2. THE CLASSIC PROBLEM
Given a piece of paper
of length L and width w,
find the largest box that
can be made by cutting
squares out of the
corners and connecting
the edges.
3. HISTORY
The box problem became popularity in 1903 when Henery
Dudeney published the problem in his puzzle column in The
Weekly Dispatch. Dudeney revised the box problem for
Cassell’s Magazine and his book Amusements in Mathematics.
There was a half-guinea prize in The Weekly Dispatch for who
ever could solve the problem.
4. SOLVING THE PROBLEM
V= L*W*H dV=12X^2-4(L+W)X+LW
Width H=X =(4(L+W)+/- sqrt((-4(L
+W))^2-4(12)(LW)))/2(12)
L=L-2X
L x =(4(L+W)+/-
sqrt(16L^2+16W^2-16LW)/
W=W-2X 24
e V=(L-2X)(W-2X)X =(4(L+W)
n V=LW-2LX-2WX+4X^2
+/-4sqrt(L^2+W^2-LW)/24
g L+W+/-sqrt(L^2+W^2-
LW)/6
V=LWX-2LX-2WX+4X^3
t
TO FIND A MAXIMUM You must then test L=1 and
h VALUE FOR FUNCTION
V(X) ONE MUST
W=1
DIFFERENTIATE, THEN 2+/-sqrt(1+1-1)/6=1/6,1/2
FIND THE ZERO OF THE
FUNCTION.
X = 1/2 so not positive
8. Additional Volume
=(a)(a)(2a)
+
(a)(2a)(3a)
=8a^3= (8x^3)/27
Total Volume Gain
=4[(8x^3)/27]
9. NEW VOLUME EQUATION
V(x)=(L-2X)(W-2X)X+4[(8x^3)/27]
V’(x)=(140x^2)/9-4(l+w)x+lw
x=(9/70)(l+w-sqrt(l^2+w^2-17lw/9))
10. ANALYSIS
To compare with Friedlander and Wilker’s construction, let’s also consider the case
that l = w = 1 (a square sheet of metal). They optimize with a volume of≈.07556.
By contrast, the text book method, which Dudeney described, would have x = 1/6
and a volume of 2/27 ≈ .07407. Frederickson’s approach has x = 3/14 and a
volume of 4/49 ≈ .08163, which beats Friedlander and Wilker by 8% and Dudeney
by 10%.
This method does lose volume when the length and width of the starting material
are not equal.
11. WORKS CITED
Frederickson, Greg N. "A New Wrinkle on an Old
Folding Problem." Maa.org. MAA Mathematical
Sciences Digital Library, 2004. Web.
