think we (students or us, whoever is working on this) needs to ask the person who poses the problem.

ReplyDeleteAnd I don't think there is anything wrong with needing to ask.

I was curious what your gut take on it was (same as mine).

Oh, but on the actual question, there are three different ways to write 24 as the sum of more than 2 consecutive integers!

(hope I haven't spoiled it for you)

Jonathan

Of course you haven't spoiled it for me. I was done with it, too arrogantly sure I had it solved.

ReplyDeleteI hadn't thought about including negatives until now. So I think I've found a second, and maybe a third...

Now I'll go see if the answer given matches mine. ;^)

There are no "answers given", 'cause I didn't bother to write any. I figured the puzzles were fairly straight-forward, and it wouldn't be too hard to check one's results with Google, if one wanted.

ReplyDeleteFor Jonathan's first question above, it seems to me that the word "consecutive" implies two or more numbers, but perhaps I'm wrong on that.

The question about making 24 with two consecutive numbers was designed to make kids think about odd and even numbers. Jonathan is right that negatives extend the possibilities for the second part of that question---there is only one way in the natural numbers, but I also found three when we include negatives.

And by the way, the related question from MTaP#22 was about natural numbers, even though it said "integers."

I see I was misunderstanding Jonathan's comment. Yep, I get 3 too, once i carefully consider extending to the negatives.

ReplyDeleteI agree that consecutive implies more than one. I guess the question would need to be written differently if we wanted to include the possibility of using just one number. I just liked how that degenerate case still followed the pattern I'd begun to see.

Can we use "i" in the first one?

ReplyDelete@ Craig: why not include 'i' in the first one? It says 'numbers' - well that might leave some room for, uuhhmm ... creativity. ;-)

ReplyDeleteIt's surely not meant to be real numbers - that would be too trivial and make up for an infinite number of solutions.

But then comes the question, how do we construct the set from which to take valid solutions, i.e. "numbers"? Should that be merely the line of positive and negative integers? Or should we take the two dimensional area of complex numbers? Well, that always depends on your frame of reference.

Now, if you think in terms of the latter - as I did too - that makes up for nice additional solutions, e.g. 4*4 + 3*3 + i*i. BTW: Did you also come up with 5*5 + i*i* + 0*0? And how about 7*7 + (5i *5i) + 0*0? Are there more of that? And would there be even more, if we included "surreal" numbers? Nice musing...

Many thanks for triggering some nice reflections about "numbers" to all of you and Sue VanHattum in the first place.

I'm curious what you have in mind when you mention surreal numbers in this context.

ReplyDeleteUuupps, sorry. I hope I didn't raise expectations I could hardly fulfill. I tried recently to understand those "surreal" numbers as explained in the book of Donald E. Knuth. Too sad, I have to admint that most of it still escapes my mind.

ReplyDeleteHowever, I just wondered, if surreal numbers would add even more 'room' for solutions for this exercise? Or would they just be equivalent to the complex solutions pointed out?

My sincere apologies if I disappointed anybody...

No problem. I studied that book myself, years ago, and found it delightful. But I still wouldn't say I have a handle on how to explain surreal numbers to anyone else.

ReplyDeleteI don't remember a connection with complex numbers, but then, I do have a terrible memory... ;^)

Connecting surreal numbers to complex numbers, both arise by starting with the real numbers and then saying "what if"...

ReplyDeleteWhat if there were a real number x such that x^2 +1 = 0? Well, there isn't! Fine, let's make up a number that has this property — call it

i— and we get the complex numbers.Likewise, what if there were a real number x such that x > 0 but x < 1/n for all positive integers n? Well, there isn't! Fine, let's make up a number...

etc., the rest of the process is formally quite similar, and we get the surreal numbers.Mathematicians call this "what if" exercise a

field extension, because the real numbers are a field (a number system in which division makes sense) and we areextendingit by adjoining our new, made up numbers.In another post (here) I said that real mathematicians ask why. (See #8.) Pat B said real mathematicians ask 'what if'. Thanks for framing that question so nicely - field extensions - I'll remember that now. :^)

ReplyDelete