Expert: Sherman D. - 4/21/2009

Question
QUESTION: Hi! My husband and I are right now doing a trivia game with our neighbors where they gave us a bunch of trivia questions to solve in a week and we did the same to them.  Most of them are involving trivia questions about things in our local area, but of course there are some good ones thrown in here in there to stump us.  This is the one question we are stumped on here.  This one actually seems like a brain teaser too.  It reads:  How many rectangles can be drawn using the following equally spaced points as corners?

. . . .
. . . .
. . . .


They probably found this trivia question on the Net.  However, my husband and I aren't coming up with the same thing for answers.  Plus, I read this to be different.  We both graduated almost 30 years ago, so we are now feeling pretty old you know.  LOL   

Any help would be appreciated.  Thanks!

ANSWER: Here they are

1100
1100
1100

0110
0110
0110

0011
0011
0011

1110
1110
0000

0000
1110
1110

0111
0111
0000

0000
0111
0111

1111
1111
0000

0000
1111
1111

1111
1001
1111

so unless i've done something wrong, thats 10 in all

1 4 by 3
2 2 by 4
3 2 by 3
4 2 by 3


Keep in mind you can't have a 2x2 or 3x3 dot size, because that would make them squares instead of rectangles.

now some may consider a square also a rectangle, because the both have 4 right angles.

how many rectangles to you 2 come up with.

if you want to get them with a trivia, this is more logic than math skill

000
000
000

3 by 3 dots, you have to connect all 9 dots, but your only allowed for straights lines. the only clue if you wish to give them is "you must think outside the box"

here is one i still like, and can fool others.

think of a number between 1 and 50

take that number and add 3, subtract 4, add 5, subtract 2, now subtract from the original number you started out with

did you get 2

you can also do it with dividing and multipling, you just have to make sure that no matter what number you pick, you'll get the same answer no matter what their number is.

---------- FOLLOW-UP ----------

QUESTION: Hi!  Thanks for a quick response and also taking your time out for this trivia question we are stumped on.  

Now we are getting confused for the number on this trivia question.  First of all, I just want to make sure it showing 12 dots as I had them on the margin and it was hard to see 3 of them by having it that way.  It should be:

  . . . .
  . . . .
  . . . .

My son managed to ask a former science teacher what he comes up with.  He had 20 for an answer.  He said that everyone forgets about the rhombus'.  

Would that change your answer of 10 to 20?  

Thanks so much for your time.  It is greatly appreciated.  Have a good one!

Patty  

Answer
a 4 by 4 dot area gives you 20 rectangles, however

at http://mathforum.org/dr.math/faq/formulas/faq.quad.html#rectangle

a rectangle is defined as

A quadrilateral with adjacent sides perpendicular
(all four angles are therefore right angles).

therefore a square would be defined as being also a rectangle, but not the other way around.

so now you have

0000
1100
1100

1100
1100
0000

0110
0110
0000

0011
0011
0000

0000
0011
0011

0000
0110
0110

1110
1010
1110

0111
0101
0111

in which i get 18.

i'm not completely certain what they will accept as being rectangular or not.