To win a bet, I need to grow a square potato. How could I grow a
square potato? I know that Japan grows square watermelons.

You are right, square melons have been grown. It would seem they have
also grown cube-shaped tomatoes and cucumbers in Abu Dhabi.

"The experiment has proved 100% successful. Now, in addition to square
melons, we also have cube-shaped tomato and cucumber. People don't
have to struggle any more to fit large round fruits like watermelon in
their refrigerators or in boxes for export." said a spokesperson" - I
have no idea who this spokesperson was.  But the quote came from here:
http://www.nexusportal.net/showthread/t-1352.html

Here is a good image of the square melons:
http://www.susannassoapbox.com/square.jpg - From susannassoapbox.com

In fact, here is a good article about Japan's square melons:
http://archives.cnn.com/2001/WORLD/asiapcf/east/06/15/square.watermelon/index.html
- From CNN

Vegetables grow wherever they can, and if they can only grow in a
little box that's where they'll grow. Japanese farmers use tempered
glass containers to produce their square melons.

The box would be put on after the vegetable starts to grow.  I would
highly recommend plastic or glass as cardboard, perhaps even wood,
would rot in the underground environment of the potato.  I would also
recommend the boxes have very small perforations to allow oxygen
exchange or otherwise you might get a rotted potato.

In your case, in order to make both seeing the potato and the boxing
easier, you may want to consider growing them in a hydroponic setting
to make access to the root more convenient.

Here are a couple of informative sites about hydroponic gardening if
you are not already familiar with the technique:
http://www.suite101.com/welcome.cfm/hydroponic_gardening - From suite101.com
http://www.hydroponicsonline.com/ - From hydroponicsonline.com

Now the potato comes in various sizes according to variety.  It may be
that you will not have to grow one to maturity, but just large enough
to fill whatever size box you use.  Your question didn't specify a
potato grown to maturity.  You may very well win the bet by presenting
a nice little pale-skinned "new" potato instead of a large brown one.

Should this trend catch on, food fights may evolve in a whole new direction.

Now, go collect that bet.

Search Google - growing square melons, square fruits and vegetables

Cheers
Digs

P.S. - If you need any clarification, I am leaving in about an hour
and will not be back till Friday.  Just so you know you will not have
been forgotten if I do not reply to the clarification request
immediately.

It might be of incidental interest that the Romans used to grow square
people, that is they had orphaned babies grow in pots to be malformed.

I can't remember where I read this, and it may not be true, but most likely is.

There was an advert that used square tomatoes being eaten by an Irishman, as a joke.