From - Sat Jul 25 18:55:24 1998
From: ilan@isgtec.com
Newsgroups: rec.puzzles
Subject: Re: French Toast puzzle
Date: Wed, 15 Jul 1998 02:46:20 GMT
Message-ID: <6oh55s$r16$1@nnrp1.dejanews.com>

Geoff Hazel wrote:
> I'm cooking French Toast with uniform, square bread slices.  I notice that
> I can just fit three pieces in a round skillet, with corners touching the
> edges, but no bent or cut or overlapped pieces.  They form a "t" in the
> pan.
>
> I wonder with a pan 10 inches in diameter, what is the size of the bread?
>
> PS I don't have the answer, just the problem.  good luck!
>

SPOILER





























+---------+---------+
|\   x    |         |
|  \      |         |
|    \    |y        |
|    R \  |         |
|        \|O        |
|        /|         |
+----+---/+----+----+
     |  /      |
     |  /      |
     | / R     |
     | /       |
     |/        |
     |/        |
     +---------+

Let O denote the centre of the circle, and R be its radius. Let x be the side
of each square, and y be the distance from the top of the "T" to the centre of
the circle.

Using the pythagoras theorem we have:

x^2+y^2 = R^2

(x/2)^2+(2x-y)^2 = R^2

Using the first equation, y can be eliminated from the second equation:

(x/2)^2+(2x-sqrt(R^2-x^2))^2 = R^2

Expanding and gathering terms yields

(13/4)x^2-4x*sqrt(R^2-x^2) = 0

This can be simplified into

(13/16)x = sqrt(R^2-x^2)

Raising to the second power and gathering terms finally yields

x = (16/(5*sqrt(17)))R

For R = 5" we get x = 16/sqrt(17)" = 3.88057".



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__/\\  //\__  Ilan Mayer
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