Math Forum Discussions

Glen Prideaux

Posts: 1
Registered: 12/6/04

Re: Word Problem help
Posted: Sep 22, 2003 3:53 PM

Plain Text

Anonymous writes:

> Hey..please help me with this word problem:
> The auditorium at Centennial High Schoool has one thousand seats.
> numbered from 1 to 1000, One day each seat was filled and the 1000
> people followed these directions:
>
> First, each person stood up.
> Next, every second person, including the person in seat two sat down.
> Then every third person including the person in seat 3, changed to the
> opposite. That is, if the person was standing, he or she sat down. If
> the person was sitting, he or she stood up.
>
> Following this, every fourth person, including the person in seat 4,
> changed to the opposite. Then, every fifth person, including the
> person in seat 5, changed to the opposite, and so on. Finally, the one
> thousandth person changed to the opposite. After this last change, was
> the one thousandth person sitting or was that person standing?
>
> Questions:
> 1. Was the person in seat 1 sitting or was that person standing?
> 2. For which of the seats 1-20 were people sitting? Standing?
> 3. Was the person in seat 1000 sitting or was that person standing?
> 4. For which of seats 1-1000 were people standing?

For the moment, forget that there are 1000 seats, and consider just
the first, say, 20. Work out a few steps, then see if you can see a
pattern of some sort, or work out what is happening.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 (step)

u u u u u u u u u u u u u u u u u u u u (1)
u d u d u d u d u d u d u d u d u d u d (2)
u d d d u u u u d d u u u d d d u u u d (3)
u d d u u u u d d d u d u d d u u u u u (4)
u d d u d u u d d u u d u d u u u u u d (5)

and so forth.

Some questions you might ask:

* What makes a person change from standing (u) to sitting (d) or vice
versa? That is, how must the person's seat number relate to the
step number for him/her to change?

* For any given seat, how many times will the person change from
sitting to standing or vice versa?

* If a person changes position n times, what is his/her final
position?

* What are the conditions that will result in a final standing
position? What are the conditions that will result in a final
sitting position?

It's a bit hard knowing how much of a hint to give without just giving
the whole thing away. Even if these questions don't make much sense,
at least do the full 20 steps for the first 20 seats and look for a
pattern. If you still can't see it, do a few more seats and steps
... perhaps up to 25 or 30.

Glen

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