## trying to find a pattern may be dangerous

Arithmetic, algebra, number theory, sequence and series, analysis, ...
hk
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### trying to find a pattern may be dangerous

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?

Alvaro
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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.

hk
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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

Alvaro
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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.

hk
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Last edited by hk on Wed Apr 05, 2006 7:52 pm, edited 1 time in total.

jdrandall123
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A better guess at a high-school problem.

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

hk
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also a fine one. But drawing?
Some nice ideas for a lesson, I think

euler
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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.

kenbrooker
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### Re:

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?"
Then what happened to taking 3 steps?
"Good Judgment comes from Experience;