The Experiment
In this experiment each person
chooses a whole number greater
than zero. The winner is the person who chose the lowest
number that nobody else chose. For example, if people
chose: 1, 1, 2, 2, 2, 3, 3, 3, 4,
4, 19, 22, 22, and 23, then the winner is the person who chose
19. If every
number was chosen by
more than one person then there is no winner.
The Purpose
The purposes of this experiment
are:
- To encourage
mathematical analysis and discussion at home at the dinner table, and
in the school hallways and playgrounds. Studies have shown that
students do better at math when it is not just part of their classroom
experience, but also a part of their social lives.
- To learn something
mathematically interesting from the results.
- To raise funds for
a worthy tzedaka.
How to Play
Anyone can play: kids,
families, staff, friends. You may choose as many numbers as you
like. Each number costs $2. The more numbers you choose,
the greater your chance of winning. Write down your numbers on a
piece of paper (don't tell anyone else your numbers!) and send them
along with your check to Rita at the school. For example, if you want to guess
all the numbers 1-100, then write that down and send a check for
$200. Numbers will be accepted through Friday, January 13,
2006.
Children can particiapte in
school by bringing their number with $2 to Rita. The winning
number will be announced after Martin Luther King Day. To
encourage people to make serious choices, 10% of all money collected
will go to the person who chose the winning number. The other 90%
will go to the fund
in memory of David Taubenfeld.
The Mathematics is Hard
Analyzing this game proves to be
very difficult even in the very simple
situation where there are just three players each restricted to
choosing one number. The
first person might reason that choosing 1, the lowest number available,
is a good idea. However, if one of the other two players reasons
the same way, then neither they nor the first player will win.
Both players therefore may
decide that choosing higher is worthwhile, but then the third player
could win just by choosing 1. Removing the restriction of
choosing just one number makes the experiment more interesting but less
easy to analyze. For example, if any player wanted to spend more
money in order to choose extra numbers, that player could guarantee a
win by choosing the numbers 1, 2, and 3. As long as the other two
players reamined with one number each, this player is sure to have the
lowest number that nobody else chose. Just as surely, the other
players could put up more money and choose more numbers themselves in
order to defend against this strategy.
The game is even harder to
analyze when there are more players involved. For example, if
1000 people were
playing then you would probably not guess anything from 1 to 10,
figuring that other people would cover these numbers. However, if
everyone
reasoned this way then the numbers 1 to 10 would make excellent
choices!
The one thing that seems
clear is that the numbers you choose should
be in some way proportional to the number of people who are playing,
and the number of choices they make. For example, if a million
people are playing, then it seems more likely that the small numbers
will be covered by other people, freeing you up to choose larger
numbers. With a million players, the ultimate winning number
might easily be a number in the thousands. On the other hand,
with only three players one would expect the winning number to be 1 or
2.
What a Mathematician
Might Ask
A mathematician would first
concentrate on the simpler version of the game where each person
chooses one number, leaving the harder version (where each person can
choose more than one number) for later investigation. For the one
number per peson game, he/she might wonder about the
relationship between
the total number of chosen numbers and the expected (or average)
winning
number. This is the kind of question that is hard to analyze, but
easy to explore via experiment. You normally think of scientists
as the ones who do experiments, but mathematicians do experiments
too! It is common for mathematicians to experiment and use the
data to help direct their search for a theory.
A Conjecture Without Much Confidence
Let n be the number of chosen
numbers, I conjecture that, in practice (with serious
participants and choices), the winning number will be around the square
root of n. I cannot
give much support for this conjecture because my reasons are vague and
not well analyzed yet, but the conjecture assumes that all players
chose just one number. I am hoping that the results of our
experiment together will support or refute this conjecture. If
the results of the experiment supports the conjecture, then I would be
willing to put a more serious effort into a proof.
Number of
Chosen Numbers
|
The Winning
Number
|
25
|
5
|
36
|
6
|
49
|
7
|
100
|
10
|
256
|
16
|
Are There Other Math Games Like This One?
Yes. There is a similar game
that you can play with just two people. It defies precise
analysis without experiment. It also makes a great diversion for
those long car rides, after the kids have asked for the 30th time "are
we there yet?". Each person simulataneously yells out a whole
number greater than zero. If your number is exactly one greater
than the other person's then you win and you get your number for a
score. For example, if you yell 4 and your buddy yells 3, then
you get 4 points. If your number is more than one higher than
your buddy's number, then your buddy wins and gets his number for a
score. For example, if you yell out 4 and your friend yells 2,
then your friend wins and scores 2 points. Of course if you both
yell out the same number then nobody scores on that round. The
games continues
until a player gets 21 points. That player is the winner. A
friend suggested that for kids, it might be easier to play this game
with cards instead of voices. Each kid holds his choice face down
until both players are ready, and then the cards are faced.
The winning and losing is
incidental to the real point of this game. The real point is to
experiment and observe the value of the highest number yelled out by
anyone in the course of the game. In a 21 point game, with
serious motivated players (frivolous choices can ruin the fun, math,
and spirit of the game), I have never seen anyone yell out more than
9. This number will of course vary with each game, but after many
games, you can make a good calculation of the average highest number
that gets yelled out. My experience gives about 4.6.
The longer the game
continues, the more likely it seems for a larger number to be called
out. You must play the game in order to understand this.
The players end up spiralling upwards trying to outdo each other by
exactly one, and at some point someone blinks and shoots back down for
a sure easy small score.
What is the relationship between
the number of points needed to win the game and the expected (average)
highest number called out during the course of the game? Nobody
yet has a coherent theory that predicts this expected high value, given
the number of points needed to win. Happy experimenting!
Results
Results were
tabulated on January 16, 2006. The winning number was 5, picked
by Marty
Sirkin . Here are some interesting statisctics.
Smallest Number Chosen: 1
Largest Number Chosen: 1,986,000
Most Commonly Chosen
Numbers:
1 and 3 (3 times)
Number of People Playing: 18
Number of Numbers Picked: 110
Smallest Number Not Chosen:
2
Number of One Digit Numbers: 10
Number
of Two Digit Numbers:
40
Conclusions
Our
conjecture was that the winning number would
be close to the square root of the number of entries. Notice that the winning number was 5 (smallest
number picked only once), the number of entries was 110, while the
number of distinct people playing was only 18. Do you think the data from our
experiment convincingly confirms
or refutes our conjecture? Can you refine the conjecture based on
this data? What
follow-up experiment would help you decide?