Ponder This

 

We could subtract 6,765 from the 262,144 to calculate the 255,379 patterns that DO have little four sided boxes in them.� It’s the 6,765 figure though that we were looking for.

That’s quite a trick.� We are using the famous Fibonacci sequence here, where each succeeding number is the sum of the previous two.� And 6,765 is the 20^th one of these.� Our snake, remember, has twenty segments, it’s just that the first two are welded together in a fixed left hand turn.

 

This is great fun.� I wish that were all there was too it.�� But, no! We have more problems.

Consider the joints (Lweld,L,R,L,L,R,L,L,R,L,L,R,R,L,R,L,R,L,R)

Or (Lweld,L,R,L,L,R,L,L,R,L,L,R L,L,R,L,L,R,L)

Here, in both cases, we don’t have any little four sided boxes, but the snake does touch itself.

And then sneaky snake goes dancin’, wigglin’ and a-hissin’
Sneaky snake goes dancin’, gigglin’ and a-kissin’
I don’t like old sneaky snake; he laughs too much you see
When he goes wigglin’ through the grass, it tickles his underneath.

If you draw out the two sequences above, you will see a twelve sided figure that closes.� It looks like the red cross emblem.� So, now we are worried about 12 sided figures that close, as opposed to 4 sided ones which we dealt with in the first part of this discussion.

Consider any of the twelve sided figures involved with the 20 segments.� Think about those that lap over so to speak.� We could have 8 segments on the head end lapping over or 8 segments on the tail end of the snake lapping over.� Or, any combination of (8H,0T). (7H,1T),(6H,2T),(5H,3T),(4H,4T),(3H,5T),(2H,6T),(1H,7T) or (0H,8T).

 

Or, maybe this is really more appropriate:

 

In any event I believe there are 655 patterns involving the closed 12 sided figure that must be subtracted from the 6,765.

"Well, sneaky snake drinks root beer, and he just makes me sick
When he is not dancin’, he looks just like a stick
Now, he doesn’t have any arms or legs, you cannot see his ears
And while we are not lookin’, he’s stealin’ all of our beer"

Now we need to do the same kind of thing for the 16 sided figures and the four 20 sided figures.

Do I really have too? Yes! Here goes.

If you draw out the little 16 sided figure and study it awhile, you will see there are 8 positions available if you want to place the snake’s head right on the figure. Then there would be four tail segments lapping over in some manner. Of these 8 possibilities, two involve one choice only for the 17^th segment, while six allow two choices for the 17th segment. Remember now, four segments lapping over.

1……………………………….2
2……………………………….3
3……………………………….5
5……………………………….8

So, 2X5 plus 6X8 =58 patterns when the snake’s head in right on the figure, or what we might call a (0H,4T) situation.

Now for the other lapping situations................
(1H,3T), (2H,2T), (3H,1T) and (4H,0T).

My count for each of these is 28,15,,24,and 18 totaling 85 patterns

This makes a total for the 16 figure problem equal to 58 plus 85 or 143.

"And then sneaky snake goes dancin’, wigglin’ and a-hissin’
Sneaky snake goes dancin’, gigglin’ and a-kissin’
I don’t like old sneaky snake; he laughs too much you see
When he goes wigglin’ through the grass, it tickles his underneath”

Now for the famous 20 sided figures.

There are four patterns by my count and 5,10,10, and 20 possible positions for the snake’s head.� So, 45 possible patterns.

So, in summary,
 6765
-655
-143
-45
5922

And 5922-26 equals the number of patents issued to IBM in the year 2010.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com