How can I find the minimum number of open rectangles in a grid?

I am really stuck on my maths number grid investigation! Help me...?

• Ok, so it is in for tomorrow, i don't know where else to turn for help! My friends and family have no clue like me so i'm going to type out the whole investigation sheet... Task statement: On a 10x10 number grid, take any 2x2 square and find the product of (multiply) the top left and bottom right numbers. Do the same with the top right and bottom left. Find the difference between the two answers. What do you notice in each case? Write down a hypothesis; that is a fact that you think might be true for other 2x2 square frame in the number grid. Check this again with another 2x2 square fame in the number grid. INVESTIGATE FURTHER! Hints: Try different sizes of square frames inside the grid, e.g 3x3, 4x4, 5x5 etc... Organise results clearly. Explain clearly any ideas or observations. Try to predict what will happen? State clearly your prediction and any justification. Generalise and prove any hypotheses using ALGEBRA. Present results, conclusions and proofs clearly, and explain what you have proved with evidence and further checks. Try to link the results with the size of the small square frame? (i.e. nth term for the result in an nxn frame) YOU MUST TRY rectangular frames e.g. 2x2, 2x3, 2x4, 2x5 etc... then 3x3, 3x4, 3x5 etc... Extension: Try different sizes of number grid with different size of frame Try to produce a general formula for an (axb) frame in a (pxp number grid) PLEASE THIS IS JUST COMPLETE GIBBERISH TO ME! this is just what is on my sheet so i hope you can make more sense of it that i could, thanks:)

• Answer:

  Have you got the grid on the paper, if not write it out. On the first line write numbers 1,2,3,4,5,6,79,10 and the next line 11,12,...,20 and so on all the way to 100. From anywhere on the grid draw a smaller grid of 2x2 so in my selection I have 25,26 on the first line and 35,36 below. Now the question says multiply the top left with bottom right and bottom left with top right and then work out the difference. (25x36)-(35x26) =900-910 =-10 Try it with more 2x2 grid and you'll notice the difference is always 10. I think from here you'll be able to do the rest yourself.