Real-World question i can't get my head around.

[JoshMaths]

I am going to feel silly when you guys give me the answer in 2 seconds but here we go.

Imagine there are 16 rectangular pieces of paper with length 2 and width 1.

You want to organise them in a big square or make them fit the smallest area possible.

How many columns do you have and how many rows?

If you need a pitiful attempt then i would say...
Let x be the total length and y be the total width then x=2y and r*c = 16 where r = rows and c = columns and we want to find min xy = min 3y²

given x = 2r and y = c then 3y² = 32 yet this minima gives y = 0 obviously, so i am stuck.

Yes i know this is year 8 maths, yes i am in University, any help much appreciative.

 

[mathman]

No matter how you arrange them the area will be 32. Since 32 is not a perfect square you can't make a square. You can make rectangles of any shape (1 x 16, 2 x 8, 4 x 4) where either direction can be all 1's or all 2's. It is also possible to mix 1's and 2's.

 

[JoshMaths]

Thanks, one constraint i might have forgot to mention is that all the pages must be vertical so you can read them.

 

[mfb]

mathman's solution can be done with vertical pages. Another nice arrangement could be 3 vertical, 6 horizontal. It requires 36 tiles instead of 32, but it gives a nice 6x6-pattern (with 4 empty spots) - the smallest possible square.