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Difference between revisions of "2010 UNCO Math Contest II Problems/Problem 11"

(Created page with "== Problem == (a) The <math>3 \times 3</math> square grid has <math>9</math> dots equally spaced. How many squares (of all sizes) can you make using four of these dots as vertic...")
 
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draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black);
 
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black);
 
draw((4,1)--(5,0)--(6,1)--(5,2)--cycle,black);
 
draw((4,1)--(5,0)--(6,1)--(5,2)--cycle,black);
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draw((3,-1)--(3,3),dash);
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draw((3,-1)--(3,3),dashed);
  
 
</asy>
 
</asy>
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(d) How many for an <math>(N+1) \times (N+1)</math>  grid of dots?
 
(d) How many for an <math>(N+1) \times (N+1)</math>  grid of dots?
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−
 
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== Solution ==
 
== Solution ==
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(a) <math>6</math> (b) <math>20</math> (c) <math>50</math> (d) <math>1\cdot n^2 + 2\cdot (n-1)^2+3\cdot (n-2)^2 + \cdots + n\cdot 1^2</math>
  
 
== See also ==
 
== See also ==
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{{UNC Math Contest box|year=2010|n=II|num-b=10|after=Last Question}}
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{{UNCO Math Contest box|year=2010|n=II|num-b=10|after=Last Question}}
  
 
[[Category:Intermediate Combinatorics Problems]]
 
[[Category:Intermediate Combinatorics Problems]]

Latest revision as of 00:58, 2 February 2023

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) $1\cdot n^2 + 2\cdot (n-1)^2+3\cdot (n-2)^2 + \cdots + n\cdot 1^2$

See also

2010 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Last Question
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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