A proof that is only based on visual elements, without any comments.
An arithmetic identity can be demonstrated by a picture showing a self-evident equality between numerical quantities. The above figure shows that the difference between
the th
pentagonal number and is equal to three times the th triangular number.
Of course, the situation depicted is a particular case of the formula (here it corresponds
to ),
but it is presented in a way that can be immediately generalized.
Another form of proof without words frequently used in elementary geometry is the
dissection proof.
Ayoub, A. B. "Proof Without Words: Arithmetic Mean-Geometric Mean Inequality." Math. Computer Educ.31, 191,
1997.Bogomolny, A. "Proofs Without Words." http://cut-the-knot.org/ctk/pww.shtml.Chilaka,
J. O. "Proof Without Words." Math. Computer Educ.30,
312, 1996.Chilaka, J. O. "Proof Without Words: A Combinatorial
Identity for ."
Math. Computer Educ.35, 43, 2001.Harvard University Instructional
Computing Group. "Proof without Words: Sum of Squared Integers." http://icg.harvard.edu/~gov3009/spring02/sumsq.pdf.Nelsen,
R. B. Proofs
Without Words: Exercises in Visual Thinking. Washington, DC: Math. Assoc.
Amer., 1997.Nelsen, R. B. "Proof Without Words: Sums of Integers
as Sums of Cubes." Math. Mag.71, 65, 1998.Nelsen,
R. B. Proofs
Without Words II: More Exercises in Visual Thinking. Washington, DC: Math.
Assoc. Amer., 2001.Morey, J. "Proof Without Words." http://www.math.ubc.ca/~morey/talk/proofwowords.html.Sher,
D. "Proof Without Words: . Math. Computer Educ.31,
190, 1997.Sher, D. "Proof Without Words: ." Math. Computer
Educ.32, 51, 1998.Wilson, J. http://jwilson.coe.uga.edu/emt725/AMGM/AMGM.1.html.Wise,
D. S. "Proof Without Words: A Generalization from Pythagoras." Math.
Mag.71, 64, 1998.
th
pentagonal number and 
is equal to three times the 
th triangular number.
Of course, the situation depicted is a particular case of the formula (here it corresponds
to 
),
but it is presented in a way that can be immediately generalized.
."
Math. Computer Educ. 35, 43, 2001.Harvard University Instructional
Computing Group. "Proof without Words: Sum of Squared Integers." 
. Math. Computer Educ. 31,
190, 1997.Sher, D. "Proof Without Words: 
." Math. Computer
Educ. 32, 51, 1998.Wilson, J.