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I have pieces of wooden planks 1.22m by 2.44m. I want to find the minimum number of planks required to construct a wooden surface of sides L length and W width.

The first intuition is that I can arrange the planks "horizontally" or "vertically" so I would calculate total number of planks N = (L/1.22 x W/2.44) or (L/2.44 x W/1.22) where the divisions needs to get rounded up to nearest integer. Since this is a simple case, I just calculate the two cases and get the minimum option.

For fun and interest, tried to express this in calculus but have failed. I don't know how to handle the discrete maths involved here. Anyone can advise or link to study material? Thanks.

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2 Answers

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Calculate $n=\lceil \frac L{1.22}\rceil$ and $m=\lceil \frac W{1.22}\rceil$, whete $\lceil\ \rceil$ denotes rounding up. if at least one of $n,m$ is even, you need $\frac{nm}2$ planks. Otherwise you need $\frac{nm+\min\{n,m\}}2$ planks. If you are allowed to mix horizontal and vertical arrangement, then $\lceil \frac{nm}2\rceil$ planks are enough.

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You need $\lceil \frac{LW}{(1.22)(2.44)} \rceil$ tiles.

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How do you plan to tile a $1.22\sqrt{2} \times 1.22\sqrt{2}$ square using one tile? –  Erick Wong Sep 10 '12 at 13:16
I just need enough tiles to cover the area. Each tile has area $1.22 \cdot 2.44$. The total area is $LW$. The problem was to construct a surface of side $L$ and $W$. Presumably the tiles can be cut. –  copper.hat Sep 10 '12 at 14:31
If you're going to take such liberties, why not use just 1 tile and shave it to get many tiles of the same surface area? –  Erick Wong Sep 10 '12 at 15:24
Well, in general you need to cut the tiles anyway. Besides, I don't need to shave the tiles, I can just use my Banach-Tarski Tile cutter. –  copper.hat Sep 10 '12 at 15:50
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