Another Maths Problem.

[Buenchico]

Since people seem to be in the mood for mathematical oddities today, take a look at this:
http://tinypic.com/view.php?pic=2ujp9p3&s=8

How many regions would you get from 6 dots all joined together?

[BlackadderV]

32? n to the power of 2?

[Buenchico]

Good try, BlackadderV, but wrong!

[jno]

any reason why it wouldn't be 32? since each dot seems to double the number created by having one dot fewer.

[BlackadderV]

Thought it must be wrong, because that is the logical arithmetic progression. Short of drawing it, I don't know.

[Buenchico]

That's I would have thought, Jno, but if you actually try drawing it you don't get 32.

[jim360]

It's 31. I think.

[jno]

I've drawn it, badly, and can't read my own writing clearly enough to see if it's 30 or 31; but it doesn't seem to be 32. Why?

[Buenchico]

OK, I'll give you the sequence and see if you can work out the next term. The number of regions doesn't go
2, 4, 8, 16, 32, 64, 128, 256. . .
as you might expect.

It actually goes
2, 4, 8, 16, 31, 57, 99, 163, . . .
but there's still a mathematical pattern, so it's possible to work out what comes after 163. Any suggestions?

BTW: This is from the first maths lesson I did each year with Year 8 (12yo) pupils, so it shouldn't be too hard ;-)

[Buenchico]

(Well done, Jim!)

[jim360]

I cheated by looking it up, although I know how to derive the result. I'm wondering if I can think of a justification... watch this space.

[jim360]

(Just to clarify, I solved the problem of 31 myself although I was expecting the answer; I looked the formula up rather than derived it.)

[Prudie]

I think it's 256 but can't work out the general term yet

[Buenchico]

Congratulations, Prudie!

(I wouldn't worry too much about the general term. I've seen a formula for it, worked out by a Professor of Applied Mathematics after I showed him the 12-year-olds way of working it out, but I didn't even understand his notation!)

[Prudie]

That's reassuring! The third line down of the differences just goes up by 1 each time (if you get what I mean).

[Buenchico]

I'm assuming that Prudie got there via the 'repeated differences' technique, that works for any sequence (as long as you know a sufficiently large number of terms to start with):
http://tinypic.com/view.php?pic=jfjb0z&;s=8

[The Builder]

2 4 8 15 26 42 64
2 4 7 11 16 22
2 3 4 5 6 7 ........

Prudie's differences ..............

[Buenchico]

Crossed posts! You obviously did use 'repeated differences'. I taught myself that trick when I was about 8 or 9 years old but I'm amazed that many maths graduates have never come across it!

[Buenchico]

PS: I've won a few pints in pubs by getting people to state which number comes after 16 when you mark those regions ;-)

[Prudie]

I actually had never come across this method during my own education - in fact I only saw it for the first time a couple of years ago when a work colleague asked me to help him to help his son with his homework and I saw it on the example sheet. I agree I can't understand why I never saw it before.

[jim360]

I think I've completely solved the problem and have some understanding about how to intuit one's way towards it. On the off-chance that anyone's interested, I might write it up neatly because it's beyond AB's capacity to present the answer (I'd need a fair few diagrams).

Anyway, the general difference between two terms in the sequence is

f(n+1) - f(n) = (n^3-3n^2+8n)/6 ; f(1) = 0

Which does indeed generate the right sequence 1, 2, 4, 8, 16, 31 ...