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MATH / There is a total of 70 squares of three sizes whose vertices are points on this…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem
There is a total of 70 squares of three sizes whose vertices are points on this rectangular grid of points. What is the value of ?
[asy]
unitsize(0.4 inch);
dot((0,0),linewidth(9bp));
dot((1,0),linewidth(9bp));
dot((2,0),linewidth(9bp));
dot((0,1),linewidth(9bp));
dot((0,2),linewidth(9bp));
dot((1,1),linewidth(9bp));
dot((2,1),linewidth(9bp));
dot((1,2),linewidth(9bp));
dot((2,2),linewidth(9bp));
filldraw((2.95,-0.05)--(3.05,-0.05)--(3.05,0.05)--(2.95,0.05)--cycle,black);
filldraw((2.45,-0.05)--(2.55,-0.05)--(2.55,0.05)--(2.45,0.05)--cycle,black);
filldraw((3.45,-0.05)--(3.55,-0.05)--(3.55,0.05)--(3.45,0.05)--cycle,black);
filldraw((2.95,0.95)--(3.05,0.95)--(3.05,1.05)--(2.95,1.05)--cycle,black);
filldraw((2.45,0.95)--(2.55,0.95)--(2.55,1.05)--(2.45,1.05)--cycle,black);
filldraw((3.45,0.95)--(3.55,0.95)--(3.55,1.05)--(3.45,1.05)--cycle,black);
filldraw((2.95,1.95)--(3.05,1.95)--(3.05,2.05)--(2.95,2.05)--cycle,black);
filldraw((2.45,1.95)--(2.55,1.95)--(2.55,2.05)--(2.45,2.05)--cycle,black);
filldraw((3.45,1.95)--(3.55,1.95)--(3.55,2.05)--(3.45,2.05)--cycle,black);
dot((4,0),linewidth(9bp));
dot((5,0),linewidth(9bp));
dot((4,1),linewidth(9bp));
dot((5,1),linewidth(9bp));
dot((4,2),linewidth(9bp));
dot((5,2),linewidth(9bp));
[/asy]
Plain-text mathematical notation (without MathML)
There is a total of 70 squares of three sizes whose vertices are points on this rectangular 3×n grid of points. What is the value of n? [asy] unitsize(0.4 inch); dot((0,0),linewidth(9bp)); dot((1,0),linewidth(9bp)); dot((2,0),linewidth(9bp)); dot((0,1),linewidth(9bp)); dot((0,2),linewidth(9bp)); dot((1,1),linewidth(9bp)); dot((2,1),linewidth(9bp)); dot((1,2),linewidth(9bp)); dot((2,2),linewidth(9bp)); filldraw((2.95,-0.05)--(3.05,-0.05)--(3.05,0.05)--(2.95,0.05)--cycle,black); filldraw((2.45,-0.05)--(2.55,-0.05)--(2.55,0.05)--(2.45,0.05)--cycle,black); filldraw((3.45,-0.05)--(3.55,-0.05)--(3.55,0.05)--(3.45,0.05)--cycle,black); filldraw((2.95,0.95)--(3.05,0.95)--(3.05,1.05)--(2.95,1.05)--cycle,black); filldraw((2.45,0.95)--(2.55,0.95)--(2.55,1.05)--(2.45,1.05)--cycle,black); filldraw((3.45,0.95)--(3.55,0.95)--(3.55,1.05)--(3.45,1.05)--cycle,black); filldraw((2.95,1.95)--(3.05,1.95)--(3.05,2.05)--(2.95,2.05)--cycle,black); filldraw((2.45,1.95)--(2.55,1.95)--(2.55,2.05)--(2.45,2.05)--cycle,black); filldraw((3.45,1.95)--(3.55,1.95)--(3.55,2.05)--(3.45,2.05)--cycle,black); dot((4,0),linewidth(9bp)); dot((5,0),linewidth(9bp)); dot((4,1),linewidth(9bp)); dot((5,1),linewidth(9bp)); dot((4,2),linewidth(9bp)); dot((5,2),linewidth(9bp)); [/asy]
Original LaTeX notation
There is a total of 70 squares of three sizes whose vertices are points on this rectangular $3\times n$ grid of points. What is the value of $n$? [asy] unitsize(0.4 inch); dot((0,0),linewidth(9bp)); dot((1,0),linewidth(9bp)); dot((2,0),linewidth(9bp)); dot((0,1),linewidth(9bp)); dot((0,2),linewidth(9bp)); dot((1,1),linewidth(9bp)); dot((2,1),linewidth(9bp)); dot((1,2),linewidth(9bp)); dot((2,2),linewidth(9bp)); filldraw((2.95,-0.05)--(3.05,-0.05)--(3.05,0.05)--(2.95,0.05)--cycle,black); filldraw((2.45,-0.05)--(2.55,-0.05)--(2.55,0.05)--(2.45,0.05)--cycle,black); filldraw((3.45,-0.05)--(3.55,-0.05)--(3.55,0.05)--(3.45,0.05)--cycle,black); filldraw((2.95,0.95)--(3.05,0.95)--(3.05,1.05)--(2.95,1.05)--cycle,black); filldraw((2.45,0.95)--(2.55,0.95)--(2.55,1.05)--(2.45,1.05)--cycle,black); filldraw((3.45,0.95)--(3.55,0.95)--(3.55,1.05)--(3.45,1.05)--cycle,black); filldraw((2.95,1.95)--(3.05,1.95)--(3.05,2.05)--(2.95,2.05)--cycle,black); filldraw((2.45,1.95)--(2.55,1.95)--(2.55,2.05)--(2.45,2.05)--cycle,black); filldraw((3.45,1.95)--(3.55,1.95)--(3.55,2.05)--(3.45,2.05)--cycle,black); dot((4,0),linewidth(9bp)); dot((5,0),linewidth(9bp)); dot((4,1),linewidth(9bp)); dot((5,1),linewidth(9bp)); dot((4,2),linewidth(9bp)); dot((5,2),linewidth(9bp)); [/asy]
level
Level 5
type
Counting & Probability
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