**ERIC Number:**EJ886050

**Record Type:**Journal

**Publication Date:**2010-May

**Pages:**8

**Abstractor:**As Provided

**Reference Count:**0

**ISBN:**N/A

**ISSN:**ISSN-0746-8342

Using Squares to Sum Squares

DeTemple, Duane

College Mathematics Journal, v41 n3 p214-221 May 2010

Purely combinatorial proofs are given for the sum of squares formula, 1[superscript 2] + 2[superscript 2] + ... + n[superscript 2] = n(n + 1) (2n + 1) / 6, and the sum of sums of squares formula, 1[superscript 2] + (1[superscript 2] + 2[superscript 2]) + ... + (1[superscript 2] + 2[superscript 2] + ... + n[superscript 2]) = n(n + 1)[superscript 2] (n + 2) / 12. More precisely, the following algebraic equivalents are derived, [image omitted]. The proofs obtained literally count squares, namely lattice squares whose vertices are in an "n" x "n" grid. For the first formula, only lattice squares aligned to the grid are counted; for the second formula, aligned "and" tilted squares are counted.

Descriptors: College Mathematics, Mathematics Instruction, Mathematical Formulas, Mathematical Logic, Validity, Teaching Methods, Algebra

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A