Finding Remainder of Shaded Squares in 6x4 Grid

In summary, the conversation is discussing a math problem involving a 6 x 4 grid where 12 of the 24 squares must be shaded in a specific way. The number of possible shadings is represented by a variable and the question asks to find the remainder when this variable is divided by 1000. The participants mention using symmetry and reflection to approach the problem and suggest seeking help on a discussion site. The conversation is ultimately moved to a different forum to avoid cluttering up the homework help forum.
  • #1
ehrenfest
2,020
1

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.
 
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  • #2
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.
 
  • #3
Dick said:
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.

Dick said:
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"?
 
  • #4
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?
 
  • #5
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.
 
  • #6
cristo suggests you confine these things to a single thread in "General Math" to avoid cluttering up the HH forums.
 
  • #7
Dick said:
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.
 
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1. What is the purpose of finding the remainder of shaded squares in a 6x4 grid?

The purpose of finding the remainder of shaded squares in a 6x4 grid is to determine the number of squares that are not fully shaded. This can be useful in various mathematical and scientific calculations involving grids and patterns.

2. How do you find the remainder of shaded squares in a 6x4 grid?

To find the remainder of shaded squares in a 6x4 grid, you can use the formula (Total number of squares in grid) - (Number of fully shaded squares). This will give you the number of squares that are partially shaded or not shaded at all.

3. Can the remainder of shaded squares be a decimal or fraction?

No, the remainder of shaded squares in a 6x4 grid will always be a whole number. This is because the number of squares in a grid and the number of fully shaded squares will always be whole numbers, therefore the difference between them will also be a whole number.

4. Why is finding the remainder of shaded squares important in scientific research?

Finding the remainder of shaded squares in a 6x4 grid can be important in scientific research as it can help in understanding patterns, analyzing data, and making calculations more accurate. It can also be used to identify discrepancies or errors in data collection or measurement.

5. Can the same method be used to find the remainder of shaded squares in grids of different sizes?

Yes, the same method can be used to find the remainder of shaded squares in grids of different sizes. The formula (Total number of squares in grid) - (Number of fully shaded squares) will work for any grid size, as long as the total number of squares and the number of fully shaded squares are accurately calculated.

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