MATH with my KIDS

An alien is thinking of a number…

…and you have to guess what it is.

You were captured by a malevolent alien, who would really like to eat you for dinner. He told you his name, but since it is unpronounceable to humans, you like to call him Evil Eddie. Evil Eddie has a morbid sense of fairness and so, before roasting you with an apple stuck in your mouth and serving you with a side of glowing, purple, French-cut string beans, has locked you in a chamber carved out of the bedrock of a strange world. Before the polycarbide door slides shut with a hiss, Eddie tells you that you he is thinking of a number. If you can guess it, he will reluctantly release you. If not, well…let’s just hope that you can guess the number.

As you look around the cavern you see, in the dim and flickering light (does it come from torches? You don’t see any.), an enormous stone bowl, filled to the brim with glass marbles. How many are there? Thousands at least. Tens of thousands, perhaps. On the other side of the chamber is an identical bowl, but this one is empty. In the middle of the room, hanging from the ceiling is a rope. You hear Eddie’s voice thundering into the room “when you think that you know the number, put that many marbles in the empty bowl. Then pull the rope.”

Fortunately you have a friend. In an adjacent chamber is a benevolent alien. You don’t know his name, so you decide to call him Manvel. By some contrived plot device that the author has not taken the time to think of, you know that Manvel:

1) can read Evil Eddie’s mind (hence, knows the number),

2) is going to try to transmit this number to you, and

3) is going to do so by tapping out the digits (through the wall) in some base (could be base 10 (that’s what we normally use), could be base 2 (called binary. We think of computers as using base 2), could be any whole number base.

You listen and hear …tap… long pause …tap tap tap tap tap… long pause …tap tap tap… then silence.

You have the digits! They are 153. As stated above, you now know the number is:

$(1\times b^2)+(5\times b)+3$

You just don’t know what $b$ is. (I’ve glossed over the issue of big endian versus little endian order, but we are assuming, by aforementioned plot device, that big endian order has been used by Manvel).

So…how many marbles do you put in the bowl?

to be continued…

More Experimental Topology and Experiments in Topology

I visited my friends Peter and Liz (I stayed a few nights) and came back with a laundry list of things to post about:

1) They have some really cool polyhedra and mathematical quilts, so sometime I am going to have to go over there with a camera and click some photos. Peter’s latest quilt project is one tiled with spidrons, which is in the design phase now.

2) Peter was a little upset that I failed to credit him for pointing out to me originally that $7\times 3=21$ and $7$ expanded base $3$ is $21$ (see synchronicity) and asked if there were any other examples of this. Anyway, thanks, Peter, for pointing this out and inspiring so much recreational math.

3) I finally carried out that idea I had that I mentioned back in the post more experimental topology about making a five pointed star. It took some trial and error (hey! that’s what experimental topology is all about. Right?) but here is what I came up with:

Start with ten strips of paper, with one end cut into points. The points should come to an angle of approximately $\frac{\pi}{5}$.

Draw a line down the center of each and tape them together so that the tips are all touching:

Now start taping opposite ends together, like so:

When you have four of the five opposite pairs taped together, the last one needs one full twist. If you’ve been proceeding in a clockwise direction when taping opposite pairs together, you should make one full twist in the last pair by turning the strip closest to you in a counter-clockwise direction. Anyway, in the end you should get something like this:

Now, cut along all of those center lines, and what do you get?

A mess, but if you untangle the mess, you should get:

A pair of stars that are linked!

Topologically this is the Hopf link.

4) Peter also loaned me a couple of books:

Experiments in Topology

I haven’t read much yet, but it has some cool stuff, like if you have a strip of paper, one inch thick, what is the shortest Moebius strip that you could make?

Also he lent me

Goedel, Escher, Bach: an Eternal Golden Braid

(gosh, that’s a really small image of the book, but, oh well). I’m about a fifth of the way into this book. Excellent so far. Won the Pulitzer Prize! I’m learning lots about the works of Goedel, Escher and (believe it or not) Bach.

That’s all for now, but soon I will have to blog about my visit to my kids’ classes for career day, which was last week….Stay tuned.

Synchronicity

I first heard the word synchronicity when I was sitting in an airport, waiting for a plane, working on this math problem (for fun) and over-hearing a man’s cell phone conversation. I thought that it was a good word for this phenomenon. Here it is:

Have you noticed that 3 times 7 is 21 and if you expand 7 base 3 it is 21. That is:

$21=3\times7$

$21_3=7$

This seems like an interesting coincidence. Could it be that this happens for any other numbers? Maybe it happens infinitely often, maybe never. Asking such questions is where the adventure begins. Of course we are implicitly working base 10 here. That is, in the first line the string of numerals “21″ is interpreted as a base 10 number. In the second one it is (explicitly) interpreted base 3. You could allow 10 to be replace by other bases also.

If we restrict our attention to base 10 (in the first line), I claim that, by exhaustive computer search, there are only five other instances of this phenomenon. They are:

$1,\!066,\!338,\!805,\!156,\!287,\!287,\!067$ $=9\times118,\!482,\!089,\!461,\!809,\!698,\!563$ and $1,\!066,\!338,\!805,\!156,\!287,\!287,\!067_9$ $=118,\!482,\!089,\!461,\!809,\!698,\!563$

$1,\!124,\!161,\!329,\!714,\!632,\!881,\!704$ $=9\times124,\!906,\!814,\!412,\!736,\!986,\!856$ and $1,\!124,\!161,\!329,\!714,\!632,\!881,\!704_9$ $=124,\!906,\!814,\!412,\!736,\!986,\!856$

$2,\!305,\!867,\!155,\!177,\!711,\!644,\!802$ $=9\times256,\!207,\!461,\!686,\!412,\!404,\!978$ and $2,\!305,\!867,\!155,\!177,\!711,\!644,\!802_9$ $=256,\!207,\!461,\!686,\!412,\!404,\!978$

$2,\!306,\!166,\!776,\!784,\!312,\!535,\!170$ $=9\times256,\!240,\!752,\!976,\!034,\!726,\!130$ and $2,\!306,\!166,\!776,\!784,\!312,\!535,\!170_9$ $=256,\!240,\!752,\!976,\!034,\!726,\!130$

$5,\!744,\!341,\!611,\!556,\!736,\!174,\!883$ $=9\times638,\!260,\!179,\!061,\!859,\!574,\!987$ and $5,\!744,\!341,\!611,\!556,\!736,\!174,\!883_9$ $=638,\!260,\!179,\!061,\!859,\!574,\!987$