Intuitively clearer proofs of the sum of squares formula

Jonathan A. CoxSUNY Fredonia Sigma Xi

December 7, 2007

Riemann sums

Area under curve ≈ sum of areas of rectanglesΔx=width of each rectangle

Area ≈ Σf(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 geometryA 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 methodDoug 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 methodHow Martin Gardner described it as a skyscraper

Winner 1: The Greek rectangle methodDoug 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 constructionDoug 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 constructionDoug 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. Induction2. Telescoping cubic sum3. Regrouping as odds4. Pyramid of cubes--fiddling to account for bits5. Three lines on chessboard (Benjamin-Quinn-Wurtz)6. Combinatorial (Benjamin-Quinn)7. Archimedes: Fitting together three pyramids8. Solving a system of 4 linear equations in 4 variables9. Fitting together six pyramids10.Greek rectangles (Martin Gardner's skyscraper)11.Integration

Sum of cubesHow many rectangles are there on a chessboard?

9 horizontal lines9 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 cubesBut the number of rectangles is also !

There are k3 rectangleswith 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….