Roxbury Math Epsilon Club Lecture3

Lecture 3

Roxbury Math Epsilon Club

Let’s start with discussing a few more points about Magic Squares. This time, let’s talk about 4 x 4 Magic Square (just to make things a bit more interesting) First question is how many numbers can go into 4 x 4 Square. The answer is almost looking at us. This number is of course 16. If we have 16 numbers, then what is the “magic number” for 4 x 4 Magic Square? Remember that for 3 x 3, the calculation was: … and the “magic sum” was: The exact same calculation for 4 x 4 is: … and the “magic sum” is: So, all the rows, columns and diagonals need to add up to 34. Let’s actually try to figure out what numbers to put in. Our first attempt will be just to put in all the numbers in their consecutive order left-to-right and top-to-bottom: 1 5 9 13

2 6 10 14

3 7 11 15

4 8 12 16

Let’s look at the diagonals…

Looks like we’re in great luck… Diagonals seem to be already finished (and we should leave all of them as they are). But what should we do with the remaining numbers to complete the square? Clearly, the top row is too little (10) and the bottom row is too big (58)… But notice, that the sum of the top and bottom row is: So, all we need to do is somehow rearrange numbers, so that each row would be equal to 34 and we’re done.

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Lecture 3

Roxbury Math Epsilon Club

One way to “equalize” would be to just interchange numbers like this: 1 14 15 4 5 6 7 8 9 10 11 12 13 2 3 16 Now first and last row are ok… but now column 2 and column 3 still need some work: They are:

Once again, they’re almost correct. We just need to take 2 “out of column 3” and “give it to column 2”. All we need to do is swap 14 and 15 as well as 2 and 3 and the columns would also have the proper sums: 1 15 14 4 5 6 7 8 9 10 11 12 13 3 2 16 Does anybody have a guess of what we should do with numbers 5, 8, 9 and 12 to make the whole thing work? Remember, we don’t want to move anything on diagonals and we don’t want to break any rows, columns which already add up to 34. The solution is of course to do “double-swapping” of numbers in the exact same way as we did with the 2, 3, 14 and 15. The final square looks like this: 1 15 14 12 6 7 8 10 11 13 3 2

4 9 5 16

Here are a few more “magical properties” of Magic Squares: Subtract all the numbers from “17”. Since inside the square we only have numbers 1 through 16, we will still get the same group of numbers after subtracting. But will the magic properties still hold. Let’s see: Suppose: Why are we supposing this? Well, if we have a row (or a column) of a 4 x 4 magic square, we know that it has to add to 34. 2

Lecture 3

Roxbury Math Epsilon Club

Now, the new group of numbers will be:

Then, Looks like we’re still getting the original magic property after the “square-wide” subtraction. Indeed, the magic is still there… 16 5 9 4

2 11 7 14

3 10 6 15

13 8 12 1

Swapping rows (or columns) about the middle. Here is one more magical property. We can swap rows of a magic square about the “middle” and we still end up with a magic square. Let’s apply this to the square above. If we start out with our square and swap 2 middle rows: 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16 We end up with another Magic Square 1 15 14 4 8 10 11 5 12 6 7 9 13 3 2 16

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Lecture 3

Roxbury Math Epsilon Club

We could have just as well swapped 2 middle columns like this: 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16 Now, we end up with yet another Magic Square: 1 14 15 4 12 7 6 9 8 11 10 5 13 2 3 16 One day (not in 5-th grade), you will learn much more about this subject. In Math, this area is called Linear Algebra and it is used just about everywhere.

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Lecture 3

Roxbury Math Epsilon Club

So, enough about Magic Squares. Let’s consider a very different kind of problem: At what time will a big hand and little hand on a clock point in the exact same direction? Let’s say the time is now 3:00 PM. As we know, the little hand points to 3 and the big hand points straight up to 12. We also know that a big hand moves faster than little hand… so at some point the big hand will catch up to the little hand. The question is: at what time will that be? 12

12

3

3

?

In order to solve this problem let’s forget for a minute that we’re dealing with a clock and pretend that the big hand and the little hand are running a race. Since the big hand is faster, the little hand gets a bit of a head-start. So a different way of asking the same question is at what point during the race will the big hand overtake the little hand. Let’s first figure out how fast each of the hands are moving. That seems easy. The little hand moves one digit over one hour, so its speed is: The big hand goes all the way around the clock in one hour. It’s the same as saying that it moves twelve digits over one hour, so its speed is: What does it mean when both hands point in the same direction? It means that they both have to be in the same place at the same time. At the time of their “meeting” the position of little hand will be: At the same time, the big hand will be at:

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Lecture 3

Roxbury Math Epsilon Club

How do we find a number such that when that number is added to 3 the result is the same as when that same number is multiplied by 12? The solution is to write this as a mathematical equation: This is the same as: So, our “magic number: is:

Let’s check to make sure:

Wait a minute… what is the hands to line up.

? This represents what part of one hour we need to wait for

But we want to see this time in minutes. Very simple, we just multiply by 60 and get the result:

We can also compute that:

In other words, what we’re saying is that the time when both hand are pointing in the exact same direction will be:

Next time you’re at home just after 3:00, wait a few minutes and you will see it with your own eyes!!!

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Lecture 3

Roxbury Math Epsilon Club

Homework… I would like everybody to think of really clever solutions to these very formidable math problems… 1) Try to figure out what time the clock will show if the little hand started at 6:00 PM instead of 3:00 PM 2) Do it again if the little hand started at 11:00 PM (this will be very simple) 3) How old am I? In twenty years from now, I will be exactly 4 times as old as I am today.

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