Time for MATH MINUTE! (provide your favorite theme music
here).
Get out
your paper & pencil, because I have a new puzzle for you!!
PUZZLE #84 If you are left standing, you win!
A. Get a
group of people and chairs together in a circle (1 chair each), we are going to
play a game. The rules are simple:
Each
person stands in front of their chair (face the chairs in so all can see what
is happening).
Starting
at some point (call it number 1) and going around the circle clockwise, every
other person must sit down. Keep this up until only one person is left standing.
He/she is the winner!
Now
comes the hard part: you have 30 seconds to choose your position ahead of time
so that you can win.
What
position do you choose?
Example:
Suppose there are 10 people. So we have a circle of 10 people and 10 chairs. 
The
people in positions 2, 4, 6, 8, 10 must sit down on the first round (hope you did
not choose an even number).
This
leaves positions 1, 3, 5, 7, 9.
Then
we must have every other position sit down starting with 1 again (since 10 was
eliminated).
That
knocks out 3 and 7, to leave 1, 5, 9. But, since 9 is left the next position to
sit down must be 1, then skip 5, and 9 must sit 
down, which leaves
5 standing.
So,
if you chose 5 you win!
Before
you jump to the conclusion that all you have to do is choose the middle number
check out the following list of results with different sizes of groups (you can
easily compute these by writing the numbers in a circle and then crossing out
every other one, starting with 2).
Size Winning
position
8 1
9 3
10 5
11 7
Compute
the results for 12, 16, 20, and 32 people.
See
any patterns?
What answer did
you get for your group? Does it fit in with the pattern above?
B. Find a
pattern for the above results.
Can
you predict the winning position for 100 people, 1000 people?
Have
fun!
Send
your comments, ideas and solutions before Monday to the email below, and
in the subject line be sure to put
MM in the subject line
Visit
us here online at:
http://www2.potsdam.edu/parksjm/MM1.1.htm
to see the results every Friday.
See you next time on MATH MINUTE! (theme music fades out here).