Finding Remainder of Shaded Squares in 6x4 Grid

[ehrenfest]

Homework Statement

In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let be the number of shadings with this property. Find the remainder when is divided by 1000.

There is a picture at this link if you do not understand the question:

http://www.artofproblemsolving.com/Wiki/index.php/2007_AIME_I_Problems#Problem_8

Homework Equations

The Attempt at a Solution

So, somehow we need to divide this into case or find some equivalent problem or something. Symmetry might help us since reflection about the middle horizontal line and reflection about the middle vertical line will preserve the property. Also, inversion of color will preserve the property. So the answer (before you take the remainder) must be a multiple of two. Hmmm...I am out of ideas.

 

[Dick]

erhenfest, the problem has a solution link. Why don't you click it if you are out of ideas? What's the point to posting it here? This is a "Homework Help" forum, I don't think this qualifies.

 

[ehrenfest]

erhenfest, the problem has a solution link. Why don't you click it if you are out of ideas?

If there were a hint button, I would click on it but there isn't and I don't want to give up completely.

What's the point to posting it here? This is a "Homework Help" forum, I don't think this qualifies.

Almost none of my questions are homework (in fact I am not even taking any math classes this summer). This is also a "coursework forum". But anyway do you think it would be better to post these in "General Math"?

 

[Dick]

That's a reasonable point, but yes, you should probably post them elsewhere. You've already posted one of these without an explicit pointer to the source, though you did quote the AIME reference. Is there a "Problem Practice" forum to discuss problem solving strategies?

 

[Dick]

You might try this. Just look at the solution for a little bit. Don't read the whole thing through. That constitutes a hint along the lines to proceed without telling you the whole story. I've done it when practicing.

 

[Dick]

cristo suggests you confine these things to a single thread in "General Math" to avoid cluttering up the HH forums.

 

[ehrenfest]

cristo suggests you confine these things to a single thread in "General Math" to avoid cluttering up the HH forums. He also suggests the following discussion site about these questions http://www.artofproblemsolving.com/F...p?c=182&cid=45

I moved the discussion to GD: https://www.physicsforums.com/showthread.php?t=237287

The link you gave does not work.

 

[Dick]

I moved the discussion to GD: https://www.physicsforums.com/showthread.php?t=237287

The link you gave does not work.

Yeah, I know. I don't seem to be able to paste it right. Good luck!

 

[cristo]

Here is the link that I found that Dick posted above: http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45