2010 UNCO Math Contest II Problems/Problem 11

Problem

(a) The $3 \times 3$ square grid has $9$ dots equally spaced. How many squares (of all sizes) can you make using four of these dots as vertices? Two examples are shown.

$[asy] for (int x=0; x<3; ++x) {for (int y=0; y<3; ++y) {dot((x,y));dot((x+4,y));} } draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((4,1)--(5,0)--(6,1)--(5,2)--cycle,black); draw((3,-1)--(3,3),dashed); [/asy]$

(b) How many for a $4 \times 4$ ?

(c) How many for a $5 \times 5$ ?

(d) How many for an $(N+1) \times (N+1)$ grid of dots?

Solution

(a) $6$ (b) $20$ (c) $50$ (d) $1\cdot n^2 + 2\cdot (n-1)^2+3\cdot (n-2)^2 + \cdots + n\cdot 1^2$

2010 UNCO Math Contest II (Problems • Answer Key • Resources)
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2010 UNCO Math Contest II Problems/Problem 11

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