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COMPLETING THE CUBEAuthor(s): BARBARA TURNERSource: The Mathematics Teacher, Vol. 70, No. 1 (JANUARY 1977), pp. 67-70Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27960708 .
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COMPLETING THE CUBE
Classic summation formulas come to life with elegant geometric models.
By BARBARA TURNER California State University
Long Beach, California 90840
MANY interesting properties of sums of numbers can be more readily discovered and comprehended by the student if there is a geometrical way to visualize them. Com
plex algebraic formulas become more un derstandable when we can literally see why they are true. One example of a formula with an elementary geometrical proof is the formula for the sum of the first positive
integers, ̂ ' ?
i - 1
A new three-dimensional geometrical
proof of the formula for /2 iS ?b tained by the process of "completing the
cube," a three-dimensional analog of a well-known two-dimensional geometrical technique used in deriving the formula for
"Completing the cube" involves a con struction whereby the space occupied by a cube is progressively filled up by rectangular boxes. This construction is il lustrated in figures 2 through 6, and is en
ticing to most students. If the rectangular boxes are of different colors, the derivation will be clearer and more enjoyable for the students.
One derivation of the formula * . n(n + \)
'=?;? l'I ?
depends on the formula -1
(i) ' = "2- '.
which has a neat two-dimensional geome trical proof, illustrated in figure 1 for the
Fig. 1. Completing the square
case = 4. Figure 1 illustrates that 1 + 2+ 3 + 4 = 42 - (i + 2 + 3). (Count the num ber of unshaded boxes, the total number of
boxes, and the number of shaded boxes). More generally, the figure would be an
square, consisting of columns of boxes each. The /th column, containing / unshaded boxes, would be completed by ? i shaded boxes. Thus, the column far
January 1977 67
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thest to the right would contain unshaded boxes (and no shaded boxes). The next col umn would contain ? 1 unshaded boxes plus I shaded box. Similarly, the column to its left would have ? 2 unshaded boxes and 2 shaded boxes, and so on. Finally, the column farthest to the left would contain 1 unshaded box and ? 1 shaded boxes. Thus, the truth of equation (1 ) is confirmed
geometrically, since the number of un n
shaded boxes, ', equals the total number
of boxes, 2, minus the number of shaded n-i
boxes, '. The geometrical proof of formula (1) in-,
volves literally "completing the square." "Completing the cube" is an analogous
process in three dimensions, which can be used to establish the formula
/ n-i n-i \
(2) ?/? = ?>-{ ?/ + 2>?)
from which can be derived the following formula for the sum of the squares of the first positive integers:
(3) ? .2 + lX2n + 1)
To complete the cube, one must have a
supply of rectangular boxes of two kinds, called rods and squares. "Rods" should be of dimensions X 1 X 1, and "squares" should be of dimensions ?X?XI (where is any positive integer). The unit of mea surement is immaterial. Rods of dimensions X 1 X 1 are called n-rods, and squares of dimensions X ? X I are called w-squares. A 1-rod and a 1-square are identi cal. The volume of an -rod is n; the volume of an n-square is n2.
If the rods and squares are color-coded
according to size, the numerical
relationships involved in the geometrical completion of the cube become clearer and the construction of the model will be more
enjoyable. Cuisenaire rods and squares are
appropriate for construction of the cube and work extremely well.
We shall illustrate completing the cube and derive equation (2) for the case = 4. The construction is illustrated in figures 2
through 6. On top of a 4-square, place a 3-square, a
2-square, and a 1-square, in that order, in the manner indicated in figure 2.
Fig. 2. Volume = l2 + 22 + 32 + 42
The configuration of squares fills up part of a 4 X 4 X 4 cube. To complete the cube, the remainder of this cube must be filled in with rods and squares.
The next step is to erect an upside-down staircase of rods on top of one side of the
configuration, placing a 1-rod on the 2 square, a 2-rod upright on the 3-square, and a 3-rod upright on the 4-square. The
resulting configuration is illustrated in fig ure 3.
Fig. 3. Volume = l2 + 22 + 32 + 42 +1+2 + 3
68 Mathematics Teacher
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At this point, the top of the 2-square is
(partially) covered by two 1-squares. Now,
completely cover the top of the 2-square by
placing on it two more 1-squares, as in
figure 4.
Fig. 4. Volume = Ia + 2* + 32 + 4f + 1 + 2 + 3 + 2 X Is
At this point, a reverse L-shaped region of the top of the 3-square is left uncovered.
Now, completely cover the top of the 3
square by placing upright on its exposed surface two 2-squares, at right angles to
each other, as in figure 5.
Fig. 5. Volume = l1 + 2e + 3e + 4* 4- 1 + 2 + 3 +
2 X 1? + 2 X 2*
At this point, a reverse L-shaped region of the top of the 4-square is still uncovered.
Completely cover the top of the 4-square by
placing upright on its exposed surface two
3-squares, at right angles to each other, as
in figure 6.
Fig. 6.
Volume = 4s Volume = I2 + 22 + 32 + 42
+ l+ 2 + 3 + 2Xl2 + 2X22 + 2X32
The configuration of squares and rods
now completely fills the space of a 4 X 4 X
4 cube. Thus, the volume of the
configuration is 4s. But, the volume is also
equal to the sum of the volumes of the rods
and squares. Thus, the volume equals
l2 + 22 + 32 + 42 + 1 + 2 + 3 + 2 X l2 + 2 X 22 + 2 X 32.
Thus,
'^ + ' + ^48? t-l ?-1 i-l
Hence,
'2 = 43- ( ' + 2'2 ) ?
This formula is the special case of (2), when = 4.
The geometrical process of "completing the cube" is the same for any number of
squares. Hence, for any integer > 1,
completing the cube gives a three-dimen sional proof of the formula (2).
From (2), the formula ^ /* g*ven by
(3) is easily derived.
January 1977 69
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In fact, since (2) holds for all integers > 1,
n+i / 2 = ( + -
[ + *2 i ?
1=1 = ?=1 1
Hence,
( + I)2 + 3?/2 = ( + I)3
- /
Thus,
(3) ( + \)(2 + 1)
Thus, completing the cube yields a three dimensional geometrical proof of (3), the well-known formula for the sum of the
squares of the first positive integers. The study of numerical relationships
through the use of geometrical models may
provide insight through which further alge braic identities are discovered without the use of geometry. For example, there is a
resemblance in the structure of equations (1) and (2), the formulas obtained by com
pleting the square and completing the cube; so it is natural to wonder if a similar pat tern continues for sums of kth powers of
integers for k > 2. The following more
general equation, suggested by (1) and (2), can be guessed, and in fact proved.
(4) '*
Equations (1) and (2) are the instances of
(4) for the cases k = 1 and k = 2. The significance of equation (4) is that it
can be used to express the sum of the kth
powers of the first positive integers in terms of the sums of the fh powers of the first positive integers for all j< k. In fact, one can derive the following formula from
equation (4).
(5) A .k _(n+
- (n + 1)*
hl k+l
((: : ??+( k + 1
Thus, a formula expressing ^ as a func
tion of is obtained once the formulas for
* are known for all j < k.
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70 Mathematics Teacher
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