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# Intuitively clearer proofs of the sum of squares formula

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```									Intuitively clearer proofs of the
sum of squares formula

Jonathan A. Cox
SUNY Fredonia Sigma Xi
December 7, 2007
Riemann sums

Area under curve ≈ sum of areas of rectangles
Δx=width of each rectangle

Σ
Area ≈ f(xi)Δx
Handy formulas for computing Riemann sums
Handy formulas for computing Riemann sums

• Sum of integers
Handy formulas for computing Riemann sums

• Sum of integers

• Sum of squares
Handy formulas for computing Riemann sums

• Sum of integers

• Sum of squares

• Sum of cubes
Handy formulas for computing Riemann sums

• Sum of integers

• Sum of squares

• Sum of cubes

• Even fourth powers!*

*D. Varberg and E. Purcell. Calculus with Analytic Geometry. Sixth Ed.
Why is   ?
Why is                           ?
Gauss Legend

1 + 2 + 3 + ∙ ∙ ∙ + 49 + 50 + 51 + 52 + ∙ ∙ ∙ + 98 + 99 + 100
Why is                           ?
Gauss Legend

1 + 2 + 3 + ∙ ∙ ∙ + 49 + 50 + 51 + 52 + ∙ ∙ ∙ + 98 + 99 + 100
Why is                           ?
Gauss Legend

1 + 2 + 3 + ∙ ∙ ∙ + 49 + 50 + 51 + 52 + ∙ ∙ ∙ + 98 + 99 + 100

50 pairs, each with sum 101
Standard proofs of sum of squares
Standard proofs of sum of squares
• Induction
Induction Proof of                       .

• Base case: Let n=1. Then

• Induction step: Assume the formula is holds
for n and show that it works for n+1.
Standard proofs of sum of squares
• Induction
• Telescoping sum of cubes
Standard proofs of sum of squares
• Induction
• Telescoping sum of cubes

These proofs are not intuitively clear!
Regrouping the sum by odds
• Every perfect square is a sum of consecutive
odd numbers.
• 36=1+3+5+7+9+11
• Write each square in the sum as a sum of odds.
• Regroup all like odds together and add these
first.
• These two different ways of summing the
squares give an equality which can be solved
for the desired sum.
Regrouping the sum by odds
Regrouping the sum by odds
Regrouping the sum by odds
• Better, but not intuitively clear
Regrouping the sum by odds
• Better, but not intuitively clear
• Involves algebraic acrobatics
Regrouping the sum by odds
• Better, but not intuitively clear
• Involves algebraic acrobatics
• Had this explanation been discovered
previously?
Regrouping the sum by odds
• Better, but not intuitively clear
• Involves algebraic acrobatics
• Had this explanation been discovered
previously?
• Martin Gardner’s skyscraper construction
(Knotted Doughnuts and other Mathematical Entertainments)
The quest for an intuitively clear proof
• Benjamin, Quinn and Wurtz give a proof by
counting squares on an n x n chessboard in 2
different ways. (College Math. J., 2006)
• Benjamin and Quinn give another “purely
combinatorial” proof in Proofs that Really Count.
• There are more than 10 different proofs!
• And………………….
• SOME OF THEM ARE INTUITIVELY CLEAR! 
• We’ll look at up to five of the remaining proofs
(as time permits).
Solving a linear system
• By Don Cohen, from www.mathman.biz
Solving a linear system
• By Don Cohen, from www.mathman.biz
• Assume the formula is a polynomial in n
Solving a linear system
• By Don Cohen, from www.mathman.biz
• Assume the formula is a polynomial in n
• First, what’s the degree of the polynomial?
Solving a linear system
•   By Don Cohen, from www.mathman.biz
•   Assume the formula is a polynomial in n
•   First, what’s the degree of the polynomial?
•   The third differences are constant, so the
formula will be cubic.
Solving a linear system
• By Don Cohen, from www.mathman.biz
• Assume the formula is a polynomial in n
• First, what’s the degree of the polynomial?
• The third differences are constant, so the
formula will be cubic.
• Want to find the 4 coefficients (variables)
Solving a linear system
• By Don Cohen, from www.mathman.biz
• Assume the formula is a polynomial in n
• First, what’s the degree of the polynomial?
• The third differences are constant, so the
formula will be cubic.
• Want to find the 4 coefficients (variables)
• First 4 sums of squares give 4 equations
Solving a linear system
• By Don Cohen, from www.mathman.biz
• Assume the formula is a polynomial in n
• First, what’s the degree of the polynomial?
• The third differences are constant, so the
formula will be cubic.
• Want to find the 4 coefficients (variables)
• First 4 sums of squares give 4 equations
• Solve system of 4 linear equations in 4 variables
Looking to geometry
Looking to geometry
A sum of squares          ≈ volume of a pyramid with
square base

Source: David Bressoud, Calculus Before Newton and Leibniz: Part II,
http://www.macalester.edu/~bressoud/pub/CBN2.pdf
“Fiddling with the bits that stick out"
• Sum of squares = volume of the pyramid
• Why not just use volume formula for a pyramid?
• It’s not a true pyramid, more like a staircase….
"A very pleasant extension to stacking oranges is to consider the
relationship between the volume of the indicative pyramid and
the sum of squares, taking cubic oranges of one unit of volume.
This, eventually, after some fiddling to account for bits that stick
out and bits that stick in, generates the formula for summing
squares." (A.W., UK)
http://nrich.maths.org/public/viewer.php?obj_id=2497
“Fiddling with the bits that stick out"

• Underlying pyramid has volume n3/3
• Add in the half-cubes (triangular prisms) above the slice
• Subtract off volumes of n little pyramids added twice
Archimedes’ proof with pyramids
(D. Bressoud -- http://www.macalester.edu/~bressoud/pub/CBN2.pdf)
Archimedes’ proof with pyramids
(D. Bressoud -- http://www.macalester.edu/~bressoud/pub/CBN2.pdf)
Winner 1: The Greek rectangle method
Doug Williams, http://www.mav.vic.edu.au/PSTC/cc/pyramids.htm

• This is the same construction that Martin Gardner
described using skyscrapers!
Winner 1: The Greek rectangle method

Doug Williams, http://www.mav.vic.edu.au/PSTC/cc/pyramids.htm
Winner 1: The Greek rectangle method
How Martin Gardner described it as a skyscraper
Winner 1: The Greek rectangle method
Doug Williams, http://www.mav.vic.edu.au/PSTC/cc/pyramids.htm

• This is the same construction that Martin Gardner
described using skyscrapers!
• Flesh out the skyscaper with a sequence of squares
on each side to make a rectangle.
• Each sequence of squares has area                     !

• The rectangle has dimensions n(n+1)/2 and 2n+1 !
Winner 1: The Greek rectangle method

Doug Williams, http://www.mav.vic.edu.au/PSTC/cc/pyramids.htm
By the way…
• The sum of integers formula can be proved
with a similar geometric construction.
(D. Bressoud -- http://www.macalester.edu/~bressoud/pub/CBN2.pdf)
Winner 2: The six-pyramid construction
Doug Williams, http://www.mav.vic.edu.au/PSTC/cc/pyramids.htm

•   6 identical “sum of first n squares” pyramids
•   Fit them together to form a rectangular prism
•   It has dimensions n, n+1, and 2n+1
•   Thus

• It’s fairly easy to see that this works for n+1 if it
works for n.
Winner 2: The six-pyramid construction
Doug Williams, http://www.mav.vic.edu.au/PSTC/cc/pyramids.htm

•   6 identical “sum of first n squares” pyramids
•   Fit them together to form a rectangular prism
•   It has dimensions n, n+1, and 2n+1
•   Thus

• It’s fairly easy to see that this works for n+1 if it
works for n. (Induction?!)
What makes a good proof?
What makes a good proof?
• It should convince the intended audience that
the statement is true.
Appendix:
Ways of proving sum of squares formula
1. Induction
2. Telescoping cubic sum
3. Regrouping as odds
4. Pyramid of cubes--fiddling to account for bits
5. Three lines on chessboard (Benjamin-Quinn-Wurtz)
6. Combinatorial (Benjamin-Quinn)
7. Archimedes: Fitting together three pyramids
8. Solving a system of 4 linear equations in 4 variables
9. Fitting together six pyramids
10.Greek rectangles (Martin Gardner's skyscraper)
11.Integration
Sum of cubes
How many rectangles are there on a chessboard?
9 horizontal lines
9 vertical lines
Choose two of each

rectangles on an 8 x 8 board
In general,
there are         rectangles on an n x n board.
Sum of cubes
But the number of rectangles is also          !

There are k3 rectangles
with maximum
coordinate k.

Source: A. Benjamin, J. Quinn and C. Wurtz.
Summing cubes by counting rectangles, CMJ, 2006.
Can we do sum of squares on the chessboard?

• The number of squares on an n x n chessboard
is
•   .

•                                             .

Perhaps choosing three total horizontal and
vertical lines determines four squares….
This approach doesn’t seem to work.
Can we do sum of squares on the chessboard?
Yes!
Can we do sum of squares on the chessboard?
Yes!
Benjamin, Quinn, and Wurtz do this.
Can we do sum of squares on the chessboard?
Yes!
Benjamin, Quinn, and Wurtz do this.

This is still lacking in intuitive clarity….

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