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trying to find a pattern may be dangerous

Posted: Wed Apr 05, 2006 6:31 pm
by hk
In a school class a problem was stated.
A pupil solved this problem by drawing and counting for small cases and found: 1,2,4,8,16 and then reasoned the next number would be 32, which was wrong.
What was the problem stated?

Posted: Wed Apr 05, 2006 7:20 pm
by Alvaro
If I had to take a guess, I would say "In how many parts can the 4-dimensional space be disected using n hyperplanes?"

Another option is "if your legs are long enough that you can go upto 4 steps at a time, in how many different ways can you go up a flight of n steps?"

Just as the first 5 terms don't determine the 6th, the first 6 terms don't determine the sequence. So the question could have been pretty much anything.

Posted: Wed Apr 05, 2006 7:33 pm
by hk
well, if you found that hyperplane thing, you can also find what problem a highschool kid can be tackling drawing. (it's the same sloane entry)
This is also a very famous one

Posted: Wed Apr 05, 2006 7:42 pm
by Alvaro
hk wrote:well, if you found that hyperplane thing, you can also find what problem a highschool kid can be tackling drawing. (it's the same sloane entry)
This is also a very famous one
I didn't "find" anything. I happen to "know" some of these things, although I can't think of the alternative problem that produces the same sequence. Ok, I just looked it up. So the problem you had in mind was this chords-dividing-a-circle thing.

However, I don't see why a highschool kid couldn't tackle the staircase problem by drawing.

Posted: Wed Apr 05, 2006 7:49 pm
by hk

Posted: Wed Apr 05, 2006 7:49 pm
by jdrandall123
A better guess at a high-school problem.


Number of divisors of n! The answer should be 30, not 32.

Posted: Wed Apr 05, 2006 7:53 pm
by hk
also a fine one. But drawing?
Some nice ideas for a lesson, I think

Posted: Wed Apr 05, 2006 10:32 pm
by euler
I guess it would be twisting the wording of the puzzle slightly to say that the "drawing" was taking numbered discs at random from a bag, but I thought people might be interested in this example of an unexpected next value...

Consider the number of different products that can be taken from the set of consecutive natural numbers, {1,2,3,...,n}.

n=1: 1 (1)
n=2: 1,2 (2)
n=3: 1,2,3,6 (4)
n=4: 1,2,3,4,6,8,12,24 (8)
n=5: 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120 (16)

If you count carefully you should get 26 for n=6.

Re:

Posted: Wed Apr 11, 2018 2:21 am
by kenbrooker
Alvaro wrote:
Wed Apr 05, 2006 7:20 pm
Another option is "if your legs are long enough that you can go upto 4 steps at a time, in how many different ways can you go up a flight of n steps?"
What happened to your 3 steps (not in the riddle)?

And Euler's immediately preceding post is
Indeed interesting...

Re: trying to find a pattern may be dangerous

Posted: Wed Apr 11, 2018 11:49 am
by hk
The last time Alvaro visited this forum was Fri, 14 Dec 2012 23:05:12.
Probably got lost in some hyperplane maze. :lol:
So I don't think he will elaborate on his flight climbing thing.

Re: trying to find a pattern may be dangerous

Posted: Wed Apr 11, 2018 4:22 pm
by kenbrooker
My "secret" intention was to facilitate helping people who joined PE.chat after 2006 (perhaps the majority?) enjoy your good/entertaining point that "trying to find a pattern may be dangerous" and, to enjoy Mr Euler's interesting, to me, reply (or Maybe I "think" too much, which also may be dangerous : )

Re: trying to find a pattern may be dangerous

Posted: Wed Apr 11, 2018 7:04 pm
by hk
I may be wrong but in my opinion Alvaro pretty much killed the thread with his gruff response.

Re: trying to find a pattern may be dangerous

Posted: Wed Apr 11, 2018 9:32 pm
by kenbrooker
NOT TO DISAGREE WITH YOU hk but I, for one, went on to enjoy your two links, and jdrandall123's and euler's thoughts, though indeed it would be hard to do a "drawing" for
a High Schooler (or maybe even for a Rocket Scientist...)...

I hope you won't give up on alerting us to any other idiosyncrasies, especially
considering my quote below...