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flooring - math question

I am installing hardwood floor.  I have planks that come in 6 ft, 4 ft, 3 ft, and 2 ft.    I have 42 rows to install, each row being 14 foot wide.  I have 32 6ft planks; 16 4ft planks; 96 3ft planks; 16 2ft planks.   I want to maximize the usage of the planks, with the least amount of waste.   Can anyone help me out?   Must be a way to do this using math.
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10 Solutions
No way to do the math without knowing the width of the planks or the room dimensions, both of which you failed to give.
Even though I am terrible at maths there is one aspect that you missed, and which one of the maths geniuses will proably need to know.  How far apart are the joists onto which the boards will be nailed or screwed?  Obviously floorboards need to be staggered, so you will end up with offcuts that can be used elsewhere.  If you are laying something like laminate sheets onto a solid floor, then forget that I mentioned this.

Did you mean 14 inches wide rather than 14 feet wide?
The width of all the "rows" would amount to 588 feet if you really meant to say feet.

Personally I would just draw this out on squared graph paper or even use Excel to make a grid.
Cornelia YoderArtistCommented:
You don't need to do any math.

For the first row, Use 6', 3', then measure the remaining length (approx. 5') and cut a plank to fit.

Place the remaining piece as the first plank in the next row.  
Continue to place planks in that row until you have less than 6' remaining (choosing lengths to make the joints not too near those in the previous row).
When you need to cut a new plank to finish a row, use the cut off piece to start the next row.
<repeat until done>
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Yes, we need more detailed dimensions.
You also should consider the arrangement of the end joints: adjacent rows are staggered, alternate rows should align as much as possible.
HLRosenbergerAuthor Commented:
The planks are being glued down on concrete, so no joists.

I understand about the staggering the joints.  

The total area is 13 feet by 13 1/2 feet, although I figure the total square footage was not an issue.  So my rows are 13 1/2 feet wide.   To me, it's the linear footage I want to maximize, since each plank is the same width.  The planks are 3 3/4 inches wide.   So, for a row I could use one 6ft plank, two 3 ft planks, and one 2 foot plank, with 6" cut off.  Or two 6ft planks and and one 2 foot plank, with 6" cut off.   (I prefer not to start a row with a 6" cutoff piece.)  I have many more 3 ft planks that I do the other lengths, so those need to be used the most.  

I have installed flooring in the past, and I understand about staggering; using the cutoff piece to start the next row, etc.  

Since its being glued down, and once has to be very careful about getting the adhesive on the top of the flooring, I want to pre-cut all pieces for all 42 rows prior to installation.  I thought maybe there are a way to maximize the linear footage.

Assuming that you need 42 rows, each row being 14 feet long, you do not have enough wood to finish the project--even if there is zero waste!

You need 42x14 = 588 lineal feet of flooring. You only have:
(32) 6 foot = 192 lineal feet
(16) 4 foot = 64 lineal feet
(96) 3 foot = 288 lineal feet
(16) 2 foot = 32 lineal feet

Total = 576 lineal feet

I believe the best pattern would be:
Sixteen rows of (2) 6 foot, (1) 2 foot
Sixteen rows of (4) 3 foot, (1) 4 foot cut in half
Six rows of (4) 3 foot, (1) 2 foot
Two rows of (4) 3 foot, (1) 4 foot cut in half
Two rows of (3) 4 foot, (1) 4 foot cut in half

Total consumption:
(32) 6 foot         Fully utilized
(96) 3 foot         Fully utilized
(22) 2 foot         Note that you only have 16 of these
(16) 4 foot         Fully utilized
If the length only needs to be 13.5 feet, then you just barely have enough material.

Cutting pattern:
Eight rows of (2) 6 foot, (1) 6 foot cut into 1.5 foot piece
Eight rows of (1) 6 foot, (1) 3 foot, (1) 6 foot cut into 4.5 foot piece (remainder from above cut)
Five rows of (3) 4 foot, (1) 2 foot cut into 1.5 foot piece (0.5 foot scrap)
Sixteen rows of (4) 3 foot, (1) 3 foot cut in half
Four rows of (4) 3 foot, (1) 2 foot cut into 1.5 foot piece (0.5 foot scrap)
One row of (1) 4 foot, (4) 2 foot, (1) 2 foot cut into 1.5 foot piece (0.5 foot scrap)

Total consumption:
(32) 6 foot         fully utilized
(16) 4 foot         fully utilized
(96) 3 foot         fully utilized
(14) 2 foot         two pieces left over
HLRosenbergerAuthor Commented:
I have more material, and more area to do.  This was just an approximation, and the square area of my living room.
Cornelia YoderArtistCommented:
Well, do what I suggested on "paper" then.   An excel sheet is perfect for this, just make the cells square, count them, and lay out the boards on the rows, using colors.  If you understand staggering, you should be able to do this in about 20 minutes in excel or on any square grid paper.
Maybe you have done this before on concrete?
If not here are some informative tips.
Note the comment:
"wood does move. Ensure you leave the proper expansion space around any stationary objects (walls, etc.). Wood flooring is a natural product and will move with varying moisture and humidity levels."

And slab considerations:
"Look more closely at the slab. Does it have a psi of 3,000 or higher? Is it porous enough? If you drop a bead of water on the slab, is it quickly absorbed? If not, there may be a sealer that must be removed or a non-porous surface (such as burnished/steel troweled concrete) that must be ground down so the adhesive can properly “grab” the concrete. It is critical that the slab is smooth and flat. Only use high-quality polymer-modified, cement-based patching and leveling compounds.
Since you need 13.5 feet per row and the only odd length planks you have are 3-footers, take 21 of them an cut in half. The now leaves you with 32 6ft planks, 16 4ft planks, 75 3ft planks, and 16 2ft planks and you need 42 12ft lengths to complete the floor. There are a number of combinations of 12 that can be created from the remaining planks. For example, 16 rows of 1 6ft, 1 4ft, and 1 2ft, leaving 26 more rows of 12 ft needed from 16 6ft planks and 75 3ft planks, which could be 8 rows of 2 6ft planks and 18 rows of 4 3ft planks, or 16 rows of 1 6ft plank and 2 3ft planks and 10 rows of 4 3ft planks. There are a number of other combinations of 12, such as 3 4ft or 6 2ft or 1 4ft, 2 3ft and 1 2ft. Note this does not include any additional cutting, which could generate more combinations of 12. I don't know of any purely mathematical way of determining the best combinations and a lot will depend on how aesthetic you want the pattern to look. At least you know you'll have enough flooring to cover the job.
>>> "a lot will depend on how aesthetic you want the pattern to look" <<<

Agreed.  Here's an example of how I was able to make the best use of all the available offcuts from my bedroom wallpaper whilst still maintaining an aesthetically pleasing pattern match ;-)
You would never notice that I had economised ... with the lights off and the drapes closed
HLRosenbergerAuthor Commented:
Thanks to all!!
Thank you HLRosenberger

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