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### Exchanging Seats in a Boat

```
Date: 06/08/2001 at 23:10:06
From: Beth
Subject: Problem solving

This is an assignment for bonus points for a class I had at college
last semester. Several different people tried to solve it by modeling
it with objects and drawing pictures, but everyone came up with

Here is the problem:

Ten women are fishing in a long, narrow boat. One seat in the center
of the boat is empty. The five women in the front of the boat want
to change seats with the five women in the back of the boat. A woman
can move from her seat to an adjacent empty seat, or she can step over
one person without capsizing the boat. What is the minimum number of
moves needed for the ten women to exchange seats?
```

```
Date: 06/09/2001 at 00:00:49
From: Doctor Schwa
Subject: Re: Problem solving

Hi Beth,

I'd approach this problem with the strategy of trying an easier
problem. That is, with zero people in the boat, zero steps are
required. That was easy! Next, with two people, it takes three steps:
one person moves to the center, then one step over, then the first
person moves to the end.

With four people, it gets a little trickier, but:
ab.cd
a.bcd
acb.d
acbd.
ac.db
cda.b
cd.ab

seems to be the best way to do it (is it the only way?)
and it takes 8 steps.

I don't see any obvious pattern yet (though +3, +5 makes me wonder if
it's +7 next, so I'll guess 15 steps next).

Let's try six people:
abc.def
ab.cdef
abdc.ef
abdce.f
abd.ecf
a.dbecf
da.becf
daeb.cf
daebfc.
daebf.c
dae.fbc
d.eafbc
de.afbc
defa.bc
def.abc

It is indeed 15 steps, so my guess of the pattern is probably right.

Now there's the question of whether you can *prove* that pattern ...
but following my guess, you'll get a number quite a bit smaller than
54 (assuming my guess is really correct).

I hope that helps. Feel free to write back if you'd like to talk more
see

Traffic Jam - Suzanne Alejandre, for the Math Forum
http://mathforum.org/alejandre/frisbie/student.jam.html

Enjoy,

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Puzzles
Middle School Algebra
Middle School Puzzles

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