The fourth dimension is a direction different from all the directions in normal space. Some say that time is the fourth dimension ... And in a sense, this is true. Others say that the fourth dimension is a hyperspace dimension quite different from time ... This is also true.
For most of this book, Rucker uses "fourth dimension" to refer to a fourth spatial dimension, though near the end of the book there is also a few chapters on time as the fourth dimension.
In order to help us imagine what happens when we go up a dimension to the fourth dimension, Rucker will often describe what happens when we go down a dimension instead, to the second dimension -- he attributes this analogy to the ancient Greek philosopher Plato. We keep this in mind as we look at Rucker's first puzzle:
Stare out your window and imagine that the objects you see are actually two-dimensional shapes embedded in the window glass. The window glass is thus a sort of two-dimensional world. Under what condition can, say, two car shapes move each other without collision?
If the two care shapes are of different size, then they can move past each other. The third dimension of distance-from-the-window is, in terms of images, represented by size. For the two-dimensional shapes moving in the window glass, size is a higher dimension. Imagine living in a three-dimensional analogue of such a world: you would be able to change your size at will, and you would be able to move "through" people of different sizes.
And so in this strange world where you can change your size at will, size would actually represent a new dimension -- the fourth dimension! In terms of Common Core transformations, we'd say that when you are apparently changing your size (a dilation), you're actually moving in the fourth dimension (a translation)! The center of the dilation is the vanishing point of the translation (as in perspective -- Lesson 1-5 of the U of Chicago text).
I know -- so far this sounds a bit confusing, but hopefully we'll be able to wrap our minds around the fourth dimension as we proceed in Rucker's book.
Meanwhile, let's look at those Conway questions from yesterday:
OPTION A: SOLVED – WINNERS WILL BE NOTIFIED WITHIN 24 HOURS
I’m thinking of a ten-digit integer whose digits are all distinct. It happens that the number formed by the first n of them is divisible by n for each n from 1 to 10. What is my number?I found this one very easy -- only because I'd already watched a Numberphile video with the answer:
Notice that the answer is actually the number mentioned in the title with an extra 0 added to the end. I was surprised that Numberphile was not mentioned in the Pizza Hut thread -- and neither do I see Pizza Hut mentioned in the comment thread at Numberphile.
OPTION B: SOLVED – WINNERS WILL BE NOTIFIED WITHIN 24 HOURS
Our school’s puzzle-club meets in one of the schoolrooms every Friday after school.
Last Friday, one of the members said, “I’ve hidden a list of numbers in this envelope that add up to the number of this room.” A girl said, “That’s obviously not enough information to determine the number of the room. If you told us the number of numbers in the envelope and their product, would that be enough to work them all out?”
He (after scribbling for some time): “No.” She (after scribbling for some more time): “well, at least I’ve worked out their product.”
What is the number of the school room we meet in?”When I first saw this problem, it reminded me of one I posted on the blog about six months ago:
Two perfect logicians, S and P, are told that integers x and y have been chosen such that 1 < x < y and x+y < 100. S is given the value x+y and P is given the value xy. They then have the following conversation.
P: I cannot determine the two numbers.
S: I knew that.
P: Now I can determine them.
S: So can I.
Given that the above statements are true, what are the two numbers?
"Nick," the author of this website, lists the numbers as 4 and 13. At first I thought that this was the answer to Conway's Option B -- that is, the room number would be 17. But then I realized that Nick's problem and Conway's are not identical:
-- For Nick, S knows only the sum and P knows only the product. For Conway, S knows not only the sum but the numbers themselves as well. And Conway's girl deduces the product, while P knows the product from the start.
-- In Nick's problem there are only two numbers. In Conway's problem, there may be more than two numbers -- how many numbers there are is part of what the girl asks for.
-- In Nick's problem there is a limit to the sum -- it must be less than 100, and this ceiling allows Nick to eliminate many possibilities. Conway doesn't list any such limit. Of course in practice, if the girl knows anything about the building she's in, she might know what the room numbers go up to and thus place a limit on the sum. But Conway mentions no such thing.
Nevertheless, the idea behind both problems is the same -- when one person says "no," that is actually providing information to the other person, and it's from this information that the other person learns something about the numbers.
Notice that in Nick's problem, neither of the numbers can be 1. Conway doesn't say anything about 1 being forbidden as one of the numbers. On the other hand, I will assume that 0 can't be any of the numbers -- otherwise the product would be 0 and then the girl can't make any conclusions about the nonzero numbers.
To solve Conway's Option B, let's see what products we can rule out. Just like Nick, we can immediately rule out any prime numbers. For suppose we have:
Member: There are n numbers with a product of p (prime).
Then the girl would know that one of the numbers is p and all the rest (i.e. n -1 of them) are 1 (note that Conway never says that the numbers in the envelope are distinct). Then the member would have to say "yes," since the girl could easily derive the room number as p + n - 1, contradicting the fact that he said "no."
Next, Nick considers products of two primes, pq. (These have a special name -- semiprimes.) If we have a semiprime, then there must be at least n - 2 copies of 1 in the envelope, with the last two being either p and q, or yet another 1 and pq.
I must admit that at this point I'm stumped. Nick's problem had an easy solution because there are so many restrictions on the numbers. Like some of the posters in the Pizza Hut thread, I'm sure that Conway must have left some restriction out -- such as the numbers being distinct, or 1 being disallowed as one of the numbers. Otherwise, even if the member says that there are just 2 numbers in the envelope and the product is a semiprime pq, the girl would not be able to distinguish between the numbers being p and q, or the numbers being 1 and pq. So the member would be forced to say "no" for any composite product, which is hardly enough information for the girl to conclude anything.
Since I'm still having problems with Option B, do we even need to look at the Option C question?
This is what I wrote last year about today's lesson:
Today's lesson is on surface areas. But recall that back at the end of the first semester, I mentioned Dan Meyer, the King of the MTBoS (Math Twitter Blogosphere), and his famous 3-act lessons. I pointed out how one of his lessons was based on surface area, and so I would wait until we reached surface area before doing his lesson. (In fact, yesterday's lesson also comes from a MTBoS member, Laura Lee. There were just so many ideas that happen to correspond to this week's lessons and I'm posting them all this week!)
Well, we've reached surface area. And so I present Dan Meyer's 3-act activity, "Dandy Candies," a lesson on surface area. Of course, I made the candy Irish potato candy, since this week is, after all, St. Patrick's Day.
Meyer includes some additional questions for teachers to ask the students, but I only included what fits onto a single student page. Those who want the extra information can get it directly from Meyer:
Meanwhile, I also added some additional questions from U of Chicago. Notice that today's lesson is supposed to correspond to Dr. Franklin Mason's Lesson 12.2, and 12.3 is tomorrow, but notice how Dr. M divides the figures of surface areas into lessons:
Surface Area: Prisms and Pyramids, then Cylinders and Cones
Volume: Prisms and Cylinders, then Pyramids and Cones
U of Chicago:
Surface Area: Prisms and Cylinders, then Pyramids and Cones
Volume: Prisms and Cylinders, then Pyramids and Cones
So Dr. M combines the two polyhedra -- prisms and pyramids -- for surface area only, while for volume he follows the U of Chicago and combines prisms and cylinders. For getting through the formulas quickly, doing the prism and cylinder formulas together makes more sense, since they really are the same formula -- indeed, the U of Chicago calls them both cylindric solids. In other words, a cylinder is just a "circular prism."
Therefore this lesson incorporates Lesson 10-1 of the U of Chicago text, which is on surface areas of prisms and cylinders.
In last year's post, I spent a great deal of time discussing the PARCC PBA (Performance-Based Assessment) and the CAHSEE (California High School Exit Exam) tests. The idea was that students would have a tough trying to get through this week's material because they are so distracted by these big tests, depending on their state.
But this year, as it turns out, neither of those tests even exists anymore! The PARCC has been reduced to the EOY (End-of-Year) exam only, and the governor has signed a law dropping the CAHSEE as a graduation requirement in California. Hopefully, this will mean that students will be able to understand this week's material more clearly. Many Common Core opponents would agree -- less time for testing means more time for teaching!